MAM2083S

MAM2085S
Vector Calculus for Engineers

Handbook 2016
Contents
Course information for students
Course outline
Semester Calendars
Tutorial exercises
Decision trees
Formula sheet
Exercises
Chapter 1: Surfaces and space curves
Chapter 2: Multivariate integration and transformations
Chapter 3: Line integrals and the vector calculus theorems
Chapter 4: Numerical applications
Appendices
Revision worksheet
Worksheet of Cartesian and parametric expressions
Worksheet on conservative vector fields and potential functions
Worksheet 1 on polar coordinates
Worksheet 2 on polar coordinates
Worksheet on areas as iterated integrals

Acknowledgements: Our thanks go to the developers of this course over the years. Many
tried and tested exercises are widely available and we have tried to draw them together in
this handbook, while including further exercises where we felt they might be necessary.

1
 Course Information
Prerequisites:
Vector Calculus for Engineers (MAM2083 and MAM2085) is a core course for second-year
engineering students. In this course we study vector-valued functions and functions of several
variables; in particular, how to integrate and differentiate such functions. This course builds
on the material covered in the first-year engineering mathematics courses.

Why two courses?

MAM2085 is exactly the same as MAM2083 except that MAM2085 has slightly more
contact time. ASPECT first year offers a lot of support. MAM2085 allows the ASPECT
community to remain together for part of second year. Evidence has shown that providing
these twin options has improved pass rates considerably.

Lectures:
MAM2083S lectures will be in first period Tuesday to Friday. The venue is RW James 3A.
The course convenor and lecturer of MAM2083S is Dr Bob Osano, M3.01.
MAM2085S lectures will be in first period Monday to Friday. The venue is Jordon 2G.
The course convenor and lecturer of MAM2085S is Dr Tracy Craig, EM3.13.
You are encouraged to visit your lecturer if you have any problems or queries concerning the
course. These courses have Vula sites which you must check at least once every two
days. Announcements posted there will be understood to have been read by every
student 48 hours after posting.

Textbook:
The prescribed textbook for the course is the 4th Metric International Edition of Calculus:
Concepts and Contexts by James Stewart. Every student is expected to have a copy of this
book. The book is no longer printed in hardcopy. Second hand copies are available if you ask
around. The e-book is also available.

The Handbook
The handbook consists of four classes of exercises
 Class exercises, covered by the lecturer on the board
 Workshop exercises, done by you in the lecture period
 Tutorial exercises, done by you in the tutorial periods
 Homework, which mostly takes the form of assignments from the textbook, but can
also include other exercises.
Workshop exercises and tutorial exercises not covered in the time available should be
completed as homework. Test and exam questions are often based on these exercises so it is
important to your success in the course that you complete everything and understand how to
do it all correctly.

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Tutorials:
Every student is required to attend one afternoon tutorial a week.
MAM2083S: These will take place 14:00-16:00 on Tuesday-Thursday afternoons in venues
to be announced.
MAM2085S: These will take place 14:00-16:30 on Tuesdays and Thursdays in venues to be
announced. Afternoon tutorials start on Tuesday 19 July.
MAM2085S: There are additional, optional, tutorials in 3rd period throughout the week.
MAM2085S students are encouraged to attend as many of those as you can.

Attendance at tutorials is compulsory: registers will be taken. Any student who arrives late
for a tutorial, or who uses the time to do other work, will be marked absent. Satisfactory
attendance at tutorials is a DP requirement.

Class Tests:
There will be two class tests [dates to be announced]. Students who miss a class test for a
valid reason will either be excused, or required to write a replacement test. If you are ill on
the day of a test, then you should present a medical certificate in person to the lecturer within
three days of your return to University. To be valid, a medical certificate must state
specifically that the patient was examined by the doctor signing the certificate and that the
patient was found medically unfit to write the MAM2083S/2085S test on the date concerned.

Examination:
The MAM2083S and MAM2085S exam will consist of a 2½ hour paper written in
October/November.

Calculators:
Calculators will not be required in class tests and examinations. If they are allowed, it must
be a basic scientific calculator, not able to do graphics or matrix calculations, or to do
differentiation or integration. This rule will be strictly enforced.

Class Records:
Your Class Record will be 0.5 × Class Test 1 + 0.5 × Class Test 2. If you are absent from a
MAM2085S Class Test without an acceptable excuse, then you will be credited with 0 for
that test.

DP Certificates:
You must earn a “Duly Performed certificate” (a DP) for MAM2085S in order to write the
final examination. The requirements are
 a Class Record of 35% or more.
 satisfactory attendance at tutorials.

Final Results:
If your Class Record is x% and you score y% on the final examination, then your final mark
will be (0.4x + 0.6y)% OR (0.25x + 0.75y)%. You need a final mark of at least 50% to pass.

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 Course outline – the short version
1. Use polar, cylindrical and spherical coordinates to describe two and three dimensional
regions and to carry out changes of coordinate systems in integration problems.

2. Describe and sketch surfaces and space curves; find tangent lines, and normal, tangent
and osculating planes; find arclength and curvature of space curves.

3. Apply the definitions of continuity and differentiability to functions of two variables

4. Derive and apply partial derivatives of multivariate functions; use the Chain Rule for
multivariate partial derivatives; derive and apply directional derivatives of functions of
two variables; derive and apply the gradient vector for functions of two variables.

5. Evaluate double, triple and surface integrals; apply double, triple and surface integrals in
contexts such as mass, area, volume and flux; change variable order or coordinate system
in double, triple and surface integrals.

6. Use the definition of linearity to prove or disprove linearity; apply linear and nonlinear
transformations to points, curves and regions; derive affine approximations; use
derivative matrices and Jacobians to carry out shifts in coordinate systems.

7. Evaluate line integrals; use line integrals in typical applications, such as arclength, mass,
area and work done; evaluate line integrals in the context of conservative vector fields
and path independence.

8. Apply Green’s Theorem, Stokes’ Theorem and the Divergence Theorem in appropriate
contexts.

9. Carry out numerical methods such as Taylor’s Theorem, Optimisation, Newton’s method
and Lagrange multipliers for functions of two variables.

4
 Course outline – the long version
Chapter 1: Surfaces and space curves
Section 1.1 Surfaces and regions
By the end of this section you should
 Understand that single-variable functions have domains which are intervals of real
numbers, while multivariate functions have domains which are 2-d regions on a
coordinate plane
 Recognise the equations and names of the typical surfaces and be able to draw them.
 Know the meaning of “trace” and “level curve” (and “level surface”)
 Be able to draw the level curves of a surface and deduce the shape of the surface from its
level curves.
 Use polar coordinates to describe regions in a coordinate plane; use cylindrical and/or
spherical coordinates to describe regions in 3-d space.
Section 1.2 Space curves
By the end of this section you should
 Be able to parametrise Cartesian equations and be able to find Cartesian relationships
between parametric equations
 Recognise typical curves by their parametric expressions (such as lines, circles and
helixes)
 Be able to sketch space curves by interpreting the projections onto the coordinate
planes/intersection of surfaces as represented by the relationships between the parametric
equations.
 Know the definition of continuity for a space curve/vector-valued function.
 Be able to differentiate and integrate vector functions.
  
 Interpret r (t ) , r (t ) and r (t ) as position, velocity and acceleration of a particle along a
path.
 
 Know when to expect that x (t ) and x (t ) will be orthogonal to one another.
 Be able to find the equations of tangent lines to curves.
 Be able to calculate arclength of curves.
 Be able to reparametrise a curve in terms of its arclength
 Be able to calculate curvature of curves.
 Know what radius of curvature is and how it relates to curvature.
 
 Be able to calculate the vectors T (t ) and N (t ) and understand what they mean.
 Be able to find the normal planes and osculating planes to curves.
Section 1.3 Limits of multivariate functions (surfaces)
By the end of this section you should
 Know that the typical surfaces (sphere, paraboloid, etc.) are continuous everywhere
 Know the definition of continuity for a multivariate function/surface.
 Be able to use the “approach along paths” method for disproving the existence of limits
for indeterminate forms.
 Be able to use the Squeeze Theorem for proving the existence of limits of indeterminate
forms.

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 Be able to use polar coordinates for proving the existence of limits of indeterminate
forms.
Section 1.4 Partial derivatives, differentiability and tangent planes
By the end of this section you should
 Be able to find partial derivatives of functions of several variables
 Be able to use partial derivatives to find equations of tangent lines to curves (where one
variable remains constant)
 Be able to find higher order partial derivatives and know the two main types of notation.
 Know that the existence of partial derivatives does not imply either continuity or
differentiability.
 Know that if a surface is differentiable at a point, it has a tangent plane at that point; you
should be able to find this tangent plane
 Know “The Elephant” and how to use it to prove or disprove differentiability
 Be able to use the Chain Rule for partial derivatives
 Be able to find directional derivatives at a point using the first principles formula.
 Know that the dot product form of the directional derivative formula is only applicable
when the function is differentiable at the point.
 Know the relationship between the directional derivatives and the tangent plane (if one
exists).
 Be able to determine the gradient vector and interpret it in terms of direction and
magnitude of greatest increase, and orthogonal relationship to level curves.
 Be able to find tangent planes using the gradient vector.

Chapter 2: Multivariate integration and Transformations

Section 2.1 Double integrals
By the end of this section you should
 Be able to evaluate double integrals over a given region
 Be able to change the order of integration
 Be able to use double integrals to calculate volumes, areas and masses.
 Be able to sketch regions and volumes as represented by double integrals
Section 2.2 Surface integrals
By the end of this section you should
 Be able to project surfaces onto coordinate planes and to describe those projections
 Know how to shift from dS (surface) integrals to dA (double) integrals.
 Be able to use surface integrals to calculate surface area and mass.
 Be able to use surface integrals to calculate flux, and be proficient with both forms of the
flux equation.
Section 2.3 Triple integrals
By the end of this section you should
 Be able to evaluate triple integrals
 Be able to change the order of integration
 Be able to use triple integrals to calculate volume and mass.

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Section 2.4 Linear transformations
By the end of this section you should
 Know the definition of a linear map and be able to use it to prove or disprove linearity.
 Shift between the function and matrix forms of linear transformations
 Be able to apply linear transformation to points
 Know the role the determinant of the matrix plays in a geometric transformation
 Know the effect a linear transformation has on lines and planes.
Section 2.5 Non-linear transformations, derivative matrices and affine approximations
By the end of this section you should
 Be able to determine the derivative matrix of a transformation/vector field
 Know what an affine approximation is and how to calculate it for a given transformation
and point.
 Know what a Jacobian is and how to calculate it
 Know the role the Jacobian plays in geometric transformations
 Know the Jacobians for transformations from Cartesian to polar, cylindrical and spherical
coordinate systems
 Be able to transform regions of integration into alternate regions through appropriate
transformations
 Be able to evaluate double or triple integrals using changed coordinates

Chapter 3: Line integrals and the vector calculus theorems

Section 3.1: Line integrals and vector fields
By the end of this section you should
 Know the basic definition of a line integral
 Be able to transform from any of ds, dx or dy integrals into dt integrals
 Be able to use line integrals to calculate arclength, mass and “curtain” area
 Be familiar with the notation of line integrals
 Be able to use line integrals to calculate work done
 Understand what is meant by “conservative vector field” and “potential function” and be
able to determine potential functions
 Understand the connection between conservative vector fields and path independence of
line integrals
 Understand all the terms in the expressions “simple closed curve” and “simply connected
region”
 Know the four equivalent statements with respect to conservative vector fields
 Know how to calculate div and curl of vector fields and know the notation for both.
Section 3.2: Green’s Theorem
By the end of this section you should
 Know the statement of Green’s Theorem, with its associated formula.
 Be able to use Green’s Theorem to turn line integrals into double integrals
 Be able to use Green’s Theorem to calculate area by transforming double integrals into
line integrals
 Know how Green’s Theorem extends to regions which are not simply connected

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 Be able to apply Green’s Theorem over regions where the integrand is undefined by
designing new boundaries for the region.
Section 3.3: Stokes’ Theorem
By the end of this section you should
 Know the statement of Stokes’ Theorem, with its associated formula.
 Know how Green’s Theorem is a special case of Stokes’ Theorem
 Be able to use Stokes’ Theorem to calculate work done
 Be able to use Stokes’ Theorem to turn line integrals into surface integrals
Section 3.4: The Divergence Theorem
By the end of this section you should
 Know the statement of the Divergence Theorem, with its associated formula.
 Be able to use the Divergence Theorem to calculate flux through surfaces

Chapter 4: Numerical applications of multivariate functions

Section 4.1: Taylor’s Theorem
By the end of this section you should
 Be able to find Taylor expansions of functions of 2 variables up to the second degree
terms
Section 4.2: Local and absolute maxima and minima
By the end of this section you should
 Be able to find stationary points of a function of 2 variables
 Be able to use the second derivative test to classify stationary points
 Be able to find absolute maxima and minima on a closed, bounded region
Section 4.3: Lagrange multipliers
By the end of this section you should
 Know under what conditions we use Lagrange multipliers
 Be able to set up and solve a system of equations using Lagrange multipliers
Section 4.4: Newton’s Method
By the end of this section you should
 Be able to use Newton’s Method for finding solutions to systems of equations.

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Semester Calendar
Topics to be covered in MAM2083S in 2016

Wk Lect Monday Tuesday Wednesday Thursday Friday

1 1-5 18-Jul 19 20 21 22
Surfaces Surfaces, Coord systems
Coord systems Regions

2 6-10 25 26 27 28 29
Space curves Space curves Tangents Arclength
Arclength Curvature

3 11-15 1 2 3 4 5
limits and limits and Partial deivatives Partial derivatives
continuity continuity Tangent planes

4 16-19 8 9 10 11 12
differentiability differentiability Chain rule
directional deriv

5 20-24 15 16 17 18 19
directional deriv directional deriv Gradient Double integrals
Gradient

6 25-29 22 23 24 25 26
Double integrals Double integrals Surface integrals Surface integrals

7 30-34 05-Sep 6 7 8 9
Triple integrals Triple integrals Linear transform. Deriv matrices
Affine approx

8 35-39 12 13 14 15 16
Jacobians, change Jacobians, change Jacobians, change Jacobians, change
of coords of coords of coords of coords

9 40-44 19 20 21 22 23
Line integrals Line integrals Line int. and work Div and curl
Cons. vector fields Green's Thm

10 45-49 26 27 28 29 30
Green's Thm Green's Thm Stokes' Thm Stokes' Thm

11 50-54 03-Oct 4 5 6 7
Divergence Thm Divergence Thm Taylor's theorem Taylor's theorem

12 55-59 10 11 12 13 14
Local max & min Abs. max & min Lagrange mult. Newton's Method

9
 Topics to be covered in MAM2085S in 2016

Wk Lect Monday Tuesday Wednesday Thursday Friday

1 1-5 18-Jul 19 20 21 22
Surfaces Surfaces Coord systems Coord systems Regions

2 6-10 25 26 27 28 29
Space curves Space curves Space curves Tangents Arclength
Arclength Curvature

3 11-15 1 2 3 4 5
Curvature limits and limits and Partial deivatives Partial derivatives
Tangent planes continuity continuity Tangent planes

4 16-19 8 9 10 11 12
differentiability differentiability Chain rule directional deriv
directional deriv

5 20-22 15 16 17 18 19
directional deriv Gradient Revision Revision Revision
Gradient

6 23-27 22 23 24 25 26
Double integrals Double integrals Double integrals Double integrals Surface integrals

7 28-32 05-Sep 6 7 8 9
Surface integrals Surface integrals Surface integrals Triple integrals Triple integrals

8 33-37 12 13 14 15 16
Linear transform. Linear transform. Deriv matrices Newton's Method Jacobians, change
Affine approx of coords

9 38-42 19 20 21 22 23
Jacobians, change Jacobians, change Jacobians, change Line integrals Line integrals
of coords of coords of coords

10 43-47 26 27 28 29 30
Line int. and work Line int. and work Div and curl Green's Thm Green's Thm
Cons. vector fields Cons. vector fields Green's Thm

11 48-52 03-Oct 4 5 6 7
Green's Thm Stokes' Thm Stokes' Thm Divergence Thm Divergence Thm

12 53-57 10 11 12 13 14
Taylor's theorem Taylor's theorem Local max and min Absolute max/min Lagrange mult.

13 58
Lagrange mult.

10
Tutorial Questions

The tutorial questions in each section are given below. In each week, the tutorial questions
assigned go up to the work covered in on the Tuesday morning of that week.

