Math 38 Mathematical Analysis III: I. F. Evidente

MATH 38
Mathematical Analysis III

I. F. Evidente
IMSP (UPLB)
Unit 3
Differentiation of Functions of Two or More Variables

1
Limits and Continuity
Partial Derivatives, Differentiability
Higher Order Derivatives, The Chain Rule
Local Linearity and Differentials
Relative and Absolute Extrema of Functions of Two Variables
Outline
Functions of Two or More Variables

Basic Ideas
Graphs of Functions of Two or More Variables
Limits of Functions of Two or More Variables

Limits Along Paths
Intuitive Notion of Limits
Basic Limit Theorems
Outline

Basic Ideas

Limits Along Paths
Recall
R R = R2 = {(x, y) | x R and y R}
Geometrically, R2 is associated with two-dimensional Euclidean space
(Euclidean 2-space)
Definition
A function of two variables f (x, y) is a rule that assigns a unique real
number to each point (x, y) in some subset D of 2 .
Examples
Example
1
2
3
f (x, y) = 6 3x 2y
f (x, y) = 4x 2 + y 2
p
f (x, y) = 9 x 2 y 2
Evaluation
Example
1
f (x, y) = 6 3x 2y .
1 Find f (0, 2)
2 Find f (x + x, y + y)
f (x, y) = 4x 2 + y 2 .
p
1 Find f ( 2, 1).
2 Find f (x + x, y)
Remark
The set D is called the domain of f , which we denote by dom f .
Remark
If no set D is specified, then D is the natural domain of f :
Remark
If no set D is specified, then D is the natural domain of f : the
largest subset of 2 containing all points (x, y) for which f (x, y) is
defined.
Remark
If no set D is specified, then D is the natural domain of f : the
largest subset of 2 containing all points (x, y) for which f (x, y) is
defined.
The set of all possible values of f (x, y) is called the range of f and is
denoted by ran f .
Examples
Example
Find and sketch the domain of the following:
1
2
3
f (x, y) =
x
2x + y
f (x, y) = 4x 2 + y 2
p
f (x, y) = 9 x 2 y 2
Remark
The graph of f (x, y) is the surface in
the equation z = f (x, y).
Example
Determine the graph of the following:
1
2
3
f (x, y) = 6 3x 2y
f (x, y) = 4x 2 + y 2
p
f (x, y) = 9 x 2 y 2
R3 (Euclidean 3-space) satisfying
Example
f (x, y) = 6 3x 2y
Example
f (x, y) = 4x 2 + y 2
Example
f (x, y) =
p
9 x2 y 2
Remark
Let be a plane and a surface in 3 . Let C be the curve of intersection

of and . We call C the trace of on the plane .
Example
1
Determine the trace of the graph of f (x, y) = 4x 2 + y 2 on the plane

z = 4, the y z -plane and the x y -plane.
Determine the trace of the graph of f (x, y) =
plane z = 0, the xz -plane and the x y -plane.
p
9 x 2 y 2 on the
Remark
Let S be a set of points in 3 . The projection of S on the x y -plane is

obtained by "dropping" all points of S orthogonally on the x y -plane. We
denote the projection of S on x y -plane by Projx y (S).
Example
Find Projx y (S) for the following:

1
2
3
S = {(1, 1, 1), (1, 0, 2), (0, 1, 1)}

p
S is the graph of f (x, y) = 9 x 2 y 2
S is the graph of f (x, y) = 4x 2 + y 2
Remark
Suppose is the graph of f . Geometrically:
The domain of f is Projx y ()
Remark
Suppose is the graph of f . Geometrically:
The domain of f is Projx y ()
The range of f is the "projection" of onto the z -axis.
Example
Use the graph of f to determine its domain and range:

1
2
f (x, y) = 4x 2 + y 2
p
f (x, y) = 9 x 2 y 2
Example
f (x, y) = 4x 2 + y 2
Example
f (x, y) =
p
9 x2 y 2
Contour Maps
Let f (x, y) be a function of two variables with graph given by z = f (x, y).
Contour Maps
Let C k be the trace of the graph of f (x, y) on the plane z = k
Contour Maps
The level curve of height k is Projx y (C k )
Contour Maps
A set of level curves for various values of k is called a contour map
of f
Contour Maps
A set of level curves for various values of k is called a contour map
of f
We get an idea about the graph of f from its contour map.
Raise the level curve of height k to z = k .
The "steepness" of the graph can be gleaned from the contour map.
Example
Draw a contour map for:

1
2
f (x, y) = 4x 2 + y 2
p
f (x, y) = 9 x 2 y 2
f (x, y) = 4x 2 + y 2
f (x, y) =
p
9 x2 y 2
Remark
All the above notions can be extended to functions of 3 or more variables.
In particular, we are also interested in functions of 3 variables.
Outline

Basic Ideas

Limits Along Paths
Limits Along Paths

Let f (x, y) be a function of two variables.
Limits Along Paths

Let f (x, y) be a function of two variables. Let C be a smooth curve on the
x y -plane with equation y = g (x) whose points are in the domain of f .
Limits Along Paths

x y -plane with equation y = g (x) whose points are in the domain of f . Let
(x 0 , y 0 ) be a point on C .
Limits Along Paths