1.1 Surfaces and regions: 2, 5, 8, 11, 15, 19, 20

1.2 Space curves: 24, 25, 27, 30, 34, 35, 39, 40, 41, 45, 46, 55, 56, 57
1.3 Limits of multivariate functions: 66, 67, 72, 73, 74
1.4 Partial derivatives, differentiability and tangent planes: 78, 79, 82, 84, 88, 92, 96, 98,
99, 108, 109, 115, 120, 124, 125, 131, 132

2.1 Double integrals: 5, 9, 10, 14, 15, 16, 20

2.2 Surface integrals: 29, 30, 31, 36, 37
2.3 Triple integrals: 41, 46, 47, 52
2.4 Linear transformations: 56, 62, 63, 70, 71, 72, 79
2.5 Non-linear transformations, derivative matrices and affine approximations: 88, 89,
98, 99, 100, 101, 102

3.1 Line integrals and vector fields: 5, 8, 9, 16, 17, 18, 26

3.2 Green’s Theorem: 32, 33, 37, 38, 42
3.3 Stokes’ Theorem: 48, 49
3.4 The Divergence Theorem: 61, 62, 63, 64

4.1 Taylor’s Theorem: 4, 5, 6

4.2 Local and absolute maxima and minima: 11, 12, 13, 21
4.3 Lagrange multipliers: 27, 28
4.4 Newton’s Method: 44

11
Decision trees
Flux decision tree

Mass decision tree

12
Work decision tree

13
MAM2085 Formula sheet
This is what your formula sheet would look like, if you had one, which you don’t.

Trigonometric identities Singe variable differentiation rules

cos 2  cos   sin 
2 2 d n
x  nx n1
 2 cos 2   1 dx
d x
 1  2 sin 2  a  a x ln a
dx
sin 2  2 sin  cos 
d f ( x)
sin 2   cos 2   1 a  a x ln a f ( x)
dx
tan 2   1  sec 2  d 1
cot 2   1  csc 2  ln x 
dx x
d
Coordinate systems sin x  cos x
dx
r 2  x2  y2 d
cos x   sin x
 2  x2  y2  z2 dx
x  r cos    cos  sin  d
tan x  sec 2 x
y  r sin    sin  sin  dx
z  z   cos  d
sec x  sec x tan x
dx
d 1
Chain Rule arcsin x 
z z x z y
dx 1 x2
 
s x s y s d
arctan x 
1
z z x z y dx 1 x2
  d
t x t y t ( f ( x) g ( x))  f ( x) g ( x)  f ( x) g ( x)
dx
d f ( x) f ( x) g ( x)  f ( x) g ( x)
Conversion factor 
2
dx g ( x) ( g ( x)) 2
 z   z 
2

 dS   1       dA
 x   y  Chain rule:
d
f ( g ( x))  f ( g ( x)) g ( x)
S R dx

Affine approximation
 x  a
 
F ( x, y, z )  F (a, b, c)  F (a, b, c) y  b 
 z c
 

Newton’s Method
   
x2  x1  ( F ( x1 )) 1 F ( x1 )

14
Definition of differentiability
f (a  h, b  k )  f (a, b)  hf x (a, b)  kf y (a, b)
lim 0
( h , k )( 0, 0 )
h2  k 2

Directional derivative
f ( x0  ha, y0  hb)  f ( x0 , y0 )
Duˆ f ( x0 , y0 )  lim
h 0 h

Curl and div

    
   , ,  F  ( P, Q, R )
 x y x 
   R Q P R Q P 
curlF    F    ,  ,  
 y z z x x y 
  P Q R
divF    F   
x y z

Taylor expansion
f ( x, y)  f (a, b)  ( x  a) f x (a, b)  ( y  b) f y (a, b) 
1
2 [( x  a) 2 f xx (a, b)  2( x  a)( y  b) f xy (a, b)  ( y  b) 2 f yy (a, b)]  h.o.t

Variable change
( x, y)
 f ( x, y)dA   f (u, v) (u, v) dA
R R

Path independence
  
 f  dr  f (r (b))  f (r (a))
C

Green’s Theorem
 Q P 
 Pdx  Qdy    x  y dA
C D

Stokes’ Theorem
    

C
F  dr   
S
 F  dS     F  nˆ dS
S

Divergence Theorem
  

S
F  dS   dV
div
E
F

15
 Chapter 1 Surfaces and Space Curves
Section 1.1 Surfaces and regions
1. What are the domains for the following functions? [Remember, the domain of a 2-variable
function is a region (sometimes a curve) in R2.]
y
a) f ( x, y)  x 2  ln y b) f ( x, y)  x  y c) f ( x, y ) 
x 1

2. What are the domains for the following functions?

1 1
a) f ( x, y )  x 3  b) f ( x, y)  ln( x 2  y) c) f ( x, y ) 
y y  2x

Stewart 9.6: 5, 7
Revision Worksheet

3. Plot the level curves of the following functions. For each of them, plot several curves and
label them sensibly. What does the surface look like?
a) f ( x, y)  x 2  y 2 b) f ( x, y)  x 2  y 2 c) f ( x, y)  x 2  y 2

4. Plot the level curves of the following functions. For each of them, plot several curves and
label them sensibly. What does the surface look like?
a) f ( x, y)  x 2  y 2  2 b) z  4  x 2  y 2

5. For each of the following functions, sketch and label the family of level curves. Use these
level curves to sketch the graphs of the functions. (“sketch the graphs of the functions” means
“sketch the surfaces”)
a) z  x 2  y 2  1 b) z  2 x 2  1 c) z  13  x 2  y 2

d) z  6  2 x 2  3 y 2 e) z  x 2  y 2  1 f) z  x 2  y 2  1
Can you identify each of these surfaces by name?

6. Sketch the level curves for each of the following functions. What do you notice? What do
the graphs of the functions look like?
a) z  ln( x 2  y 2 ) b) z  arctan( x 2  y 2 )

Stewart 9.6: 17, 19, 23

Stewart 11.1: 19, 21, 23, 25, 27, 29, 35-40, 41, 43

16
Typical Quadric Surfaces
Name Diagram Equation
Ellipsoid x2 y2 z2
This becomes a sphere   1
a2 b2 c2
if a  b  c Example:
16 x 2  4 y 2  9 z 2  144

Sphere x2  y2  z 2  r 2
This is a special case of Examples:
the ellipsoid. Sphere: x 2  y 2  z 2  4
x 2  y 2  z 2  2z
Hemisphere: z  1  x 2  y 2

Elliptical paraboloid z  ax 2  by 2
This becomes the Examples:
(more common) z  2x 2  y 2
circular paraboloid if
ab f ( x, y)  8  x 2  2 y 2

Hyperbolic paraboloid z  ax 2  by 2
(saddle) Example:
f ( x, y)  x 2  y 2  3

Cone z 2  ax 2  by 2
This is a circular cone Examples:
if a  b Circular cone: z 2  x 2  y 2
(Half) cone: z  x 2  y 2

Hyperboloid of one x2 y2 z2
sheet   1
a2 b2 c2
x2 y2
or z  c 2  2  1
a b
Example:
z 2  x2  y 2 1

17
Hyperboloid of two x2 y2 z2
sheets    1
a2 b2 c2
x2 y2
or z  c 1  2  2
a b
Example:
z   2x2  2 y 2  1
Parabolic sheet / y  ax 2  bx  c
Parabolic trough / Any variables here are fine, but
Parabolic trough / there must just be two of them.
“Gutter”
Basically, a 2d Examples:
parabola equation, but z  2x 2  1
extended out to make a
z  y2
surface in 3d
z  5  4( y  1) 2

Cylinder x2  y2  r 2
Take a 2d circle or (x  k)2  ( y  l)2  r 2
ellipse and extend the
Any two variables will do.
curve out into 3d to
Example:
form a surface
orthogonal to a x 2  ( y  1) 2  1
coordinate plane.
[Diagram credits: http://encyclopedia2.thefreedictionary.com, http://tutorial.math.lamar.edu,
http://www.dtpm.unipa.it, Rossman lab]

18
7. Sketch the solid in the first octant that is bounded by the graphs of the given equations.
a) x 2  z 2  9, y  2 x, y  0, z  0
b) 2 x  y  z  4, x  0, y  0, z  0
c) z  x 3 , x  4 y 2 , 16 y  x 2 , z  0

8. Sketch the region in R3 bounded by the graphs of the given equations:

a) z  0, z  y, x 2  1  y b) y  2  z 2 , y  z 2 , x  z  4, x  0

Stewart 9.6: 9, 11, 13, 15

Coordinate systems
Rectangular coordinate system x, y, z
2-d polar r, θ
Cylindrical coordinate system r, θ, z
Spherical coordinate system ρ, θ, φ
Note the order – order is important.
r 2  x2  y2
 2  x2  y2  z2
x  r cos    cos  sin 
y  r sin    sin  sin 
z  z   cos 
9. Given the point Q( 2 , 6 , 2 2 ) in rectangular coordinates, express Q in cylindrical and
spherical coordinates.

10. Given the point A(3, 3 , 3 3 ) in cylindrical coordinates, express A in rectangular and
spherical coordinates.

11. A point P has spherical coordinates ( 8 , 4 , 3 ) find both the Cartesian and the cylindrical
coordinates of P.

12. Describe the following regions in polar coordinates.

a) The region in the third quadrant inside the circle x 2  y 2  4
b) The region inside the upper semicircle of x 2  ( y  1) 2  1
c) The line x = 3

13. Describe the following regions in polar coordinates.

a) The line y = 2.
b) The right-hand semi-circular region inside x 2  ( y  1) 2  1
c) The region inside the circle x 2  ( y  2) 2  4 and below the line y = 2.

19
14. The domain of the function f ( x, y)  ln((16  x 2  y 2 )( x 2  y 2  4)) is
(A) 4  r 2  16 (B) 4  r 2  16 (C) 2  r 2  4
(D) 2  r 2  4 (E) none of A-D

15. The region strictly inside the circle x 2  y 2  1 can be described in polar coordinates by
0  r  1 , 0    2 .
a) Describe the region strictly inside the circle x 2  ( y  1) 2  1 in polar coordinates.
b) Describe the region strictly inside the circle ( x  1) 2  y 2  1 and strictly to the right of the
line x  1 in polar coordinates.
c) Describe the region strictly inside the circle ( x  1) 2  ( y  1) 2  1 in polar coordinates.
[Note: “strictly” means “not including the boundary”, so “strictly inside” means not including
the circle itself, just the region inside.]

16. Describe the region within the sphere x 2  y 2  z 2  2 z and above the plane z = 1 in
spherical coordinates and cylindrical coordinates.

17. Describe the region between the hemispheres z  1  x 2  y 2 and z  4  x 2  y 2

using spherical coordinates (straightforward) and cylindrical coordinates (trickier).

18. A surface in R3 is described in spherical coordinates by  sin 2   cos  . Sketch the

surface and give its equation in both rectangular and cylindrical coordinates.

19. Express the circle x 2  y 2  z 2  32, z  4 in spherical and cylindrical coordinates.

20. In each of the cases, sketch the region enclosed by the two given surfaces in R3, and then
express this region in terms of spherical coordinates.
a) z  x 2  y 2 , z  4  x 2  y 2

b) z  x 2  y 2 , z  1

Stewart 9.7: 3, 5, 7, 9
Stewart 9.7: 11, 13, 15, 17, 19, 21, 27, 31, 33
Stewart Appendix H1: 13, 15, 17, 19, 33, 39
Worksheets 1 and 2 on polar coordinates (see Appendices)

20
Section 1.2 Space Curves
Vector notation: Unit vectors are denoted in this handbook like this: n̂ , whereas vectors

which are not necessarily unit vectors are denoted like this: n . You are welcome to only use
the second notation. Alternatives are “bold” text, usually shown in handwriting by
underlining: n or n.

21a) Find a Cartesian equation for the curve described parametrically by


r (t )  (2 cos t , 3 sin t ), t  [0,  ]
b) Find a parametric description for the curve with Cartesian equation xy = 1, x > 0.

22. Find a Cartesian equation for the curve described parametrically by


r (t )  (t 2 , t 3 ), t  R

23. Find a parametric description for the curve with Cartesian equation ( x  1) 2  ( y  1) 2  1 .


24. For each of the following, sketch the curve in R2 parametrised by r (t ) . Determine the
Cartesian equation for each curve.
 
a) r (t )  (cos t , sin t ) 0  t  2 b) r (t )  (t , 1  t 2 ) 0  t  1

c) r (t )  (ln t , 1  (ln t ) 2 ) 1 t  e

What do you observe? Could this curve be parameterised by r (t )  (e t , 1  e 2t ) ? Explain.

25. An ellipse is shown in the diagram below. Express the equation of the ellipse in Cartesian
form as well as in parametric form.

Worksheet on Cartesian and parametric expressions (see Appendices)

26. Sketch the following space curves by first considering all three projections onto the
coordinate planes.
 
a) r (t )  (t , 1  t , 3  2t ), t  R b) r (t )  (cos t , sin t , 3t ), t  R
 
c) r (t )  (cos t , sin t , 3), t  [0,2 ] d) r (t )  (t , t 2 , t 3 ), t  R

21
27. Consider the following curves in R3:
C1  {(cos t , cos t ,2 cos t ) : t  R}
C 2  {(t ,t , t 2 ) : t  R}
C3  {(cos t , cos t , cos 2 t ) : t  R}
C 4  {(cos t , sin t , cos 2 t ) : t  R}
a) Describe and sketch each of these curves.
b) What is the relationship between C2 and C3?
c) Which of the curves intersect? Find the points of intersection.
d) Calculate the length of the curve C1.
e) Does the point (2, 2, 4) lie on the curve C2?
f) Write down a Cartesian equation for the plane containing C1, C2 and C3.

Stewart 10.1: 1, 3, 5, 7, 9, 11, 19-24, 25, 35

28. Determine where the curve intersects the surface in the cases below.

a) curve: x ( )  (2, 3,2)   (0, 2,2),   R surface: 6 x  3 y  4 z  12

b) curve: r (t )  (t , t , 3t ), t  R
2 3
surface: z  2 x 2  y 2

29. For each of the following, find the intersection (if any) of the curve with the surface:

a) r (t )  (2  3t , t , 1  2t ), t  R x  3y  z  7  0

b) r (t )  (2  3t , t , 1  2t ), t  R x  y  z 1  0

c) r (t )  ( t , 1t , t ), t  0 x 2  y  2z  4
 1
d) r (t )  ( 1t , 1  1t , 12 t 2 ), t  0 z
1 x  y2
2

 1
e) r (t )  ( 1t , 1  1t , 12 t 2 ), t  0 x  y  2z  1
x2


30. Let r (t )  (2 sin t , 2 cos t , 5t ) , where t  0 .

a) Sketch the curve in R3 parametrised by r (t ) .
b) Does this curve intersect the cylinder x 2  y 2  4 ? If so, at which points?
c) Does this curve intersect the upper half-cone z  x 2  y 2 ? If so, at which points?

31. Let a > 0. Does the curve parametrised by


r (t )  (a sin t cos 2t , a sin t sin 2t , a cos t ), t  R
lie on a sphere in R3 with its centre at the origin? If so, find the radius of the sphere.

32. Find parametric equations for the curve of intersection of the surfaces z  x 2  y 2 and
x  z  2.

22
33. Find parametric equations for the curve of intersection of the surfaces, and sketch the
curve of intersection:
a) z  x 2  y 2  1 , z  13  x 2  y 2
b) z  x 2  y 2  1 , z  2x 2  1

34a) Sketch the surfaces z  x 2  y 2 and 2 y  z  0 on the same set of axes.

b) Do these two surfaces intersect? If so, describe and sketch the curve of intersection.
c) Describe the surface {( x, y, z)  R 3 : x 2  ( y  1) 2  1} . What is the relationship between
this surface and the curve of intersection found in (b)?
d) Write down parametric equations for the curve of intersection.

35a) Sketch each of the following surfaces:

i) z  2 x 2  y 2 ii) 4 x 2  2 y 2  z  9 iii) z  9  x 2  2 y 2
b) Describe the intersection of the surfaces (i) and (ii) in part (a). Find parametric equations
for the curve of intersection.
c) Describe the intersection of the surfaces (i) and (iii) in part (a). Find parametric equations
for the curve of intersection.
d) Describe the intersection of the surfaces (ii) and (iii) in part (a). Find parametric equations
for the curve of intersection.
e) Are any of the curves in (b), (c) or (d) planar curves?

36. For each of the following, describe the curve of intersection of the two given surfaces,
and give parametric equations for this curve.
a) z  8  x 2  2 y 2 z  x 2  2y 2

b) z  9  x 2  y 2 z  cot  x 2  y 2 , 0    2

Stewart 10.1: 37, 39, 43

   
37. If r (t )  (t , ln t , 1t ) , find r (1) , r (1) and r (1) .

Stewart 10.2: 9, 11, 13, 45, 49, 51


38a) Given the curve C described parametrically by r (t )  (2t , 3t 2 , 1  t ) , find the equation of
the tangent line to C at the point (2, 3, 2) .

b) Given the curve C described parametrically by r (t )  (t 2 , 2  3t , 1  t 3 ) , find the equation
of the tangent line to C at the point (4, 8,  7) .


39. Given the curve C described parametrically by r (t )  (cos t , sin 2 t , 1  sin t ) , find the
equation of the tangent line to C at the point (0, 1, 0) .

23
40. Show that the curves parametrised by

r1 (t )  (e t  1, 2 sin t , ln(1  t )), t  1

and r2 (t )  (1  t , t 2  1, t 3  1), t  R
intersect at the origin. What is the angle of intersection?


41. The position of a moving particle at time t is given by r (t )  tiˆ  t 2 ˆj  t 3 kˆ .
a) How far from the origin is the particle after 2 seconds?
b) Obtain the position of the particle after 2  t seconds.
  
c) Compute r (2  t )  r (2) . Call this vector a .
d) What happens as t  0 ? Was this expected? Draw a picture to illustrate.
1     
e) Compute (r (2  t )  r (2)) . Call this vector b . Compare the directions of a and b .
t

f) Find lim b . What does this represent? Draw a picture to illustrate.
t 0

g) Find the direction of the tangent line to the curve given by r (t ) at t = 2.
h) Find the velocity of the particle at t = 2.
i) What is the speed of the particle at t = 2?
j) Find the acceleration of the particle at t = 2.

Stewart 10.2: 5, 21, 23

Stewart 10.4: 9, 11, 17


 2

42. If r (t )  (cos t , sin t , t ) , evaluate  r (t )dt .
0

Stewart 10.2: 33, 35, 37


43. Find the length of the curve described parametrically by r (t )  (2t , t 2 , 13 t 3 ), 0  t  1 .

44. Evaluate the length of the indicated portions of the curves below.
a) x  2 cos t , y  2 sin t , z  5t 0  t  

b) r (t )  tiˆ  23 t 2 kˆ 0  t  8
3


c) r (t )  (0, cos 3 t , sin 3 t ) 0  t  2

45. One of the rotor blades of a helicopter is 10m long and rotates through a full revolution
every second. How far does the tip of this blade travel in a second if the helicopter ascends at
the rate of 1m per second?

24
46. A particle moves along a curve C so that its position at time t is given by

r (t )  (t cos t , t sin t , t ) , t  R
a) Find the coordinates of the point A on C which corresponds to t = 0.
b) Does the point ( 2 , 2 , 2 ) lie on C? Why/why not?
c) At what time does the particle reach the point B(0, 2 , 2 ) ?
d) Observe that C lies on the cone z 2  x 2  y 2 . Can you verify this? Use this observation to
sketch the curve C.

e) Find r (t ) and hence the speed of the particle at the point B.
f) Write the vector equation of the line tangent to C at the point B.
g) Calculate the distance that the particle has travelled in moving from A to B. [The following
u 2 a2
 a 2  u 2 du    ln u  a 2  u 2  K ]
2
formula might be useful: a u
2 2
 
h) Calculate the angle between the vectors r (0) and r (0) .
 
i) Is the velocity vector r (t ) always orthogonal to the acceleration vector r (t ) ? Calculate
 
the angle between the vectors r ( 2 ) and r ( 2 ) .

Stewart 10.3: 1, 3, 5


47. Reparametrise the curve of r (t )  cos 2tiˆ  2 ˆj  sin 2tkˆ with respect to arclength measured
from the point where t = 0 in the direction of increasing t.


48. Find the curvature of the twisted cubic r (t )  (t , t 2 , t 3 ) at a general point and at (0, 0, 0).
What is the radius of curvature of the curve at (0, 0, 0)?


49. Find the curvature of r (t )  (t , 12 t 2 , t 2 ) .

50. Determine the curvature of the given curves at the indicated points.

a) The curve of r (t )  (e t cos t , e t sin t , 2) , at the point where t = 0.
b) The curve of y  1  x 2 , at (0, 1), (1, 0) and (2, −3).

51. The path of a freeway and exit ramp are superimposed on a

rectangular coordinate system in such a way that the freeway coincides
with the x-axis. The exit ramp begins at the origin O and its path follows
a curve of the form y  Ax n from O to the point P(3, 1), and then runs
along the arc of a circle of radius 3 2 units.
Find the value of A and n so that the curvature  (x) is continuous at x = 3
(that is, the road is designed so that there are no abrupt changes or discontinuities in
curvature). Recall that for a plane curve y  f (x) , the curvature is given by

25
 f ( x)
 ( x)  3
1  ( f ( x)) 2 2

[Hint: Once you have set up your equation and simplified it as much as you can, it is much
easier to solve by inspection than to solve analytically.]
Stewart 10.3: 17, 21, 23, 27, 29, 31, 33


52. Find the normal plane and osculating plane for r (t )  (t , t 2 , t 3 ) at (1, 1, 1).


53. Find a Cartesian equation for the osculating plane to r (t )  (cos t , sin t , cos t ) at the point
where t  4 .


54. Show that the osculating plane at every point on the curve r (t )  (t  2, 1  t , 12 t 2 ) is the
same plane. What can you conclude about the curve?

55. A particle moves along a curve C so that its position at time t is given by

r (t )  (t , t 2 ,1) , t  R
a) What does the curve C look like?

b) Find the unit tangent vector T (t ) at time t. What does the vector represent?

c) Calculate T (t ) . What does this vector tell us?
 
d) Calculate the dot product T (t )  T (t ) . Did you expect this answer?
e) Find a Cartesian equation for the osculating plane to C at t = 2. Did you expect this result?


56. Let C be the curve parametrised by r (t )  (sin t , cos t , sin 2 t ) , where 0  t  2 .
a) Show that the point P(1, 0,1) lies on C.
b) Let Q be the point on C corresponding to t  4 . Find the coordinates of Q.
c) Note that the curve C lies on the cylinder x 2  y 2  1 and on the “parabolic gutter” z  x 2 .
Use this to sketch the curve C, showing the positions of the points P and Q.
 
d) Find r (t ) and r (t ) for all t  [0, 2 ] .
 
e) Use the fact that r (t )  r (t ) is a normal to the osculating plane to find a Cartesian
equation for the osculating plane to C at the point Q.
f) Two possible formulas for calculating  (t ) are
 
T (t ) r (t )  r (t )
 (t )   and  (t )   3 .
r (t ) r (t )
Use one of these two methods to find the curvature of C at the point Q.

26
 
57. Let C be the curve parametrised by r (t )  (cos t , sin t  1, 2  2 sin t ) , where 0  t  2 .
a) Find three different surfaces on which C lies.
b) Sketch the curve C.
c) Find a vector equation for the line tangent to C at the point P(1,1, 2) .
d) Find the curvature of C at the point P.
e) Find a Cartesian equation for the osculating plane to C at the point P.

58. The position of a particle at time t is given by


r (t )  (sin t , 1  cos t , (1  cos t ) 2 )
a) What is the position of the particle at time t  2 ?
 
b) Find r ( 2 ) and r ( 2 ) . What do these vectors represent?

c) Give the equations of two surfaces which contain the curve C parametrised by r (t ) .
d) Sketch the curve C, showing the two surfaces you found in (c). Indicate clearly on your
 
diagram the position P of the particle at time t  2 and the vectors r ( 2 ) and r ( 2 ) .
e) Find the osculating plane of the curve C at the point P corresponding to t  2 .
f) Calculate the curvature of C at the point P.

59. Let C be the curve of intersection of the paraboloid z  x 2  y 2 with the cylinder
x 2  ( y  1) 2  4 .
a) Give parametric equations for C.
b) Find the osculating plane to this curve at a general point ( x, y, z ) on C.
c) Find the curvature of C at a general point ( x, y, z ) on this curve.
d) At which points on C is the curvature a minimum? What is the curvature at these points?

Stewart 10.3: 47

Section 1.3 Limits of multivariate functions (surfaces)

x2  y2
60. Does lim exist?
( x , y ) ( 0 , 0 ) x 2  y 2

xy
61 Does lim exist?
( x , y ) ( 0 , 0 ) x  y 2
2

xy 2
62. Does lim exist?
( x , y ) ( 0 , 0 ) x 2  y 4

2x 2  y 2
63. Does lim exist?
( x , y ) ( 0 , 0 ) x 2  2 y 2

27
 xy 4
64. Does lim exist?
( x , y ) ( 0 , 0 ) x 2  y 8

xy  2 x  y  2
65. Does lim exist?
( x , y )(1, 2 ) x  y 2  2 x  4 y  5
2

66. By finding two different paths along which the function approaches different values,
3 y 2  2x 2
show that lim does not exist.
( x , y ) ( 0 , 0 ) x 2  y 2

4x 2 y 3
67a) Show that tends to 0 if ( x, y) approaches (0, 0) along any straight line through
x3  y9
the origin.
4x 2 y 3
b) Show that lim does not exist.
( x , y ) ( 0 , 0 ) x 3  y 9

68. By finding two different paths along which the function approaches different values,
y2  x2
show that lim does not exist.
( x , y ) ( 0 , 0 ) x 2  y 2

69. By considering what happens along the lines x = 0 and y = 0, show that the function
 sin( x 2  y )
 ( x, y )  (0,0)
f ( x, y )   x  y

 1 x y0
is not continuous at (0, 0).

3x 2 y
70. Does lim exist?
( x , y ) ( 0 , 0 ) x 2  y 2

xy 2
71. Prove that the limit lim exists.
( x , y ) ( 0 , 0 ) x 2  y 2

xy
72. Does lim exist?
( x , y ) ( 0 , 0 )
x2  y2

28
  x4  y4
 ( x, y )  (0,0)
73. Let f ( x, y )   x 2  y 2

  x y0
a) Does lim f ( x, y) exist? Prove your claim.
( x , y )( 0, 0 )

b) For which value of λ is f continuous at (0, 0)?

Stewart 11.2: 1, 5, 7, 9, 11, 13, 15, 27, 33

xy ( x  2 y )
74. Let f ( x, y ) 
x2  y2
a) What is the domain of f?
b) Find the limiting value of f ( x, y) as ( x, y) approaches (0, 0) along
i) the x-axis ii) the y-axis iii) the parabola y  x 2
c) Can you conclude from your answers to (b) that lim f ( x, y) exists? Give a reason for
( x , y )( 0, 0 )

your answer.
d) If lim f ( x, y) does exist, what will its value be? Give a reason for your answer.
( x , y )( 0, 0 )

e) Show carefully that lim f ( x, y) does exist and has the value you found in (d).
( x , y )( 0, 0 )

3x 2 y
75. Show that the limit lim exists by using polar coordinates instead of the
( x , y ) ( 0 , 0 ) x 2  y 2

Squeeze Theorem. Important: the polar method does not work with all surfaces; there have to
be no discontinuities. If in doubt, use the Squeeze Theorem, it always works.

Stewart 11.2: 35, 37

Section 1.4 Partial derivatives, differentiability and tangent planes

x
76. Differentiate from first principles to find the partial derivatives of f ( x, y )  .
y

77. Differentiate from first principles to find the partial derivatives of f ( x, y)  xy .

78. Let g ( x, y)  x 2  1
a) Sketch the surface z  g ( x, y) .
b) Onto your sketch in (a) mark the point P(1, 1, 2) . From your sketch, figure out the values
(possibly approximate) of g x (1, 1) and g y (1, 1) .
c) Use the definition of the derivative to find g x (1, 1) and g y (1, 1) . How do these values
compare to your answers in (b)?

29
 1  x if y  0
79. Let f ( x, y )   .
 1 if y  0
a) Sketch the surface z  f ( x, y) .
b) Use your sketch in (a) to find f x (0,0) , f y (0,0) , f x (1,0) and f y (1,0) .
c) Now use the definition to find f x (0,0) , f y (0,0) , f x (1,0) and f y (1,0) and compare your
answers with those found in (b).

z z
80. Find and for the following functions.
x y
y
a) z  xy  x 2 y b) z  cos x sin y  3 y c) z 
x  tan x

81. Find f x ( x, y) and f y ( x, y) for the following functions.

1
b) f ( x, y)  e x
2
a) f ( x, y)  ln( xy )  y
c) f ( x, y)  x y  y x
x

82. The ideal gas law PV  nRT involves a constant R, the number n of moles of the gas, the
volume V, the Kelvin temperature T and the pressure P.
a) Express each of P, T and V as a function of the remaining two variables.
V T P
b) Find , and .
T P V
 V  T  P 
c) Calculate     . Did you expect this result?
 T  P  V 

Stewart 11.3: 11, 21, 25, 27, 29, 39, 41, 45

83. Use partial derivatives to find the parametric equations of the line tangent to the curve of
intersection of the graph z  x 2  3y 2 with the plane y = 2 at the point x = 1, y = 2.

84. Let C be the curve of intersection of the surface z  8  3( x  2) 2  4( y  1) 2 with the

plane x  3 .
a) Describe, and then sketch, the curve C.
b) Does the equation z  5  4( y  1) 2 describe the curve C? If not, what does this equation
describe?
c) Use partial differentiation to find a vector equation for the line tangent to C at the point
(3, 2,1) .

30
85. Let f ( x, y)  8  x 2  2 y 2 .
a) Sketch the surface z  f ( x, y) .
b) Which level curve passes through the point (2, 1)?
c) Sketch and describe the curve of intersection C of the surface z  f ( x, y) with the plane
x  2.
d) Use partial differentiation to find a vector equation of the tangent line to C at the point
(2, 1, 2) .
e) Find parametric equations for the curve C and hence find a vector equation of the tangent
line to C at the point (2, 1, 2) .

86. Let f ( x, y)  arctan( x 2  y 2 )

a) Draw and label a few of the level curves of f.
b) Sketch the surface z  f ( x, y) .
c) Find parametric equations of the curve of intersection C of z  f ( x, y) and x = 2.
d) Find a vector equation of the tangent line to C at the point where x = 2 and y = 1.

Stewart 11.3: 87, 89

87. For the following functions, find f xx ( x, y) , f xy ( x, y) , f yx ( x, y) and f yy ( x, y) .

x
a) f ( x, y)  x 2 e xy b) f ( x, y )  c) f ( x, y)  x 2 y 3
y

2z 2z 2z 2z

88. For the following functions, find , , and .
x 2 yx xy y 2
x2
a) z  sin(xy ) b) z  x cos y
c) z 
y  y2

Stewart 11.3: 51, 53, 55, 57, 85

 xy  x 3
 ( x, y )  (0,0)
89. Consider f ( x, y )   x 2  y 2 .

 0 x y0
What is lim f ( x, y) ? Is f continuous at (0, 0)? What is the relationship between partial
( x , y )( 0, 0 )

derivatives and continuity? (and hence differentiability)

 y2
90. Find the tangent plane to z  e x
2
at (1, −1, 1).

91. The equation of the tangent plane to the surface z  xe xy at the point (2, 0, 2) is
(A) x  4 y  z  4 (B) x  4 y  z  0 (C) x  4 y  z  4
(D) x  4 y  z  4 (E)  x  4 y  z  4

31
 5  ex
92. Find a Cartesian equation for the tangent plane to the surface z  at the point
y3
(0, 1, 4).

Stewart 11.4: 1, 3, 5, 11, 17

Differentiability
If f ( x, y) is continuous at (a, b) and if both partial derivatives exist at (a, b) then f is
differentiable at (a, b) if and only if
f (a  h, b  k )  f (a, b)  hf x (a, b)  kf y (a, b)
lim 0.
( h , k )( 0, 0 )
h2  k 2
In MAM2085, we call this expression “The Elephant”. (Ndlovu?)

93. Is z  x 2  y 2 differentiable at (0, 0)?

 xy
 ( x, y )  (0,0)
94. Is f ( x, y )   x 2  y 2 differentiable at (0, 0)?

 0 x y0

 x2 y
 ( x, y )  (0,0)
95. Is f ( x, y )   x 2  y 2 differentiable at (0, 0)?

 0 x y0
(Use the result of Exercise 75.)

 x2 y
 ( x, y )  (0,0)
96. Let f ( x, y )   x 4  y 2

 0 x y0
a) To what value does f ( x, y) approach as ( x, y) tends to (0, 0) along the x-axis?
b) To what value does f ( x, y) approach as ( x, y) tends to (0, 0) along the y-axis?
c) To what value does f ( x, y) approach as ( x, y) tends to (0, 0) along the parabola y  x 2 ?
d) Is f continuous at (0, 0) ?
e) Is f differentiable at (0, 0) ?

32
  x( x 2  y 2 )
 ( x, y )  (0,0)
97. Consider f ( x, y )   x 2  y 2 .

 0 x y0
a) Show that f is continuous at (0, 0).
b) Show that the partial derivatives do exist at (0, 0).
c) Show that f is not differentiable by using “The Elephant”.

 ( y  x) 3
 ( x, y )  (0, 0)
98. Let g ( x, y )   x 2  y 2

 0 ( x, y )  (0, 0)
a) Is g continuous at (0, 0)? Explain. [Hint: try using polar coordinates]
b) Find g x (0, 0) and g y (0, 0) .
c) Is g differentiable at (0, 0)? Explain.

 4 if x  0 or y  0
99. Let g ( x, y)   2 .
x  y
2
otherwise
a) Find g (0, 0) , g (1, 0) and g (1,1) .
b) Sketch the surface z  g ( x, y) .
c) Find lim g ( x, y) (if it exists).
( x , y )( 0, 0 )

d) Does lim g ( x, y) exist?

( x , y )(1,1)

e) Is g continuous at (0,0)? Explain.

f) Find g x (0,0) .
g) Find g x (1,0) , g y (1,0) , g x (1,1) and g y (2,0) .
h) Is g differentiable at (0,0)? Explain.
i) Does the surface z  g ( x, y) have a tangent plane at (0,0)?

 2 x  y if x  0 or y  0
100. Let f ( x, y)   2 .
x  y
2
otherwise
a) Does lim f ( x, y) exist?
( x , y )(1, 0 )

b) Is f differentiable at (1,0)?
c) Do f x (1, 0) and f y (1, 0) exist? If so, find them.
d) Is f differentiable at (0,0)?

101. Let h( x, y)  xy
a) Show that hx (0,0)  hy (0,0)  0
b) Prove that h is continuous but not differentiable at (0,0).

33
  x 2 ( x  y)
 ( x, y )  (0,0)
102. Let f ( x, y )   x 2  y 2 .

 0 x y0
a) Find f x (0,0) and f y (0,0) .
b) Is f continuous at (0, 0)?
c) Is f differentiable at (0, 0)? Explain.

 x 2  y 2 if x  0
103. Let g ( x, y )  
 y
2
if x  0
a) Find the partial derivatives g x (0, 2) and g y (0, 2) , if they exist.
b) Is there a tangent plane to the graph of z  g ( x, y) at the point P(0, 2, 4) ? Give full
reasons for your answer.

Chain Rule
If z  f ( x, y) and x  g (t ) and y  h(t ) are differentiable functions, then
dz z dx z dy
  (1)
dt x dt y dt
If z  f ( x, y) and x  g (s, t ) and y  h(s, t ) are differentiable functions, then
z z x z y
 
s x s y s
(2)
z z x z y
 
t x t y t
Proof of (1)
dz f ( x(t  h), y (t  h))  f ( x(t ), y (t ))
 lim
dt h 0 h
f ( x(t  h), y (t  h))  f ( x(t ), y (t  h))  f ( x(t ), y (t  h))  f ( x(t ), y (t ))
 lim
h 0 h
f ( x(t  h), y (t  h))  f ( x(t ), y (t  h)) f ( x(t ), y (t  h))  f ( x(t ), y (t ))
 lim  lim
h 0 h h 0 h
f ( x(t  h), y (t  h))  f ( x(t ), y (t  h)) x(t  h)  x(t )
 lim 
h 0 x(t  h)  x(t ) h
f ( x(t ), y (t  h))  f ( x(t ), y (t )) y (t  h)  y (t )
 lim 
h 0 y (t  h)  y (t ) h
z dx z dy
 
x dt y dt

dz
104. We are given that z  x 2 y 3  3xy 5 and that x  sin t , y  cos 2t . Find when t  6 .
dt

34
105. We can show that familiar single variable differentiation processes can be expressed in
multi-variable chain rule format.
a) product rule z  xy, x  f (t ), y  g (t )
b) logarithmic differentiation z  u v , u  x, v  x
c) implicit differentiation F ( x, y)  0 , y is defined implicitly in terms of x

106. Given z  x 2  2 xy , x  t ln s and y  2t  s , find

z z 2z
a) and b)
s t s 2

107. If f ( x, y, z )  xe y  z , where x  2uv , y  u  v and z  u  v then f u evaluated at the

3

point (u, v)  (3,  1) is

(A) 64e 4 (B) 64e 4 (C) 32e 4 (D) 32e 4 (E) None of A-D

108. Let z  f (u, v) have continuous second order partial derivatives. Suppose that
y
u( x, y)  xy and v( x, y ) 
x
z z z
a) Find in terms of and .
x u v
2z
b) Find an expression for x 2 in terms of u and v and the partial derivatives of z with
x 2
respect to u and v.

109. The radius r and altitude h of a right circular cylinder are increasing at rates of 0.01
cm/min and 0.02 cm/min respectively. Use the chain rule to find the rate at which the volume
is increasing at the time when r = 4 cm and h = 7 cm. At what rate is the total surface area
changing at this time?

Stewart 11.5: 5, 7, 11, 21, 23, 25, 27, 37, 43, 45, 47, 49, 51

35
Directional derivatives
The directional derivative of f at ( x0 , y0 ) in the direction of a unit vector uˆ  (a, b) is
f ( x0  ha, y0  hb)  f ( x0 , y0 )
Duˆ f ( x0 , y0 )  lim
h 0 h
if this limit exists.

110. Find the directional derivative of f ( x, y)  e x ln y at P(2, 1) in the direction of Q(3, 4) .

[Note: if given two points, the direction vector must be determined. Often the direction vector
is given. Read the question carefully.]

 x( x 2  y 2 )
 ( x, y )  (0, 0)
111. Consider f ( x, y )   x 2  y 2 .

 0 x y0
It can be shown that f is continuous but not differentiable at (0, 0) [Can you show this?].
Show that all directional derivatives exist at (0, 0).

 x2 y
 ( x, y )  (0, 0)
112. Consider f ( x, y )   x 2  y 2 .

 0 x y0
It can be shown that f is continuous but not differentiable at (0, 0) [Can you show this?].
a) Show that all directional derivatives exist at (0, 0).
b) Use the directional derivative to find the direction of the tangent line at (0, 0) to the curve
of intersection of f and y = x.

 x3  y3
 ( x, y )  (0, 0)
113. Consider f ( x, y )   x 2  y 2 .

 0 x y0
It can be shown that f is continuous but not differentiable at (0, 0) [Can you show this?].
Show that all directional derivatives exist at (0, 0).

36
  xy  x 3
 ( x, y )  (0, 0)
114. Consider f ( x, y )   x 2  y 2 .

 0 x y0
It can be shown that f is not continuous and hence not differentiable at (0, 0) [Can you show
this?].
a) Find the partial derivatives at (0, 0)

b) Show that the directional derivative in the direction v  (1, 1) does not exist.

 y 2 ( x  y)
 if ( x, y )  (0,0)
115. Let g ( x, y )   x 2  y 2 .

 0 if ( x, y )  (0,0)
a) Does lim g ( x, y) exist? Explain.
( x , y )( 0, 0 )

b) Is g continuous at (0,0)? Why?

c) Find g x (0,0) and g y (0,0) .
d) Using your answers to (c), what equation would you expect the tangent plane at (0,0,0) to
have?
e) Show that all the directional derivatives exist at the point (0,0).
f) Write down the directional derivative of g at (0,0) in the direction (1,1).
g) Does your answer to (f) make you want to change your answer to (d)?
h) Do you think that g is differentiable at (0,0)?
i) Use the definition of differentiability to check whether your answer to (h) is correct.


1  x  y
2 2
y0
116. Let f ( x, y )   .
 1 y
 y0
x

a) Does lim f ( x, y) exist? Explain.

( x , y )( 0, 0 )

b) Find f x (0, 1) and f y (1, 0) .

c) Is f differentiable at (0, 0)? Explain.
d) Is f differentiable at (1, 0)? Explain.
e) Find the directional derivative of f at (0, 0) in the direction (1, 2).
f) Is f differentiable at (0, 1)? Explain.

117. Use the definition of the directional derivative to find the directional derivative of the
function
 x 2 ( x  y)
 ( x, y )  (0,0)
f ( x, y )   x 2  y 2

 0 x y0
at the point (0,0) in the directions: a) (1, 0) b) (0, 1) c) (1, 2)
d) Do all the directional derivatives at (0,0) exist?

37
  x  y x  0 or y  0
118. Let f ( x, y )   .
 2 otherwise
a) Is f continuous at (0, 0)?
b) Is f continuous at (1, 1)?
c) Find f x (1, 0) and f y (1, 0) .
d) Find f x (2, 0) and f y (2, 0) .
e) Find the directional derivative of f at the point (2, 0) in the direction (1, 1).
f) Is f differentiable at (0, 0)?
g) Is f differentiable at (2, 0)?
h) Is f differentiable at (1, 1)?

 2 if x  0 or y  0
119. Let f ( x, y)   2
x  y
2
otherwise
a) Find the directional derivative of f at the point (2, 1) in the direction (3, 4).
b) What is the minimum value of the directional derivative of f at the point (2, 1)? In which
direction must we move to obtain this minimum value?
c) Find the directional derivative of f at the point (0, 0) in the direction (3, 4).

Stewart 11.6: 11, 17

 x3 y
 ( x, y )  (0, 0)
120. Let f ( x, y )   x 3  y 3

 0 ( x, y )  (0, 0)
a) Use the definition of the derivative to find the partial derivatives of f at (0, 0).
b) Find the directional derivative to f at (0, 0) along the line y = x.
c) Use your answers to (a) and (b) to demonstrate that f is not differentiable at (0, 0).

121. Given the function f ( x, y)  x 2  y 2  3

a) Sketch the level curves for this function.
b) Use these level curves to sketch the graph of this function.
c) In what direction from the point (1, 2) does the function f ( x, y) increase most rapidly?
Indicate this direction on the sketch you made in (a).
d) In which directions from the point (1, 2) does the function f ( x, y) remain constant?

122. Find the maximum rate of change of f ( x, y)  ln( x 2  y 2 ) at (1, 2) and the direction in
which maximum rate of change occurs.

123. Suppose that f ( x, y)  ye  x  xe  y .

a) Find the maximum rate of change of f at the point (0, 0).
b) In which direction(s) is the rate of change of f at (0, 0) equal to −1?

38
124. An engineering student is standing on the slopes of a curiously smooth and symmetric
mountain. Consulting her map, she notices that if she introduces a suitable coordinate system,
39( x 2  y 2 )
then the height (in metres) of a point with coordinates ( x, y) is h( x, y)  2000  .
1000
She is standing at the point with coordinates (100, 200).
a) What is her present elevation?
b) How many metres is she below the top of the mountain?
c) In which direction from where she is standing does the ground slope upwards most
steeply?
d) In which direction from where she is standing does the ground slope downwards most
steeply?
e) In which directions from where she is standing can she walk without changing her
elevation?

125. Let f ( x, y, z)  yz 2  z 1  x and let P be the point (3,2,1).

a) Find the level surface of f on which P lies.
b) Find the gradient of f at P.
c) Assuming that f is differentiable at P, write down a Cartesian equation for the plane that is
tangent at P to the level surface through P.
d) What is the rate of change in f at P in the direction of the vector (5, 0,1) ?
e) In which direction is the rate of change in f at P a maximum? What is the greatest rate of
change?
f) Is there a direction in which the rate of change in f is equal to 7? Explain.
g) What is the directional derivative of f at P in the direction (0,0,1)?

126. Let g ( x, y, z)  x  2 y  3z
a) Show that the point Q(3,1, 1) lies on the surface g ( x, y, z)  4 .
b) Find g (Q) . Did you expect this?
c) Find the directional derivative of g at Q in the direction of the vector (1, 1,1) . Was this
result expected?

Stewart 11.6: 7, 9, 23, 25, 27, 29, 35

127. Find the equation of the tangent plane to the surface described by
xy  2 yz  xz 2  10  0 at (5, 5, 1) .

39
128. Find the equation of the tangent line to the curve of intersection of z  5x 2  8 y 2 and
2 x 2  3 y 2  1  z  0 at ( 1
3
, 0, 53 )
a) by using gradients (“grad” in other words)
b) by using parametric equations

129. Find the tangent plane and normal line of the surface f ( x, y, z )  x 2  y 2  z  9 at the
point P(1, 2, 4) .

130. Find the equation of the tangent line to the curve of intersection of
f ( x, y, z )  x 2  y 2  2  0 and g ( x, y, z)  x  z  4  0 at (1, 1, 3) .


131. Show that the curve r (t )  (t 2 , 3t , 2 t ), t  0 pierces the surface 2 x 2  y 2  12 z 2  13 at
right angles at the point (1,3,2).

132. At what point on the paraboloid y  x 2  z 2 is the tangent plane parallel to the plane
x  2 y  3z  1 ?

133. Show that the surface x 2  2 yz  y 3  4 is perpendicular to each member of the family
of surfaces x 2  1  (2  4a) y 2  az 2 at the point (1,1, 2) , where they all intersect (where
a  R ).

134. Let S be the surface z  8  x 2  2 y 2 and let P be the point ( 2 , 2 , 2 ) . Let

 
r1 (t )  (2 cos t , 2 sin t , 2 cos t ) and r2 (t )  ( 2 , 3 cos t , 6 sin t ) .

a) Show that r1 (t ) lies on S.

b) Show that r2 (t ) lies on S.

c) Find the direction of the tangent line to r1 (t ) at P.

d) Find the direction of the tangent line to r2 (t ) at P.
e) Use (c), (d) and the cross product to find the tangent plane to S at P.
f) Let f ( x, y, z)  x 2  2 y 2  z 2 . Show that P lies on the level surface f ( x, y, z)  8 . Find
f (P) and hence the tangent plane to S at P.

135. Which of the following statements are true and which are false? For those that are true,
provide a proof. For those that are false provide a counterexample.
a) If f x (a, b) and f y (a, b) both exist, then f is differentiable at (a, b) .
b) If f x (a, b) and f y (a, b) both exist, then f is continuous at (a, b) .
c) If f x (a, b) and f y (a, b) both exist, then all the directional derivatives of f at (a, b) exist.
d) If all the directional derivatives of f at (a, b) exist , then f is differentiable at (a, b) .

40
e) If f is not continuous at (a, b) , then f x (a, b) and f y (a, b) do not exist.
f) If all the directional derivatives of f at (a, b) exist , then the surface z  f ( x, y) has a
tangent plane at (a, b) .
g) If f x (a, b) and f y (a, b) both exist, then the surface z  f ( x, y) has a tangent plane at (a, b) .
h) If f x (a, b)  f y (a, b) , then f is differentiable at (a, b) .
i) If f x (a, b) and f y (a, b) both exist, but f x (a, b)  f y (a, b) , then f is not differentiable at
(a, b) .
j) If f is differentiable at (a, b) then all the directional derivatives of f at (a, b) have the same
value.
k) If all the directional derivatives of f at (a, b) are zero, then f is differentiable at (a, b) .
l) If at least one of the directional derivatives of f at (a, b) is non-zero, then f is not
differentiable at (a, b) .
m) If all the directional derivatives of f at (a, b) are non-zero, then f is not differentiable at
(a, b) .
n) If a curve C lies on a surface S, then the osculating plane to C at the point P is the same as
the tangent plane to the surface S at P.

Stewart 11.6: 41, 43, 47, 55, 57

41
 Chapter 2: Multivariate integrations and
Transformations
Section 2.1: Double integrals
Integration terminology: The terms “repeated integral” and “iterated integral” refer to a
double or triple integral where the limits of integration are shown and the dA or dV has been
broken down into dxdy or whatever is appropriate. Example:
1 

   y sin zdxdydz
0 0 0

1. Evaluate  y sin( xy )dA where R  {(x, y) : 1  x  2, 0  y   } .

R

 ( x  y)dA where R is the region in R bounded by the graphs of y  2x 2 and

2
2. Evaluate
R

y  1 x 2


2 5
3. Evaluate   cos ydxdy .
 1
6

1 x2
4. Evaluate R 1  y 2 dA where R  {(x, y) : 0  x  1, 0  y  1} .

1 2
xe x
5. Evaluate 0 1 y dydx .

Stewart 12.2: 3, 9, 17, 19, 21

Stewart 12.3: 5, 9, 17, 19, 21

1 1

  sin( y
2
6. Evaluate )dydx .
0 x

2 4

  ye
x2
7. Evaluate dxdy .
0 y2

42
 2 4 y

8. If we interchange the order of integration in   ( x  y)dxdy , the integral becomes

0 1

2 4 y 4 y 2

(A)   ( x  y)dxdy
0 1
(B)   ( x  y)dxdy
1 0

22 2 4 x 2
22 2 4 x 2
(C)   ( x  y)dydx    ( x  y)dydx
1 0
(D)   ( x  y)dydx    ( x  y)dydx
2 0 1 0 2 1

22 2 4 y

(E)   ( x  y)dydx    ( x  y)dydx

1 0 0
2

9a) Can you integrate cos( x 2 ) with respect to x?

2 4

  y cos( x
2
b) Evaluate )dxdy .
0 y2

1 2 0 2

  e dxdy   e
2
x x2
10. Evaluate dxdy .
0 2y 1  2 y

Stewart 12.3: 41, 43, 45, 47, 49, 60

11. Find the volume of the solid which lies under the graph of z  x 2  y 2 and above the
region in the xy-plane bounded by y  2 x and y  x 2 .

12. Find the mass and centre of mass of the lamina of density  ( x, y)  y , which occupies
the region R, bounded by y  9  x 2 and y = 0.

13. Find the mass of the triangular lamina with vertices at (0, 0), (4, 0) and (4, 2), if the
density is given by  ( x, y)  xy .

14. Let R be the region in the first quadrant bounded by the graphs of the parabolas y  2x 2 ,
y  9  x 2 and the line x  0 .
a) Find the points of intersection of the parabolas.
b) Sketch the region R.
c) Express the area of the region R as a repeated integral, integrating first with respect to y,
and then with respect to x.
d) Express the area of the region R as a repeated integral, integrating first with respect to x
and then with respect to y.
e) Use either (c) or (d) to calculate the area of the region R.
f) Now let R′ be the region bounded by the parabolas y  2x 2 and y  9  x 2 . Find the area
of the region R′.

43
  xy dA . (Think carefully about the function f ( x, y)  xy 2 !)
2
g) Evaluate
R

15. Consider the solid bounded by the graphs of x 2  y 2  9 , z  0 and z  x 2 y .

a) Which of the following represents the volume of this solid?
3 0 3 9 x 2
i) 4 x  x
2 2
ydydx ii) ydydx
0  9 x 2 3  9  x 2

3 9 y 2 3 9 y 2

iii) 2  x
2
ydxdy iv) 4 x
2
ydxdy
3 0 0 0

b) Evaluate all of the above.

1 2x
16. Let I    f ( x, y )dydx .
0 x

a) Give two possible physical interpretations of I if f ( x, y) is non-negative over the region in

question.
b) Express I as a sum of repeated integrals, integrating first with respect to x and then with
respect to y.
c) Evaluate I if f ( x, y)  1  x 2 .

Stewart 12.1: 13, 17, 18

Stewart 12.2: 25, 27, 29
Stewart 12.5: 1, 3, 5, 7
Worksheet on areas as iterated integrals (see Appendices)

2 3

  (4  y
2
17. Sketch the solid whose volume is described by )dxdy .
0 0
1 x

  xy
2 2
18. Sketch the region in R over which the integral dydx is evaluated.
0 x2

3 y

19. Sketch the solid whose volume is described by   (3  y)dxdy

0 y

20. For each of the following, describe and sketch a solid whose volume is given by the
repeated integral:
2 4 y 2 1 2 x 2

  (5  x  2 y)dxdy   (2  x  y 2 )dydx
2
a) b)
2  4 y 2 0 x

44
Stewart 12.2: 23

Section 2.2: Surface integrals

21. Given the surface S, project it onto the indicated coordinate plane. Sketch the projection
with all boundaries and intercepts clearly defined.
a) S is that portion of the parabolic sheet y  1  2 x 2 which lies in the first octant between the
planes x = 0, x = 2, z = 4 and z = 8. Project onto the xz-plane.
b) S is that portion of the plane z  6  3x  2 y which lies in the first octant. Project onto the
xy-plane.

22. Given the surface S, project it onto the indicated coordinate plane. Sketch the projection
with all boundaries and intercepts clearly defined. S is that part of the surface of
z  4  x 2  y 2 in the first octant. Project onto all three coordinate planes.

Surface integral conversion factor

Let G( x, y, z ) be a function defined for all ( x, y, z )  S , where S is a portion of the graph of
z  f ( x, y) . The surface integral of G over S is
2
 z   z 
2

S G ( x, y , z ) dS  R G ( x, y , f ( x , y )) 1       dA
 x   y 
where R is the projection onto the xy-plane.
Alternately, if S can be described as a portion of y  g ( x, z) or x  h( y, z ) , we can project it
onto the xz- or yz-planes respectively and the surface integral would be
 y   y 
2 2

S G( x, y, z)dS  R G( x, g ( x, z), z) 1   x    z  dA

2
 x   x 
2

or  G( x, y, z )dS   G(h( y, z ), y, z ) 1       dA
S R  y   z 
2
 z   z 
2

At its simplest, S dS  R 1       dA

 x   y 

23. Let S be that portion of the parabolic sheet y  1  2 x 2 which lies in the first octant

 xz
2
between the planes x = 0, x = 2, z = 4 and z = 8. Evaluate dS .
S

24. Evaluate  xzdS where S is that part of the parabolic gutter

S
y  8  2 x 2 lying in the first

octant and below the plane z = 5.

Stewart 13.6: 9, 11

45
25. Find the area of that part of the portion of the surface z  3x  y 2 which lies above the
triangle in the xy-plane with vertices (0, 0, 0), (0, 1, 0) and (1, 1, 0).

26. Evaluate the mass of that part of the plane 2 x  2 y  z  2 which lies in the first octant, if
surface density at a point is equal to x  y  z .

27. Find the area of that part of the plane 2 x  5 y  z  10 which lies above the triangle with
vertices (0, 0), (0, 6) and (4, 0).

28. Let Ω be the portion of the cylinder x 2  z 2  4 that lies in the first octant between the
planes x = 0, y = 0 and x  y  4 . Calculate the mass of Ω if the surface density at each point
on Ω is  ( x, y, z)  xz .

29. Let S be that portion of the plane 6 x  3 y  2 z  6 that lies above the triangular region
with vertices (1,0,0), (0,0,0) and (0,2,0).
a) Sketch the surface S.
b) Calculate the area of S.
c) Find the mass of S if the surface density at any point ( x, y, z )  S is  ( x, y, z)  4 z .

30. Let S be that portion of the surface y  1  4 x 2 which lies in the first octant between the
planes z  0 and z  3 . Find the mass of S if the density at any point on S is equal to the
distance from that point to the yz-plane.

31. Calculate the surface area of that part of the cylinder y 2  z 2  1 which lies in the first
octant between the planes x = 0, y = 0, z = 0 and y  2  x .

32. Find the area of the surface z  2 xy where 0  x  a and 0  y  b .

Stewart 12.6: 1, 3


33. Suppose that a certain fluid has a velocity field F ( x, y, z )  (0, z, 3) . Find the flux of this
fluid through the portion of the plane z  6  3x  2 y which lies in the first octant.

34. Evaluate the flux through the surface of the cube with one corner the origin and the

opposite corner the point (1, 1, 1) if F ( x, y, z )  3xiˆ  2 yˆj  zkˆ .

46
35. Find the flux of ziˆ  x 2 kˆ upward through that part of the surface z  x 2  y 2 lying above
the square R defined by  1  x  1 and  1  y  1 .


36. Suppose that the velocity of a fluid at each point ( x, y, z ) is given by v ( x, y, z)  ( y, x, 6) .
a) Let S1 be that part of the sphere x 2  y 2  z 2  4 that lies above the plane z = 1. Find the
flux of the fluid passing through S1 in the upward direction.
b) Now let S2 be the entire surface of the sphere x 2  y 2  z 2  4 . Find the flux of the fluid
passing through S2 in the outward direction.


37. Find the flux of the field F ( x, y, z )  z 2 iˆ  xˆj  3zkˆ outward through the surface cut from
the parabolic cylinder z  4  y 2 by the planes x = 0, x = 1 and z = 0.

Stewart 13.6: 21, 31

Section 2.3: Triple integrals

 xyz
2
38. Evaluate dV where R is the rectangular box given by
R

R  {( x, y, z) : 0  x  1,  1  y  2, 0  z  3} .

39. Evaluate  zdV where R is the solid tetrahedron bounded by the four planes x = 0, y = 0,
R

z = 0 and x  y  z  1 .

40. Evaluate the following triple integrals

1  1 2 x 2 x  y
a)    y sin zdxdydz
0 0 0
b)    dzdydx
0 0 0

1
41. Calculate  12  2 x  2 y dV , where E is the solid region in the first octant lying inside
E

the cylinder x  y 2  4 , and below the plane x  y  2 z  6 .

2

Stewart 12.7: 3, 5, 11

42. Find the volume of the solid in the first octant bounded by the graphs of z  1  y 2 ,
y  2 x and x = 3.

43. Find the mass of the solid bounded by the planes x  z  1 , x  z  1 , y = 0 and the
surface y  z , if the density is  ( x, y, z)  2 y  5 .

44. Find the volume of the solid bounded by z  x 2 , z = x, y = 0 and x  y  2 .

47
45. Compute the mass of the solid unit cube, lying in the first octant with one corner at the
origin, if the density of the material is proportional to the square of the distance from the
origin. Take the constant of proportionality to be equal to 𝜋.

46. Let D be the solid in the first octant enclosed by z = 0, y = 0, x = 3, y = x and z  4  y 2 .

Find the mass of D if the density at each point ( x, y, z ) is x 2 .

47. Let B be the solid region enclosed by the hemisphere z  4  x 2  y 2 and the paraboloid
3z  x 2  y 2 , and let S be the portion of the hemisphere that lies inside the paraboloid.
a) Express the volume of B as a repeated integral, integrating with respect to x, y and z. Use
the order of integration that you think is best.
b) Express the volume of B as a double integral over a region R in the xy-plane. Hence write
the volume as a repeated integral, integrating with respect to x and y.
c) Express the surface area of S as a double integral over R. Hence write this area as a
repeated integral, integrating with respect to x and y.
d) Find the mass of S if the surface density at each point ( x, y, z ) on S is given by
 ( x, y, z)  z .
e) Express the mass of B as a repeated integral if the density at each point ( x, y, z ) in B is y .

48. Find the volume of the region D bounded by the circular paraboloid y  x 2  z 2 and the
elliptic paraboloid y  16  3x 2  z 2 .

49. Express the volume of the region bounded by z  x 2  y 2 and z  3  2 x 2  2 y 2 as a

triple integral with respect to the variables x, y and z.

50. Express the volume of the solid enclosed by z  x 2  2y 2 and z  4  x 2  y 2 as a

repeated integral of the form    dzdydx . Do not evaluate.
Stewart 12.7: 19, 21, 27

48
51. The figure shows the region of integration for the integral
1 1 x 2 1 x
I    f ( x, y, z)dydzdx .
0 0 0

Rewrite it as an equivalent integral in the five other orders.

52a) Sketch a solid whose volume is given by the repeated integral

( 6 y )
3 18 y
2
3

 
0 y
1dzdxdy
0

b) Now express the volume of the same solid as the sum of repeated integrals of the form
  

  1dxdydz
  

53. Sketch and describe the region in R3 whose volume is given by

2 4 x 2 4 x 2  y 2

 
0 x
 dz dydx .
0

Express the volume as a repeated integral of the form

   dzdxdy
Stewart 12.7: 29, 31, 33

Section 2.4: Linear transformations

 
54. Show that T ( x )  (2 x, x  y) is linear.


55. Prove that T ( x )  ( x  2, 3 y) is not a linear transformation.

56. Decide which of the following are linear transformations. Give reasons.
 x
 x  x  y    x  y
a) T : R  R defined by T    
2 2
 , b) T : R 3  R 2 defined by T  y     ,
 y   y  4 z  xz 
 
 x y 
 x   
c) T : R 2  R 3 defined by T     y  2 x  ,
 y  y 
 
1  3  0    1 1  2 
d) T : R 2  R 2 satisfying T      , T      , T      .
 0  4 1  1  1  3 

57. Which of the following transformations from R2 to R2 are linear transformations?

 x   x 1  x   x2   x   2x  y 
a) f      b) f     
 c) f     
 y   y  3  y   sin y   y   x  3y 
49
58. Given the transformations in function form, find the matrix which represents them.
 
a) T ( x )  ( x  2 y, x  4 y) b) T ( x )  (3x  y, 2 x)

c) T ( x )  ( x  z, x  y, y  z)

59. Given the transformations in function form, find the matrix which represents them.
  
a) T ( x )  (5x, x  2 y) b) T ( x )  ( y, x  7 y) c) T ( x )  (3z, 2 x, x  y  z)

60. Use the given mapping to transform the indicated points. Plot the points on a diagram,
with arrows to show how they have been transformed.

a) F ( x )  ( x  y,  x) P(1, 0) Q(2,  2) R(0, 0)
b) F ( x, y)  (2 x  y, x  2 y) P(2, 3) Q(1, 2) R(0, 0)

61. The diagram alongside has the points (−1, −1), (0, 1), (1,1), (3,1),
(3,3), (1, 3) and (2,5) plotted. Apply the transformation
 2x 
T ( x, y )    to the points shown in the diagram.
 x  y
 Note how some of the points in the diagram lie on a line. See how
they remain in a line under the transformation.
 Also see how some of the points form a square. What does this
square turn into under the transformation?

 3  1
62. Let T : R 2  R 2 be the linear map represented by the matrix A    .
1 3 
a) Find the image under T of the square with vertices at (0, 0), (1, 0), (1, 1) and (0, 1).
b) Calculate the area of this image.

1 2
63. Let T be the transformation represented by the matrix   and let R be the region
 3 4 
bounded by the lines y = 0 and
x = 1 and the curve y  x 2 . Find the area of the image of R under T.

64.For each of the situations below (i) plot the given points, (ii) transform the points using the
transformation given (and plot them), (iii) determine the ratio of the areas of the initial
rectangle and transformed parallelogram and (iv) evaluate the determinant of the matrix
representing the transformation. You should notice a relationship between your answers to
(iii) and (iv).

a) (0, 0), (2, 0), (2, 3), (0, 3) T ( x )  (2 x,  y)
b) (1,  2), (1, 1), (4, 1), (4,  2) F ( x, y)  ( x  y, 2 x)
  2y 
c) (1, 2), (3, 2), (3, 3), (1, 3) G ( x )   
 x  2y

50
 1 2 
65. A region R in the xy-plane is mapped by the matrix A    to the region R′, and R′
 4  3
 0  1
is in turn mapped to the region R′′ by the matrix B    .
 4 1 
a) Express the area of R′′ in terms of the area of R.
b) Which matrix maps the region R to the region R′′?

 1 0 1
 
66. Apply the transformation represented by the matrix  2 1 0  to
 1 1 3
 

a) the line x ( )  (1, 0, 0)   (2, 1, 2) ,

b) the plane x (,  )  (1, 0, 0)   (2, 1, 2)   (1,1, 1) and
c) the circle x 2  y 2  1 , z = 0.


67. Find the image of the line x (t )  (1, 2, 3)  t (2,4, 1) under the linear map represented by
 1 2 3
 
the matrix  4 0 1  .
 1 1 2
 

 1 2 3
 
68. Let A   0  1 0  .
 1 2 3
 
Find the images, under the linear map represented by A, of each of the planes
a) x  y  3z  3
b) x  y  z  3

69a) What is the image of the circle x 2  y 2  1 under the linear map represented by the
matrix
 a 0
  ?
 0 b
b) Use your answer to part (a) to find the area of the ellipse
x2 y2
 1
a2 b2

51
 1 1 3
 
70. Let T : R  R be the linear map represented by the matrix A   2 4 1  .
3 3

4 2 7
 
a) Find det(A).
b) Let B be the rectangular box in R3 bounded by the planes x = 1, x = 4, y = 3, y = 5, z = 4
and z = 5. What is the volume of B? What is the volume of the image of B under T? What can
you conclude about the image of B under T?
c) Find the image under T of the plane x  3 y  2 z  0 .

71. Let T : R 2  R 2 be the function defined by T ( x, y)  (2 x,3 y) for all ( x, y)  R 2 .

a) Show that T is a linear map.
b) Write down the 2×2matrix that represents T.
c) What is the image under T of the square with vertices (0,0), (1,0), (1,1) and (0,1)?
d) Find the image under T of the circle x 2  y 2  1 .

72. Let F : R 2  R 2 be the function defined by F ( x, y)  ( x, y 2 ) for all ( x, y)  R 2 .

a) Show that F is not a linear map.
b) Which points in R 2 remain unchanged under F?
c) What is the image under F of the line y = x?
d) Find the image under F of the square with vertices at (1,1), (−1,1), (−1,−1) and (1,−1).
[This transformation is actually non-linear. To determine its effect on the square in (d) you
first need to parametrise each line segment in the square and then transform that
parametrisation. Finally interpret your result as a line or curve of some kind.]

 1 1 0
 
73. Find the image under the linear map induced by the matrix A   1 2 0  of the plane
 2 3 0
 
 2 x  y  1 (you may leave your answer in vector form if necessary).

1 2 3
 
74. Let A   0  1 2  .
1 4 5
 
a) Find the image of the plane z = 1 under the linear map represented by A.
0
 
b) Find A 0  and the normal to the plane onto which A maps z = 1.
1
 
c) Does the normal to the plane z = 1 get mapped to the normal of the image plane?
d) Does A preserve lengths and angles?
e) Find detA.

52
 x2 y2
75a) Show that the ellipse   1 can be given parametrically by
a2 b2
x  a cos t , y  b sin t , 0  t  2 , a, b  R .
b) Find the image of the ellipse in (a) under the transformation represented by the matrix
 1a 1

A   1 b
.
a  b1 
c) Find detA.
x2 y2
d) Use the results of (b) and (c) to find the area inside the ellipse   1.
a2 b2

76. Find the image of the plane 2 x  4 y  z  0 under the linear map represented by each of
the following matrices:
 1 2 3  1 2 3
   
a)  4 0 1  b)   3 10 1 
 1 0 2  1 2 3 
  

77. Find the Cartesian equation(s) of the image of the plane x  7 y  7 z  4  0 under the
1 2 3
 
linear map represented by the matrix  0  1 2  .
1 2 3
 
[For this exercise, consider “Cartesian equation” to mean “the equation with no parameters in
it”]

78. Suppose that a linear map T : R 3  R 3 is such that

T (0, 1, 0)  (1, 2, 1) , T (0, 1, 1)  (3, 4, 5) and T (1, 1, 1)  (2, 1, 1) .
Find the matrix which represents the transformation.

79a) Find a vector equation for the plane P  {( x, y, z)  R 3 : x  3 y  4 z  1} .

b) Suppose that a linear transformation T : R 3  R 3 is such that
T (1, 0, 0)  (1,1,1) , T (1,1, 0)  (2, 2, 4) and T (1, 0,1)  (1, 3, 0)
What is the image of the plane P under T?

Section 2.5: Non-linear transformations, derivative matrices and affine

approximations
80. Find the derivative matrices of the following transformations.
  
a) T ( x )  ( x  y, 3x  y) b) F ( x )  ( x 2  y, x  y 3 ) c) G( x )  (sin x, cos( xy ))
 x2 y
    xe y 
d) M ( x )   y 2  e) A( x, y, z)  ( xy, y  z, z 2 ) f) W ( x, y, z )   

 xz 2   ln z 
 

53
81. Find the derivative matrices for each of the following transformations. Evaluate each of
the derivative matrices at (0, 0) and (1, 2).
 
a) T ( x )  ( xy 4 , e y ) b) F ( x )  ( y 2 , ln( x  1)) c) G( x, y)  (3x  y,  4 y)

Affine approximation
If F: R 2  R 2 is a transformation with F ( x, y)  ( f ( x, y), g ( x, y)) and f and g are
differentiable functions, then the affine approximation of F at (a, b) is
 x  a
F ( x, y)  F (a, b)  F (a, b)  .
 y  b
This expression extends easily to higher dimensions.
 x  a
 
F ( x, y, z )  F (a, b, c)  F (a, b, c) y  b 
 z c
 

82. Let F ( x, y, z)  ( xyz , xy 2 , yz ) . Use the affine approximation to F ( x, y, z ) at (1, 2, 3) to

find the approximate value of F (1.1, 2.1, 2.9) .

83a) Let G : R 2  R 2 be given by

 3x  4 y 
G( x, y)   
 2x  y 
Find the derivative matrix G (2, 3) and then use the affine approximation to G( x, y) at (2, 3)
to estimate the value of G(1.9, 2.9) . Compare this estimate with the exact value.
b) Let F : R 3  R 3 be given by
 x2  y2  z2 
 
F ( x, y , z )   xyz 
1 x  y  z 
 
Find the derivative matrix F (1, 1, 1) and then use the affine approximation to F ( x, y, z ) at
(1, 1, 1) to estimate the value of F (0.8, 1.0, 1.1) . Compare this estimate with the exact value.

 (u, v)  ( x, y )
84. F ( x, y)  (u, v) where u  x  y and v  2 x  y . Find and .
 ( x, y )  (u, v)

 (u, v)  ( x, y )
85. F ( x, y)  (u, v) where u  x  2 y and v  x  y . Find and .
 ( x, y )  (u, v)

Stewart 12.9: 3, 5

54
  xyz 
 
86. Let F   xy 2  . Find the approximate volume of the image of a small sphere of radius r
 yz 
 
centred at (a) (1, 2, 3) and (b) (3,  1, 4) .

87. Let F : R 2  R 2 be defined by F ( x, y)  (u( x, y), v( x, y)) where u( x, y)  x 3  y 3 and

v( x, y)  x 2  2 y 2 .
a) Find the derivative matrix F ( x, y) .
b) Use the affine approximation for F ( x, y) about the point (−1,1) to estimate the value of
F (0.9, 0.9) . Compare your answer with the actual value of F (0.9, 0.9) .
c) Estimate the area of the image in the uv-plane of a circle in the xy-plane with centre (−1,1)
and radius 101 .
d) What would be the approximate area of this image if the centre of the circle in the xy-plane
were at (0,0)?

88. Let F : R 2  R 2 be defined by F ( x, y)  (u( x, y), v( x, y)) where u( x, y)  x  3 y and

v( x, y)  3x  y . Let R be the triangular region in the xy-plane with vertices at (0,0), (1,0)
and (0,2).
a) Find the image R′ of R in the uv-plane.
b) Calculate the area of R′ using basic geometry.
 (u, v)
c) Calculate and compare it with the ratio of the areas of R′ and R.
 ( x, y )
d) Find the affine approximation for F ( x, y) about the point (2,5). How good is this
approximation? Why?

89. Let H : R 2  R 2 be given by

 x2  y2 
H ( x, y )   

 xy 
a) Find F (0,0) . Does this tell you if F is a linear transformation?
b) Find the image under H of the square with vertices at at (0, 0), (1, 0), (1, 1) and (0, 1).
c) Find the approximate area of the image under H of a disc with area 10 -3 square units
centred at (1, 3).

90. Let R be the region in the xy-plane bounded by the lines x  y  6 , x  y  2 and y = 0.
Suppose that R is the image of a region R′ in the uv-plane under the transformation x  u  v ,
y  u v.
a) Find the region R′.
 ( x, y )
b) Calculate and compare this value with the ratio of the areas of R and R′.
 (u, v)

55
91. Let S be the image under the transformation u  xyz , v  x  y  z and w  x 2 y of a
sphere of volume 10-6 cubic units centred at (1, 2, 0). Estimate the approximate volume of S.

Stewart 12.9: 7, 9

Coordinate changes and Jacobians

When carrying out a double or triple integral, we can change the coordinate system by using
a Jacobian.
( x, y)
R f ( x, y)dA  R f (u, v) (u, v) dA
 (old )
Note that the Jacobian is .
 (new)
Since a Jacobian is a determinant, it can be either positive or negative. If you choose to use
the Jacobian to convert between two areas or volumes (as we do in multiple integrals) you
need to take its modulus to ensure it is positive.

Frequently used Jacobians are the polar ones:

dA  dxdy  rdrd
dV  dxdydz  rdzdrd   2 sin ddd

2 8 x 2
1
92. Use polar coordinates to evaluate  
0 x 5  x2  y2
dydx .

93. Evaluate  sin( x  2 y) cos( x  2 y)dA where R is the region enclosed by the triangle with
R

vertices (0,  ) , (2 , 0) and (0, 0).

94. Evaluate  xydA where R is the region bounded by

R
xy  1 , xy  5 , y  x 2 and y  4x 2 .

 y dA , where R is the region in the first

3
95. Use a suitable transformation to evaluate
R

quadrant bounded by the graphs of xy  1 , xy  5 , 2 x  y 2 and 4 x  y 2

96. Let R be the region in the first quadrant of the xy-plane bounded by the lines y = x and
y  x  1 and the parabolas y  3  x 2 and y  5  x 2 . Use a suitable transformation to

 ( x
 y )( y  x )
 y)(2 x  1)e ( x
2
2
evaluate dA .
R

56
 4 x 2 4 x  y
2 2
1
97. Let I     xdzdydx
0 3x 0

a) Sketch the region of integration in R3.

b) Evaluate I using cylindrical coordinates.

98. Let R be the region in R2 given by x 2  y 2  a , where a is a positive real number. Then

 e
( x2  y 2 )
dA is
R

(D) 2 (1  e a )
2
(A) 2 (1  e  a ) (B) 0 (C)  (1  e  a ) (E) None of A-D

99. Find the volume of the solid bounded by z  1  x 2  y 2 and z = 0.

 y
2
100. Use cylindrical coordinates to evaluate dV , where R is the solid region lying
R

inside the cylinder x  y  1 between the hemisphere z  4  x 2  y 2 and the plane z = 0.

2 2

101. Let R be the region in the first quadrant of the xy-plane bounded by the graphs of xy = 3,
xy = 5, y = x and y = 2x. Suppose that R is transformed into a region R′ in the uv-plane by the
y
mapping u( x, y)  xy , v( x, y )  .
x
a) Sketch the region R′.
 (u, v)
b) Find .
 ( x, y )
2 2
 y  y
c) Express the double integral    sin  dA as a repeated integral with respect to u and
R 
x x
v.
d) Evaluate this repeated integral.

x y

 e
x y
102. Evaluate dA where R is as shown below left.
R

57
  (3x  4 y
2
103. Evaluate )dA where R is the region shown above right.
R

104a) Find the mass of the hemisphere z  9  x 2  ( y  3) 2 if the density at each point on
this surface is equal to the height of that point above the xy-plane.
b) Express the surface z  9  x 2  ( y  3) 2 in spherical coordinates.

 e
( x2  y 2 )
105. Evaluate dA where R  {( x, y)  R 2 : x, y  0, x 2  y 2  a} .
R

106. Let R be the solid region lying inside the sphere x 2  y 2  z 2  9 above the upper half-
cone z  x 2  y 2 .
a) Sketch the region R and describe it using spherical coordinates.
b) Calculate the volume of the region R.
1
c) Find the mass of R if the density at each point ( x, y, z ) is .
x2  y2

107. Use spherical coordinates to find the mass of the solid region bounded by
x 2  y 2  ( z  2) 2  4 , given that the density at each point in the region is equal to the
distance from the origin.

108. Let Q be the solid region lying inside the cone z  x 2  y 2 and between the

hemispheres z  1  x 2  y 2 and z  4  x 2  y 2 . If the density at each point ( x, y, z)  Q

1
is  ( x, y, z )  , calculate the mass of Q.
x  y2  z2
2

109. Let R be the image of a region R′ under the transformation x  u 2 vw , y  uv 2 w and

1
z  uvw2 . Express  3
4
dV in terms of the volume of R′.
R ( xyz )

110. Find the volume of the solid within the cone z  x 2  y 2 and within the sphere
z  x2  y2  z 2 .

111. Find the volume of the solid under z  x 2  y 2 , above the xy-plane and inside
x 2  y 2  2x .

58
112. Find the surface area of that portion of the sphere x 2  y 2  z 2  8 which lies in the first
octant inside the cylinder x 2  y 2  8 y .

4 4 y y2

  ye
x
113. Express dxdy as a repeated integral using polar coordinates. (You need not
2  4 y y 2

evaluate this integral.)

114. Use polar coordinates to evaluate  ( x  y)dA where R is the region in the xy-plane
R

bounded by the graph of x  y  2 y  0 . 2 2

115. Let R be the solid region in the first octant bounded by the sphere x 2  y 2  z 2  8 .
a) Find parametric equations for the intersection of the surfaces x 2  y 2  z 2  8 and
2z  x 2  y 2 .
2
 y 2  z 2 )3 2
b) Find the mass of R if the density at each point ( x, y, z ) is e( x
c) Find the volume of that part of R that lies within the paraboloid 2 z  x 2  y 2 .
d) Find the surface area of that part of the sphere x 2  y 2  z 2  8 lying in the first octant and
within the cylinder x 2  y 2  2 2 y .

116. Given the triple integral

3 9 y x2

 
3 x 2  y  x 2
x 2  z 2 dzdydx

a) How would you change the order of integration to make this easier to handle?
b) Evaluate this integral.

117. Find the volume of the solid region that lies inside both x 2  y 2  z 2  16 and
x2  y2  4y  0 .

118. Find the volume of the solid enclosed by x 2  y 2  z 2  a 2 and

z 2 sin 2   ( x 2  y 2 ) cos 2  .

119. Find the mass of the solid region bounded by the parabolic surfaces z  16  2 x 2  2 y 2
and z  2 x 2  2 y 2 if the density of the solid is  ( x, y, z )  x 2  y 2 .

1 2 y y2

120. Express   dxdy as a repeated integral using polar coordinates. (Don’t evaluate this
0 0

integral.)

59
121. What is the mass of the paraboloid z  1  x 2  y 2 , 1  z  5 , in the first octant, if
density is equal to the distance from the xy-plane?

122. Let P be the portion of the paraboloid z  4  x 2  y 2 lying between the planes z = 0 and
z = 3.
a) Calculate the area of the surface P.

b) Calculate the flux of the vector field v ( x, y, z)  ( x 2 , y 2 , z 2 ) through P if the unit normal at
each point on P is chosen to point outwards.

123. Let R be the solid region inside the cylinder x 2  y 2  2 y between the hemisphere
z  4  x 2  y 2 and the plane z = 0.
a) Sketch the region R.
b) Write down the volume of half of the given hemisphere.
c) Use cylindrical coordinates to find the volume of the region R.
d) Compare your answers to parts (b) and (c). Are they plausible? If they are not, then try
again.

124. Calculate the volume of the solid which lies below the hemisphere z  1  x 2  y 2 ,
inside the cylinder x 2  y 2  y  0 and above the plane z = 0.
[There is a potential stumbling block in this exercise. You need to use the fact that

 2 
3
 (cos  ) 2 d   cos  d   cos  d
2 3 3

0 0 
2

Can you see why?]

125. Let S1 be the paraboloid z  x 2  ( y  1) 2 and S2 the plane z  5  2 y .

a) Calculate the volume of the region R enclosed by S1 and S2.

b) Find the flux of F ( x, y, z )  ( y, x, 3z ) through the portion of S1 that lies below S2, in the
downward direction.

Stewart 12.9: 15, 17, 19, 21, 23, 25, 27

Stewart 12.4: 7, 9, 13, 15, 21, 23, 27, 29, 35
Stewart 12.8: 1, 3, 7, 9, 11, 13, 15, 17, 19, 31, 35, 37

60
Chapter 3: Line integrals & vector calculus theorems
Section 3.1: Line integrals and path independence
1. Evaluate  xyds where C is given by
C
x  t 2 , y  2t , t  [0, 1] .

2. Evaluate  sin xdx where C is the arc of the curve x  y 4 from (1,1) to (1, 1) .
C

 x dx  y dy  z 2 dz where C consists of the line segments from (0, 0, 0) to

2 2
3. Evaluate
C

(1, 2,1) and from (1, 2,1) to (3, 2, 0) .

4. Evaluate the following line integrals


a)  xyds where C is described by r (t )  (t 2 , 2t ), 0  t  1 .
C

b)  x sin yds where C is the line segment from (0, 3) to (4, 6).
C

c)  sin xdx  cos ydy where C consists of the top half of the circle x 2  y 2  1 from (1, 0) to
C

(1, 0) and the line segment from (1, 0) to (2, 3) .

5a) Evaluate  ( x 2  y 2 )ds , where C is

C

i) the straight line joining (0,0) to (2,2),

ii) the portion of the circle x 2  ( y  2) 2  4 in the first quadrant joining (0,0) to (2,2).
b) Give two possible physical interpretations of each of the line integrals in (a).

Stewart 13.2: 1, 3, 7, 9, 11, 19, 21

6. Find the area of a vertical curtain whose base in the xy-plane is the portion of the parabola
y  x 2 from x = 0 to x = 1, and whose height above the point ( x, y, 0) is xy.

7. Let f ( x, y)  x 2 y and let S be that part of the cylinder x 2  y 2  9 which lies in the first
octant between the surfaces z = 0 and z  f ( x, y) .
a) Make a rough sketch of S.
b) Use a line integral to calculate the area of S.
c) A piece of wire has the shape x  3 cos t , y  3 sin t , 0  t  2 .Find the mass of this wire
if the density at the point ( x, y) is given by x 2 y .
d) Calculate the area of the projection of S onto the yz-plane.

61
8. Let S be that part of the cylinder x 2  y 2  16 which lies in the first octant between z  0
and z  2 y  1 .
a) Find the area of S using a line integral.
b) Find the area of S using a surface integral.
c) Find the area of the projection of S onto the yz-plane.
d) Find the area of the projection of S onto the xz-plane.

9. Let S be that part of the cylinder x 2  y 2  9 which lies in the first octant between the
surfaces z = 0 and z  13 x 2 y .
a) Make a rough sketch of S.
b) Use a line integral to calculate the area of S.

Stewart 13.2: 33, 35


10a) Let F ( x, y)  (4 xy, 3x 2 ) . Calculate the work done in moving a particle from (0, 0) to
(1, 1), first along y = x and then along y  x 2 .

b) As above, except with F ( x, y)  (4 xy, 2 x 2 ) . What do you notice?


11. Find the work done by the force field F ( x, y)  x 2 iˆ  ye x ˆj on a particle which moves
along the parabola x  y 2  1 from (1, 0) to (2, 1).

Stewart 13.2: 39, 41

12. Which of the following vector fields are conservative? What are their potential functions?
  
a) F ( x, y)  (cos x, e xy ) b) F ( x, y)  ( sin x, sin y) c) F ( x, y)  (4 xy, 2 x 2 )

13. Which of the following vector fields are conservative? What are their potential functions?
  
a) F ( x, y)  ( ye x , e x ) b) F ( x, y)  ( y  1y , x  xy ) c) F ( x, y, z )  (2 xy, x 2  z 2 , 2 yz )

Stewart 13.3: 5, 7
Worksheet on conservative vector fields and potential functions (see Appendices)

62
Path independence of certain line integrals

Let C be a smooth curve given by the vector function r (t ), a  t  b . Let f be a differentiable
function of two or three variables whose gradient vector f is continuous on C. Then
  
 f  dr  f (r (b))  f (r (a))
C

So the line integral of a conservative vector field is path independent and can be evaluated
using only the vector field’s potential function.
Proof
b
  
 f  dr   f (r (t ))  r (t )dt
C a
b
 f dx f dy f dz 
     dt
a  x dt y dt z dt 
b
d 
 f (r (t ))dt [Chain rule]
a
dt
 
 f (r (b))  f (r (a)) [F.Thm.Calc.]

This theorem is extendable to piecewise smooth curves by dividing C into finitely many
smooth curves and adding the resulting integrals.


14. Find the work done along r (t )  (e t sin t , e t cos t ) from (0, 1) to (0,  e ) through

F ( x, y)  (3  2 xy, x 2  3 y 2 )
a) done directly
b) using an easier path
c) using a potential function

15. In Ex13 above, two of the vector fields were conservative. For those cases, evaluate the
  

line integral F  dr where C is the curve described by r (t )  t 2 iˆ  t 2 ˆj , 1  t  2 .
C

16. Let C be the upper half of the ellipse

( x  5) 2 ( y  1) 2
 1
4 9
a) Sketch the curve C.
b) Suggest a parametrisation for C.
c) Evaluate  (3x  4 y)dx  (4 x  2 y 2 )dy using this parametrisation.
C

d) Is there a quicker way to evaluate this line integral? Does this method give the same
answer as in (c)?
e) If C were the lower half of the ellipse, what would be the value of the integral? Explain.

63
 
17. Let C be the curve in R2 parametrised by r (t )  (e t , cos( 2 (1  t ))) where 0  t  1 .
a) Show that the line integral  (2 x  y)dx  ( x  2 y)dy is path independent.
C

b) We can evaluate this integral either by (i) changing the path, or (ii) finding a potential
function. Check that these two methods give the same answer.

18. Let C be the “twisted cubic” parametrised by


r (t )  (t , 3t 2 , 6t 3 ) , 0  t  1
a) Calculate the length of the curve C.
b) Evaluate the integrals  1dx ,  1dy and  1dz .
C C C

c) Use the answer to (b) to calculate the work moving a particle along C through the constant

force field F ( x, y, z )  (1,1,1) . Can you suggest another way to get this answer?
d) Find the work done moving a particle along C through the force field

G( x, y, z)  (2 xy  3z, x 2  2e z , 3x  2 ye z )

19. Evaluate  2( xy  e 2 x  y )dx   ( x 2  e 2 x  y )dy , where C is

C
 
a) the curve parametrised by x (t )  1  2 cos t , y(t )  2  3 sin t , 0  t  2 .
 
b) the curve parametrised by x (t )  1  2 cos t , y(t )  2  3 sin t , 0  t   .

  
20. Evaluate 
C
F  dr where F ( x, y)  (2 y  6 xy,2 x  3x 2  3 y 2 ) and the curve C is any path

from (−1,−1) to (2, 1) in the xy-plane.

21. The value of  ( x 2 y cos x  2 xy sin x  y 2 e x )dx  ( x 2 sin x  2 ye x )dy around the
C
2 2 2
“hypocycloid” x 3
y 3
 2 3 is
(A) 0 (B) 
4
(C) 
2
(D) 2 (E) None of A-D

Stewart 13.3: 15, 17, 19, 23

Stewart 13.3: 31, 33, 35

Curl and Divergence

    
   , ,  F  ( P, Q, R )
 x y x 
   R Q P R Q P 
curlF    F    ,  ,  
 y z z x x y 
  P Q R
divF    F   
x y z

64
  
22. Find the curl and divergence of F if F  xe y ˆj  ye z kˆ

 
23. If f ( x, y, z )  x 2  3xyz , find (a) F  f and (b) curlF

 
24. Find divF and curlF for each of the following vector fields.
 
a) F  ( x 2 y, x  z, y  z 2 ) b) F  (e x , e y , e z )
 
c) F  cos yiˆ  sin zˆj  tan xkˆ d) F  y 2 iˆ  xyˆj  yz 3 kˆ

  
25. We know that curlF  0  F is a conservative vector field. For each of the vector fields
below, use this test to determine which are conservative. In those cases, find f, a potential

function for F .
 
a) F  (cos y,  x sin y  sin z, y cos z) b) F  ( y, x 2 z, z )
   x 1 y
c) F  sin xiˆ  cos yˆj  z 2 kˆ d) F  ln yiˆ     ˆj  2 kˆ
 y z z

26. Let F ( x, y, z )  (2 xe x , sin y, z 3 ) .
2


a) Calculate   F .

b) Find the work required to move a particle through the force field F along the curve

parametrised by r (t )  (3 cos t , 4 sin t , t ) , where 0  t  2 .

27. If A  3xz 2 iˆ  yzˆj  ( x  2 z)kˆ then div(curl(A)) is

(A) x  y (B) x (C) z (D) 0 (E) None of A-D

28. Let C be the part of the helix

x  cos(2 t ) , y  sin(2 t ) , z  4t
that joins A(1, 0, 0) to B(1, 0, 4) .
a) Sketch the curve C.
  
b) Calculate  F  dr where F ( x, y, z )  ( x, y, z ) . What is a physical interpretation of this line
C

integral?

c) Calculate   F . What can you conclude?
d) Let D be the closed curve which consists of the path C followed by the straight line joining

B to A. What is the work done in moving a particle through the vector field F along D?

e) Find a potential function f for F and then calculate f ( B)  f ( A) . Does this give the same
answer as in (b)?

Stewart 13.5: 1, 3, 12, 15, 17, 19, 31

65
Section 3.2: Green’s Theorem
Let C be a positively oriented, piecewise-smooth, simple closed curve in the plane and let D
be the region bounded by C. If P and Q have continuous partial derivatives on an open region
that contains D, then
 Q P 
C Pdx  Qdy  D  x  y dA

29. Evaluate the line integrals along the given curves (all positively oriented).
x dx  xydy
4
a)
C

b)  (3 y  e sin x )dx  (7 x  1  y 4 )dy

C

c)  ( x 2  y 2 )dx  (2 y  x)dy
C

d)  ( x 5  3 y)dx  (2 x  e y )dy
3

y dx  3xydy
2
e)
C

30. Use Green’s Theorem to calculate  (2 xy  y  x 4 )dx  ( x 2  e y )dy if C is the unit circle,
C

oriented positively.

31. Use Green’s Theorem to calculate  2 xydx  (e xy  x 2 )dy where C is the perimeter of the
C

unit square with vertices at (0, 0), (0, 1), (1, 1) and (1, 0), oriented anticlockwise.

66
32. Evaluate the line integral
 (x  2)dx  xydy
2

along the closed path C with anticlockwise orientation. The path

C is formed by two straight line segments x  [0, 2] , y  [0, 2] and
by the circle x 2  y 2  4 , joining (2,0) and (0,2). Give a physical
interpretation of this line integral.

33. Evaluate
 xe
3 x
dx  ( x 3  2 x 2 y 2 )dy
C

using Green’s Theorem, where C is the boundary of the region enclosed by the circles
x 2  y 2  4 and x 2  y 2  9 oriented positively.

34. Let C be the circle x 2  y 2  1 in the xy-plane, with an anticlockwise orientation.

Evaluate
 (2 x  y 3 )dx  ( x 3  y 3 )dy
3

Using two different methods:

a) parametrising the circle C and doing the line integral,
b) using Green’s Theorem.

Stewart 13.4: 5, 7, 9, 17, 19, 21, 22, 23

Green’s Theorem for Areas

 Q P 
Area = R 1dA  R  x  y dA if P  0, Q  x OR P   y, Q  0 OR P   12 y, Q  12 x
etc.
Therefore Area =  xdy    ydx    ydx   xdy etc.
C C
1
2
C
1
2
C

x2 y2
35. Evaluate the area of the ellipse 2  2  1 using Green’s Theorem.
a b

36. Use Green’s Theorem to find the area of the ellipse x  3 cos  , y  4 sin  where
0    2 .

67
37. Let C be the curve in R2 described by the parametric equations
x(t )  t 2 , and y(t )  t 3  3t , where t  R . Note that the curve C
encloses a region R in the xy-plane. Use the formula
A( R)  1
2  xdy  ydx to calculate the area of R.
C

38. Find the area of the region bounded by the hypocycloid (shown
alongside)

r (t )  (cos 3 t , sin 3 t ) , 0  t  2 .

39. Find  ( x 3  y 3 )dx  ( x 3  y 3 )dy if C is the boundary of the region

C

between the circles x 2  y 2  1 and x 2  y 2  9 .

y x
40. Find x
C y
2 2
dx  2
x  y2
dy if C is the square path shown.

y x 1
41. Evaluate  ( x  1)
C
2
 4y 2
dx 
( x  1) 2  4 y 2
dy along the curves

a) x  y  16
2 2

b) 4 x 2  4 y 2  1
[First find out where the integrand is undefined. Second, find out whether that point is within
your region. Third, if it is, draw a circle (or ellipse – in this problem an ellipse is better)
around it and redefine your region and your boundary. Suggestion: make that troublesome
point the centre of your little circle (or ellipse).]

y x
42. Evaluate x
C y
2 2
dx  2
x  y2
dy where C is

a) the ellipse x 2  2 y 2  8
b) the ellipse ( x  4) 2  2( y  3) 2  8 .

Section 3.3: Stokes’ Theorem

Let S be an oriented piecewise-smooth surface that is bounded by a simple, closed,

piecewise-smooth boundary curve C with positive orientation. Let F be a vector field whose
components have continuous partial derivatives on an open region in R3 which contains S.
Then
    
 F  dr   
C
 F  dS 
S
   F  nˆ dS
S

68
 
43. Find work done by the field F  (3z, 4 x, 2 y) along the boundary of S, where S is that part
of the paraboloid z  9  x 2  y 2 lying above z = 0. Do the calculation three times using the
methods:
a) doing the line integral, using parametrisation of the boundary curve C.
b) Using Stokes’ Theorem, using the surface S.
c) Using Stokes’ Theorem, recognising that C is the boundary of a simpler surface, and using
that simpler surface instead of S.

44. Evaluate  zdx  xdy  ydz where C is the intersection of x

C
2
 y 2  1 and y  z  2 .

Solve this by doing the line integral as well as by using Stokes’ Theorem.


45. Evaluate  (3z 2  cos x)dx  ( x  e 2 y )dy  ( y 3  sin z )dz where C is r (t )  (cos t , sin t , 1) ,
C

0  t  2 .

46. Let F ( x, y, z )  (e x  2 xz , 2 x, y 2 ) and let D be the portion of the paraboloid

2

x 2  y 2  2 z  17 that lies above the hemisphere z  25  x  y .

2 2

a) Find parametric equations for the curve of intersection C of the paraboloid and the
hemisphere.

b) Calculate   F ( x, y, z ) .

c) Find    F  nˆ dS where n̂ is the upward-pointing unit normal to D at each point

( x, y, z)  D .


47. Let F ( x, y, z )  ( y,  x, z ) and let S be the surface of the paraboloid z  x 2  y 2 lying
between z = 0 and z = 4.

a) Evaluate  curlF  nˆdS where n̂ is a unit outer normal to S.
S
 
b) Evaluate   dr where C is the curve of intersection of S with z = 4.
C
F

c) Compare your answers to (a) and (b).

 
 ( x, y, z)  (6 yz , 5x, yze x ) and P is the portion of the
2
48. Calculate   F  ˆ
n dS , where F
P

paraboloid z  14 x 2  y 2 lying between z = 0 and z = 4, orientated upwards.

69
49. Let C be the circle ( x  1) 2  ( z   ) 2   2 in the plane y = 0. Suppose that C is orientated
in an anticlockwise direction on the xz-plane. Calculate the work required to move a particle

once around C through the force field G( x, y, z)  (e  x cos x, z  e y sin y, 2 x  e z ) .

50a) Calculate the curl of the vector field F ( x, y, z)  (e x  2 y  3z, x  e 2 y  3z, x  2 y  e 3 z ) .

b) Let P be the portion of the paraboloid z  x 2  y 2 that lies inside the cylinder
x 2  ( y  1) 2  4 . At each point on P, let n̂ be the downward-pointing unit normal. Find


P
  F  nˆ dS , where F ( x, y, z)  (e x  2 y  3z, x  e 2 y  3z, x  2 y  e 3 z ) .


51. Let F ( x, y, z )  (2 y  4 xy 2 z,  4 x 2 yz , 1  z 2  2 x 2 y 2 ) and let C be the curve of
intersection of the surfaces S1: z  4  x 2  y 2 and S2: z  2 y  4 .
a) Find parametric equations for the curve C.
b) Find a vector equation for the tangent line to C at the point (0, 2, 0).

c) Calculate curlF .

d) Find the work done in moving a particle around C through the force field F .

Stewart 13.7: 1, 3, 5, 7, 9, 13, 15, 17, 19

Section 3.4: The Divergence Theorem

Let E be a simple solid region and let S be the boundary surface of E, given with positive

(outward) orientation. Let F be a vector field whose component functions have continuous
partial derivatives on an open region that contains E. Then
  
  dS   divF dV
F
S E


52. Find the flux of the vector field F  ( xy, y 2 z, z 3 ) through the surface of the unit cube
0  x  1 , 0  y  1, 0  z  1 . Do this directly (using the standard flux surface integral
method) as well as by using the Divergence Theorem.


53. Find the flux of F ( x, y, z)  ( x, 2 y, 3z ) through x 2  y 2  z 2  4 .


54. Find the flux of F ( x, y, z )  ( xy, y 2  e xz , sin( xy )) through the surface of the region
2

bounded by z  1  x 2 , z = 0, y = 0, x = 0 and y  z  2 .


55. Use the Divergence Theorem to calculate the flux of F ( x, y, z)  x 2 yziˆ  xy 2 zˆj  xyz 2 kˆ
across the surface of the box enclosed by the planes x  0, x  a, y  0, y  b, z  0, z  c
where a, b and c are positive numbers.

70
56. Use the Divergence Theorem to calculate the flux of

F ( x, y, z)  ( x 3  y 3 )iˆ  ( y 3  z 3 ) ˆj  ( z 3  x 3 )kˆ across S where S is the sphere with centre
the origin and radius 2.


57. Find the flux of F ( x, y, z )  ( z, y, x 2 ) through the upper half of the sphere
x 2  y 2  z 2  1.


58. Find the flux of F ( x, y, z )  ( y, x, 2) through the upper half of the sphere
z  a2  x2  y2 .

59a) Calculate the divergence of the vector field F ( x, y, z )  ( xz 2 , 13 y 3 , x 2 z  3) .

b) Find the flux of the vector field F ( x, y, z )  ( xz 2 , 13 y 3 , x 2 z  3) through the hemisphere

z  4  x 2  y 2 , using the outwards-pointing unit normal.


60. Let F ( x, y, z)  ( x 2  y 2 , y 2  z 2 , 1  2 xz  yz ) represent the flow of a fluid. Find the flux
through the surface S where S is the upper half of the sphere x 2  y 2  z 2  1 oriented
upwards.

61. Let H be the hemisphere z  4  x 2  y 2 in R3. At each point ( x, y, z)  H , let

nˆ ( x, y, z ) be the outward pointing unit normal to the surface. Let R be the interior of the
hemisphere, that is, the solid region bounded between H and the plane z = 0.
 
a) Note that if F ( x, y, z)  (2 x  z, y  3x, x 2  z) then   F  4 . Comment on the following
calculations:
 

H
F  ˆ
n dS   dV   4dV  4V ( R) 
R
  F
R
64
3 

b) If you don’t agree with this answer, then how would you calculate   nˆdS ?
H
F


62. Let the velocity of a fluid be given by v ( x, y, z)  ( x 2  y 2 , y 2  z 2 ,1  2 xz  2 yz ) . Find
the flux of this fluid through the hemisphere z  1  x 2  y 2 , where the upward-pointing
unit normal is chosen at each point.

71
63. Let B be the solid region bounded by the cylinder x 2  y 2  4 and the planes z = 0 and
z = 3. Let A be the surface of B. At each point ( x, y, z)  A , let nˆ ( x, y, z ) be the outward-

pointing unit normal. Given the vector field F ( x, y, z)  (2 x  y 2 , 2 y, 2 z  xy ) , calculate

  nˆdS in two different ways:
A
F

a) directly
b) using the Divergence Theorem.

64. Let R be the region in the xy-plane bounded by the circle x 2  y 2  4 , z = 0. Let P be the
part of the plane z  2 y  3 that lies above R. Suppose that f ( x, y) is a nice smooth function
with f ( x, y)  2 y  3 for all ( x, y) with x 2  y 2  4 and with f ( x, y)  2 y  3 for all ( x, y)
with x 2  y 2  4 . Let S be the portion of the surface z  f ( x, y) that lies above R. You are
given that the volume of the solid region Q lying between P and S is 10 cubic units. A certain

fluid has velocity vector F ( x, y, z )  (2 x, z, y) . Calculate the flux of this fluid passing
through S in the upward direction.

65. Let B be the solid region lying within x 2  y 2  z 2  25 , x  0 and z  0 and let S be the
curved portion of the surface of B.
a) Find the mass of the surface S if the density function is given by  ( x, y, z)  2 z .
b) Find the mass of the solid region B if the density at the point ( x, y, z ) is equal to the
distance from the z-axis.

c) Use the Divergence Theorem to calculate the flux of F through the surface S in the
outwards direction, if the velocity field of the fluid is given by

F ( x, y, z)  (2 y  3x, 2 x  3xy, 2 xy  3z) .

66. Let S be the surface of the region R which is bounded by x 2  y 2  4 , z = 0 and z = 3,


and let F ( x, y, z)  (2 x  y 2 , 2 y, 2 z  xy ) .
a) Sketch the region R.

b) Calculate  F  nˆ dS directly, where n̂ is a unit outer normal to S.
S

c) Calculate   nˆ dS using the Divergence Theorem.
S
F

72
67. Let S be the surface of the sphere x 2  y 2  z 2  4 lying above the plane z = 0 and let

F ( x, y, z)  (2 x  z, y  3x, x 2  z) .
a) Comment on the following calculation:
 
 F  nˆ dS   divFdV
S R

where n̂ is a unit outer normal to S and R is the region bounded by S. Thus


  nˆ dS   4dV  4Vol R  4  12  43  (2)  643 
3
F
S R

b) Now evaluate  F  nˆ dS .
S

Stewart 13.8: 5, 7, 9, 11, 13, 17, 25, 26, 27, 31

73
Chapter 4: Numerical applications
Section 4.1: Taylor’s Theorem
If f (x) , a continuous and differentiable function, has a power series expansion, it looks like:
f ( x)  f (a)  f (a) x  12 f (a) x 2  16 f (a) x 3  h.o.t

f ( n ) (a)
 ( x  a) n
n 0 n!
If f ( x, y) has a power series expansion, it looks like:
f ( x, y)  f (a, b)  ( x  a) f x (a, b)  ( y  b) f y (a, b) 
1
2 [( x  a) 2 f xx (a, b)  2( x  a)( y  b) f xy (a, b)  ( y  b) 2 f yy (a, b)]  h.o.t

1. Find the Taylor expansion of f ( x, y)  e xy up to second degree terms, about the point
2

(2, 1) .

2. Find the Taylor expansion up to 2nd degree terms for f ( x, y)  x ln( x  y) about the points
i) (0, 1) and ii) (1, 0) .

3. Find the Taylor series expansions (up to the second degree terms) for cos(xy ) about the
points (a) (1, 4 ) and (b) (0, 3 ) .

4. Let f ( x, y)  x 3  y 3  3x 2  3 y 2  8 .
a) Find the Taylor series expansion of f up to the second degree terms about the point (0,0).
b) Write down the Taylor series expansion, up to the third degree terms, of f about the point
(0,0). (You should not have to do any calculations.)
c) Find the Taylor series expansion, up to the second degree terms, of f about the point (1,2).

5a) Find the Taylor series expansion (up to the second-order terms) for the function
x2
f ( x, y )  about the point (2, 1).
y
b) Sketch the level curves for this function.
c) In what direction from the point (2, 1) does the function f ( x, y) decrease most rapidly?
Indicate this direction on the sketch you made in (b). What is the relationship between this
direction and the level curves you sketched in (b)?
d) Find a Cartesian equation for the plane tangent to the graph of z  f ( x, y) at the point
P(2,1, 4) .

6. Here is some practice in single-variable Taylor expansions:

The term of degree five in the Taylor series for e x around −1 is
( x  1) 5 ( x  1) 5 e( x  1) 5  e( x  1) 5
(A) (B) (C) (D) (E) None of A-D
e  5! 5! 5! 5!
74
Section 4.2: Local and absolute maxima and minima
To find local maxima and/or minima, first set all partial derivatives to zero to find the
stationary points. Next, carry out the second derivative test for each of those points:
f xx f xy
D  f xx  f yy  ( f xy ) 2
f yx f yy
If D > 0 and f xx  0 then (a, b, f (a, b)) is a local minimum.
If D > 0 and f xx  0 then (a, b, f (a, b)) is a local maximum.
If D < 0 then (a, b, f (a, b)) is a saddle point
If D = 0 then we need more information. Try a variety of cross-sections.
Warning: Course notes from other lecturers or from previous years might have a different
way of calculating D. Be careful when reading those notes, the conclusions will be the same,
but the process of getting there might differ.

7. Find and classify the stationary points of f ( x, y)  x 3  y 3  6 xy .

8. Find and classify the stationary points of f ( x, y)  y 2  2 x 4  3x 2 y .

9. Find and classify the stationary points of f ( x, y)  x 3  3xy 2  3x 2  3 y 2  4 .

10. Find and classify the stationary points of f ( x, y)  x 3  xy 2 .

11. Find and classify the stationary points of f ( x, y)  2 x 3  x 2 y  y 2 .

12. Let g ( x, y)  2( x 2  y 2 )e ( x  y ) .
2 2

a) Find all the stationary points of g. Do not try to classify them using the second derivative
test.
b) For which ( x, y) is g ( x, y)  0 ?
c) What happens to g ( x, y) as x and y both tend to infinity? Explain.
d) Try to sketch (and describe) the graph of the surface z  g ( x, y) .

1
13. Let g ( x, y )  .
1 x  y2
2

a) Sketch the graph of z  g ( x, y) .

b) Find and classify all the stationary points of g.
c) Find the Taylor series expansion, up to the second degree terms, for g about (i) the point
(0,0), (ii) the point (1,0).
d) Write down a Cartesian equation for the plane tangent to the graph of z  g ( x, y) at the
point (1, 0, 12 ) .

75
14. Find and classify the stationary points of f ( x, y)  x 3 y  12 x 2  8 y .

15. Find and classify the stationary points of f ( x, y)  x 4  4 xy  2 y 2 .

16. Verify that f ( x, y)  sin x  sin y  sin( x  y) has a stationary point at ( ,  ) . Show that
although the second derivative test is inconclusive at ( ,  ) , the function f has neither a
maximum nor a minimum at that point. [HINT: Consider the values of f (   ,    ) for
both positive and negative values of ε, where ε is a small value.]

17. Find and classify the stationary points of f ( x, y)  xy  x 2 y  xy 2 .

18. Find all the stationary points of the functions

a) f ( x, y)  ( x  y) 2  y 4
b) g ( x, y)  y 2  4 x 8  5x 4 y
c) h( x, y)  3xy  x 2 y  xy 2
Can you classify these stationary points? Explain.

Stewart 11.7: 1, 5, 7, 9, 11, 13

19. Find the absolute maximum and minimum values of f ( x, y)  2  2 x  2 y  x 2  y 2 on

the triangular region in the first quadrant bounded by the lines x = 0, y = 0 and y  9  x .

20. Let f ( x, y)  x 3  y 2  3xy  y .

a) Locate the stationary points of f.
b) Use the second derivative test to classify the stationary points of f.
c) Let R be the region {( x, y) : 0  x  2, 0  y  2} . Find the (global) maximum and
minimum of f over R.

21a) Find the stationary points of f ( x, y)  x 2  4 xy  y 3  4 y .

b) Classify these stationary points as local maxima, local minima or neither.
c) Find the absolute maximum and minimum values of f on the triangular region with vertices
A(1,1) , B(7,1) and C (7,7) .

Stewart 11.7: 27, 29, 37, 39, 41, 45, 47

76
Section 4.3: Lagrange multipliers
To find the maximum and minimum values of f ( x, y, z ) subject to the constraint
g ( x, y, z)  k [assuming that these extreme values exist and g  0 on the surface
g ( x, y, z)  k ]:
a) Find all the values of x, y, z and λ such that
f ( x, y, z)  g ( x, y, z) and g ( x, y, z)  k
b) Evaluate f at all the points that result from step (a). The largest of these values is the
maximum value of f, the smallest is the minimum value of f.

22. Find two positive real numbers whose sum is 80 and whose product is a maximum, using
the method of Lagrange multipliers.

23. Find the dimensions to maximise the volume of a rectangular box with no top, with a
surface area of 108 cm2.

24. Let C denote the first octant arc of the curve in which paraboloid 2 z  16  x 2  y 2 and
the plane x  y  4 intersect. Find the points on C which are closest to and farthest from the
origin. Find the minimum and maximum distances from the origin to C.

25. Use Lagrange multipliers to find three positive real numbers whose sum is 1000 and
whose product is a maximum. [Note that in Exercise 39 in Stewart 11.7, you were asked to
solve essentially the same problem without using Lagrange multipliers.]

26. Find maxima and/or minima of f ( x, y)  x 2  y 2 subject to x  2 y 2  1 .

27. Use Lagrange multipliers to find the points on the curve

1 1
 1
x y
that are nearest to and furthest from the origin.

28a) Use Lagrange multipliers to find the maximum and minimum values of x 2 y  y subject
to the constraint x 2  2 y 2  7 .
b) Use your result from (a) to find the absolute maximum and minimum values of x 2 y  y
over the region {( x, y) : x 2  2 y 2  7} .

29. Prove that the product of the sines of the angles of a triangle is greatest when the triangle
is equilateral.

30. Let C be the curve of intersection of the surfaces x  y  0 and y  z 2  6 . Use

Lagrange multipliers to find the points on C (if any)that are nearest to and furthest from the
origin.

77
31. Find maxima and minima of x 2  y 2 subject to x 2  y 3  4 .

32. If an open rectangular box is to have surface area A, then what dimensions will make the
volume a maximum?

33. A container with a closed top is to be constructed in the shape of a right circular cylinder.
If the surface area is to be a fixed value S, then what dimensions will maximise the volume?

34. Use Lagrange multipliers to find the maxima and minima of f ( x, y, z )  z  x 2  y 2

subject to the constraints x  y  z  1 and x 2  y 2  4 .

35. Use Lagrange multipliers to find the dimensions of the rectangular box of maximum
volume, with faces parallel to the coordinate planes, that can be inscribed in the ellipsoid
16 x 2  4 y 2  9 z 2  144 .

36a) Use Lagrange multipliers to find the points on the curve of intersection of the surfaces
x  y  1  0 and y 2  z 2  1 that are nearest to and furthest from the origin.
b) Draw a sketch to ensure that you have the correct absolute maximum and minimum in (a).

37. Find the maximum and minimum values of f ( x, y, z)  xyz subject to the constraint
x 2  2 y 2  3z 2  6 .

38. Use the method of Lagrange Multipliers to find the shortest distance from the origin to
the hyperbola x 2  8xy  7 y 2  225 .

39. Use Lagrange multipliers to find the points on the curve x 2  2 x  y 2  4 y  4 that are
nearest to and furthest from the point (−2, 2) in the xy-plane.

40. Use Lagrange multipliers to find the points on the graph of xy 3 z 2  16 that are closest to
the origin.

41a) Use Lagrange multipliers to find the extreme values of x 2  y 2 subject to the constraint
1 1
  1.
x y
1 1 x 1
b) Sketch the graph of   1 by noting that y   1 .
x y x 1 x 1
c) On the same set of axes sketch the family of level curves of x 2  y 2  c .
d) Use your sketch in (c) to decide whether the point obtained in (a) is an absolute maximum
or absolute minimum.

78
Stewart 11.8: 1, 3, 5, 7, 9, 11, 15, 17

Section 4.4: Newton’s Method

If F: R 2  R 2 is a transformation with F ( x, y)  ( f ( x, y), g ( x, y)) and f and g are

differentiable functions, then, if x1 is an approximate solution to F ( x, y)  (0, 0) , a second
approximation is
   
x2  x1  ( F ( x1 )) 1 F ( x1 )

x 2  2 y 2  4
42. Find a better approximation to the solution of  given the first
 xy  1
  2  2 
approximation x1    . Could   have been taken as a first approximation?
 0  2
43. Starting with first estimate ( x1 , y1 )  (2,1) , use Newton’s Method to find the next
approximation ( x2 , y 2 ) to a solution of the system of equations
x2 y  4
x 3  y 3  10

44a) Show that x = 2, y = 1 is a solution to the system of equations

x2  y2  5
4y  x2
b) By sketching a pair of curves, show that the system of equations
x2  y2  6
3y  x2
has exactly one solution in the first quadrant.
c) Using x = 2, y = 1 as an initial approximation, use Newton’s Method to find another
approximation to a solution of the system of equations in (b).

45. Use Newton’s method to find the next approximation to a solution of the system of
equations
x  2y  z  6
x2  y2  z 2  6
xyz  2
starting with ( x1 , y1 , z1 )  (1, 1, 2) .

79
46a) Use a pair of graphs to find approximately where the solutions of
x3  y  1
x2  y2  4
lie.
b) Use Newton’s method to find a second approximation to the solution that lies in the fourth
quadrant, starting with ( x1 , y1 )  (1,  2) .
c) Could you have started with the initial approximation ( x1 , y1 )  (0,  2) ?

47. Sketch the graphs of x 3  y 3  1 and x 2  2 y 2  4 on the same set of axes.

b) Use your sketch to determine the number of solutions to the system of equations
x3  y3  1
x2  2y2  4
c) Is ( x, y)  (1, 1) a reasonable approximation to one of these solutions?
d) Starting with ( x1 , y1 )  (1, 1) use Newton’s method to find a second approximation
( x2 , y 2 ) to this solution.
e) Could you have started with the initial approximation ( x1 , y1 )  (0, 0) ?

80
Revision Worksheet

1. Below on the left there is a list of curve equations in Cartesian form and on the right is a
list of curve descriptions. Match them up and then draw rough graphs of each of them. (Just a
get a rough idea of shape, like axis intercepts, radius, asymptotes, whatever is necessary.)
Curve equations Curve descriptions
(a) 3x  4 y  12  0 (A) Circle
(b) 3x 2  4 y 2  12  0 (B) Parabola
(c) 3x  3 y  6 x  12 y  12  0
2 2
(C) Polygon
(d) 3x 2  4 y 2  12  0 (D) Ellipse
(e) 3x  4 y  12  0
2
(E) Hyperbola
(f) 3x  4 y  12  0 (F) Line
[Recognising what sort of curve you have just by looking at its equation is an important skill
throughout this course.]

2. Below on the left there is a list of curve equations in parametric form and on the right is a
list of curve descriptions. Match them up and then draw rough graphs of each of them. (Just a
get a rough idea of shape, like axis intercepts, radius, asymptotes, whatever is necessary.)
Curve equations Curve descriptions
a) x  2  2t , y  3  t , t  (, ) (A) Semi-circle
b) x  1  3 sin t , y  2  3 cos t , t  [0,  ] (B) Line
c) x  3 cos t , y  4 sin t , t  [0, 2 ] (C) Line segment
d) x  1  4t , y  2t , t  [0, 1] (D) Ellipse
[Recognising what sort of curve you have just by looking at its equation is an important skill
throughout this course.]

3. In each of the cases below, a set of points is described (by either a single equation or two
considered together). Describe the set of points in each case using terms such as: line, plane,
sphere, single point.
a) y  1
b) x  y  1
c) 2 x  3 y  z  1
d) x  2 and y  1
e) x 2  y 2  z 2  9
f) z  2 and x 2  y 2  z 2  9
g) x  3 and x 2  y 2  z 2  9
[Recognising curves and surfaces as sets of points is useful, particularly the intersections of
curves and surfaces.]

81
4. The points O, A and B have coordinates (0, 0, 0) , (2, 1, 1) and (1, 3,  1) respectively.
a) The line l passes through A and B. For this line, find
(i) a vector equation
(ii) parametric equations
b) Use your answer to (a)(ii) to give parametric equations for the line segment with endpoints
A and B.
c) Does the point (4, 5, 3) lie on l.
d) What is the angle between vectors OA and OB ?
e) Find a non-zero vector perpendicular to both OA and OB .
f) Write down a Cartesian equation for the plane P which passes through A, B and O.
[In first-year maths, lines and planes were described parametrically. In this course we shall
describe other types of curves and surfaces parametrically.]

5. Solve the following integrals.

x2 1
 xe dx x e c) 
x 2 x3
a) b) dx dx
x 1
 4
x 1
d)  2  cos xdx x x 2  1dx
3
dx e) f)
x 1 
 4


g)  (cos 2  ) 2 d [Be very careful. There is a big trap.]
3

[Integration is an extremely important skill in this course. You should begin this course well
prepared in all your first-year integration skills.]

82
Worksheet on Cartesian and Parametric expressions
The following curves are described parametrically. Find a Cartesian expression for each
curve.
Examples
2
 y
1. x(t )  t 2  1 , y(t )  2t , t  R x     1 , or x  14 y 2  1
2
2 2
 x  y
2. x(t )  2 cos t , y(t )  3 sin t , t  [0, 2 ]     1
2  3
Exercises
1. x(t )  cos t  1 , y(t )  sin t , t  [0, 2 ]
2. x(t )  e t , y(t )  e t , t  R

3. r (t )  (t 2 , t 3 ) , t  R

4. r (t )  (t , e t ) , t  R
5. x(t )  t  1 , y(t )  ln t , t  0
6. x(t )  4t 2  2 , y(t )  2t , t  R

7. r (t )  (e t , sin(2e t )) , t  R
8. x(t )  cos 6 cos t , y(t )  cos t , t  R
9. x(t )  sin t cos t , y(t )  sin 2t , t  R
10. x(t )  tan t , y(t )  cot t , t  (0, 2 )

11. r (t )  (t 2  2t , 1  t ) , t  R

The following space curves are described parametrically. Find the xy-projection, yz-
projection and xz-projection of each curve in Cartesian form. (Alternately: find three surfaces
on which the curve lies.)
Examples

1. r (t )  (t , t 2 , t 2 ), t  R y  x2 , z  x2 , z  y

2. r (t )  (cos t , sin t , t ), t  R x 2  y 2  1 , x  cos z , y  sin z

Exercises (all of these have t  R )

 
1. r (t )  (cos t , sin t , cos 2 t ) 6. r (t )  (cos t , sin t , 2  sin t )
 
2. r (t )  (e t , e 2t , t ) 7. r (t )  (2 cos t , 3 sin t , sin 2 t )
 
3. r (t )  (sec t , tan t , sec 2 t ) 8. r (t )  (2 cos t , 2 sin t  2, cos 2 t )
 
4. r (t )  (t  1, t  1, t 2 ) 9. r (t )  (t 3  1, 3t 3 , t 3  4)
 
5. r (t )  (ln t , ln t 2 , ln t 4 ) 10. r (t )  (cos t , sin t , cos 2t )

83
Solutions to Worksheet on Cartesian and Parametric expressions
The solutions given below are in jumbled up order. If you have the answer correct, you
should be able to find it somewhere on the list.

First set of exercises – Parametric  Cartesian

x  y2  2
x  y, x  0
x  y2 1
x 2
3
y,  1  y  1
( x  1)  y 2  1
2

y  ln( x  1), x  1
1
y , x0
x
y  sin( 2 x), x  0
1
x  ( y 3 )2 , yR
y  2 x,   x  1
2
1
2

y  ex

Second set of exercises – Projections onto coordinate planes

y  x2 x  ez y  e2z
x 2 ( y  2) 2 x2 ( y  2) 2
 1 z z 1
4 4 4 4
x2  y2  1 z  2x 2  1 z  1 2y2
y  2x z  4x z  2y
y  x2 z  ( x  1) 2
z  ( y  1) 2
x2  y2  1 z  x2 z  1 y2
x2  y2  1 (2  z ) 2  x 2  1 z  2 y
y y
x 1 z  x 5 z 4
3 3
y2 1  x2 z  x2 y 2 1  z
x2 y2 x2 y2
 1  z 1 z
4 9 4 9

84
Worksheet on Conservative Vector Fields and Potential Functions
Determine which of the following vector fields are conservative vector fields. For those, find
a potential function.

Example
 
F ( x, y)  ( y, x  2 y) . Yes, F is conservative.

F ( x, y)  f ( x, y) where f ( x, y)  xy  y 2

Exercises

1. F ( x, y)  (2 x  y, 2 y  x)

2. F ( x, y)  (3x  y, 2 x  4 y)

3. F ( x, y)  ( y cos( xy )  e x , x cos( xy ))
  1 
4. F ( x, y )   e sin x cos x, 
 y  1 

5. F ( x, y)  ( xy, xy )

6. F ( x, y)  ( x 2  y, x  x 2 )

7. F ( x, y)  (( y  1) x y , x y 1 ln x)

8. F ( x, y)  (e x , e x y )

9. F ( x, y)  (2 x  y 2 , x 2  2 xy  3)

10. F ( x, y)  (sec x tan x, sec 2 y)

 
For each of the conservative vector fields above, evaluate   dr where C is described by
C
F

r (t )  (t , t 2 ), t  [1, 2] .
[Hint: you do not need to compute a line integral.]

 
Solutions of  F  dr for the conservative vector fields
C

(in jumbled order)

31
sec 2  tan 4  sec1  tan1
sin 8  e 2  sin 1  e
25
e sin 2  ln 5  e sin1  ln 2

85
Worksheet 1 on Polar Coordinates
Describe the following shaded regions in R2 in polar coordinates.
All circles have radius r = 1 unless shown otherwise.

Examples

0  r 1 0  r  2 sin 
 2  
 
2 0  

Exercises

86
Solutions to Worksheet 1 on Polar Coordinates
The solutions given below are in jumbled up order. If you have the answer correct, you
should be able to find it somewhere on the list.

1  r  2, 0    
2

0  r  4 sin  , 
2  
0  r  2 cos  ,  2    
2

cos   sin   sin 2  r  cos   sin   sin 2 , 0    2

0  r  1, 
3   
2

0  r  1, 0    
0  r  1, 0    
4

0  r  2 sin  , 
4   3
4

(0  r  2 cos  , ( 2     4 )  ( 4    2 ))  (0  r  sec ,  4    4 )
csc  r  2 sin  , 
4   3
4

0  r  2 2 sin  , 0    
(0  r  2 sin  , 0    4 )  (0  r  csc , 
4    2 )
sec  r  2 cos  ,  4    
4

87
Worksheet 2 on Polar Coordinates
Describe the following volumes in R3 in the specified coordinate system.

1. The region bounded by the sphere x 2  y 2  z 2  4 [spherical]

2. The region bounded by the sphere x 2  y 2  z 2  4 [cylindrical]
3. The region contained by the hemisphere z  6  x 2  y 2 and the paraboloid z  x 2  y 2
[cylindrical]
4. The region contained by the hemisphere z  4  x 2  y 2 and the cone z  x 2  y 2
[spherical]
5. The region contained by the hemisphere z  4  x 2  y 2 and the cone z  x 2  y 2
[cylindrical]
6. The region below the hemisphere z  4  x 2  y 2 and above the plane z = 1 [cylindrical]

7. The region below the hemisphere z  4  x 2  y 2 and above the plane z = 1 [spherical]
8. The region bounded by z = 0, x 2  y 2  4 , z  x 2  y 2  2 [cylindrical]
9. The region inside the cylinder x 2  y 2  4 , below the cone z  x 2  y 2 and above z = 0
[cylindrical]
10. The region above the cone z  x 2  y 2 and within the sphere x 2  y 2  z 2  4 z
[spherical]
11. The region below the hemisphere z  4  x 2  y 2 , inside the cylinder x 2  y 2  2 y and
above z = 0 [cylindrical]

Solutions to Worksheet 2 on Polar Coordinates

The solutions given below are in jumbled up order. If you have the answer correct, you
should be able to find it somewhere on the list.

 0r 2   0r2   0r2 

     
 0    2   0    2   0    2 
  0  z  2  r 2   4  r2  z  4  r2 
r  z  4  r
2
    

 sec     2   0r 3   0 2 

     
 0    2   0    2   0    2 
 0     2   0  
r  z  6  r
2
 3    

88
 0r2   0 2   0    4 cos  
     
 0    2   0    2   0    2 
 0 zr   0     0   
   4   4 

 0r 3   0  r  2 sin  
   
 0    2   0   
  0  z  4  r 2 
1  z  4  r
2
  

89
Worksheet on Areas as Iterated Integrals
The area of a region R is A   dA . Express the areas of the following regions as iterated
R
   
integrals of the form   dxdy and
 
  dydx .
 

90
Solutions to Worksheet on Areas as Iterated Integrals
The solutions given below are in jumbled up order. If you have the answer correct, you
should be able to find it somewhere on the list.

1 y 1 1 x 2 1 x 2 1 2 y

  dxdy
0 y2
  dydx
0 0
  dydx    dydx
0 0 1 0
  dxdy
1 y

3 x 2 y 3 363 x 2 1 y

  dydx   dxdy
0 y
  dydx   dxdy
0 0 0 x2 0 y
2

36 12 3
y
9 y 3 3 1 2x 2 2

  dxdy    dxdy
0 0 9 0
  dxdy
0 y
  dydx    dydx
0 x 1 x

1 2 1 2 2 2 1 2 1 2 x 2

  dydx    dydx    dydx

2  x 1 1 1 x
  dxdy
0 y
  dydx
0 x

1 x 1 y 2 2 y
2
1 x 1 1 y

  dydx
0 x2
  dxdy    dxdy
0 0 1 0
  dydx
0 x2
  dxdy
0 0

91