(x 0 , y 0 ) be a point on C . The limit of f (x, y) as (x, y) approaches the
point (x 0 , y 0 ) along the curve C is:
lim
(x,y)(x 0 ,y 0 )
f (x, y) = lim f (x, g (x))

xx 0
Limits Along Paths

(x 0 , y 0 ) be a point on C . The limit of f (x, y) as (x, y) approaches the
point (x 0 , y 0 ) along the curve C is:
lim
(x,y)(x 0 ,y 0 )
f (x, y) = lim f (x, g (x))

xx 0
In the case that C has equation x = g (y), then

lim
(x,y)(x 0 ,y 0 )
f (x, y) = lim f (g (y), y)

xy 0
Examples
Example
1
Find
x2
along the path y = 0, x = 0 and y = x .
(x,y)(0,0) x 2 + y 2
Find
x9 y
along the path y = x and y = x 2 .
(x,y)(0,0) (x 6 + y 2 )2
lim
lim
Recall
For functions of one variable:
Recall
There two ways of approaching x 0 : from the left and from the right
Recall
Concept of one-sided limits: only two types
Recall
Concept of one-sided limits: only two types
Since the limit is the unique value that the y -values approach as the
x -values approach x 0 from both sides, lim f (x) exists if and only if
xa
lim+ f (x) = lim f (x)
xa
xa
For functions of two variables:


The point (x, y) may approach (x 0 , y 0 ) via an infinite number of paths.

Intuitively, the limit of f (x, y) as (x, y) approaches (x 0 , y 0 ) is the
unique value that f (x, y) approaches as (x, y) approaches (x 0 , y 0 ) via
all these possible paths!

This makes the computation of limits of functions of two variables
much more complex.

This makes the computation of limits of functions of two variables
much more complex.
We restrict our computation of limits to very simple types of
functions: polynomial and rational functions.
Theorem
1
lim
(x,y)(x 0 ,y 0 )
2
c =c
If f (x, y) is a polyomial function, then
lim
(x,y)(x 0 ,y 0 )
f (x, y) = f (x 0 , y 0 )
Examples
lim
(x,y)(1,2)
2x 3y =
Examples
lim
(x,y)(1,2)
2x 3y = 2 1 3 2 =
Examples
lim
(x,y)(1,2)
2x 3y = 2 1 3 2 = 2
Examples
lim
2x 3y = 2 1 3 2 = 2
lim
1000 =
(x,y)(1,2)
2
(x,y)(0,0)
Examples
lim
2x 3y = 2 1 3 2 = 2
lim
1000 = 1000
(x,y)(1,2)
2
(x,y)(0,0)
Theorem (Properties of Limits)

If
lim
(x,y)(x 0 ,y 0 )
1
f (x, y) = L and
lim
(x,y)(x 0 ,y 0 )
2
If c
R, then (x,y)(x
lim
c f (x, y) = c L .
,y )
lim
(x,y)(x 0 ,y 0 )
4
g (x, y) = M , then
f (x, y) g (x, y) = L M
0
lim
(x,y)(x 0 ,y 0 )
f (x, y) g (x, y) = L M .
If M 6= 0, then
lim
(x,y)(x 0 ,y 0 )
f (x, y)
L
= .
g (x, y) M
Remark
As a consequence of the last two theorems, if h(x, y) =
function where f and g are polynomial functions,
lim
(x,y)(x 0 ,y 0 )
provided that g (x 0 , y 0 ) 6= 0.
h(x, y) =
f (x 0 , y 0 )
g (x 0 , y 0 )
f (x, y)
is a rational
g (x, y)
Examples
x 2 2y + 2
=
(x,y)(0,0) x y + 1
lim
Examples
x 2 2y + 2 2
= =2
(x,y)(0,0) x y + 1
1
x2
=
(x,y)(0,0) x 2 + y 2
lim
lim
Examples
x 2 2y + 2 2
= =2
(x,y)(0,0) x y + 1
1
x2
=theorem does not apply!!!
(x,y)(0,0) x 2 + y 2
lim
lim
Announcements
Midterm Exam: February 3, Monday, 7-9 AM, MBLH, conflict need to
sign up with respective recit teachers
Scope of Midterm: until todays lesson
Announcements
No recitation class tomorrow
Announcements
Math, Physics (Materials Science) and Chem Majors are invited to the
IMSP Research Colloquium
Tilings, Quasicrystals and Pinwheels
Speaker: Dr. Dirk Frettleoh from Bielefeld University, Germany
To celebrate the International Year of Crystallography
Announcements
Math, Physics (Materials Science) and Chem Majors are invited to the
IMSP Research Colloquium
Tilings, Quasicrystals and Pinwheels
Speaker: Dr. Dirk Frettleoh from Bielefeld University, Germany
To celebrate the International Year of Crystallography
Take home SW 3-1 due on February 5, Wednesday (will be uploaded

tomorrow)
Get Chapter 1 and 2 SW bluebook from your recitation teachers,
Chapter 2 Quiz (if available)

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MATH 38

Mathematical Analysis III

Differentiation of Functions of Two or More Variables

Limits and Continuity

Partial Derivatives, Differentiability

Higher Order Derivatives, The Chain Rule

Local Linearity and Differentials

Relative and Absolute Extrema of Functions of Two Variables

Functions of Two or More Variables

Limits of Functions of Two or More Variables

Functions of Two or More Variables

Limits of Functions of Two or More Variables

R3 (Euclidean 3-space) satisfying

Let be a plane and a surface in 3 . Let C be the curve of intersection

Determine the trace of the graph of f (x, y) = 4x 2 + y 2 on the plane

Let S be a set of points in 3 . The projection of S on the x y -plane is

Find Projx y (S) for the following:

S = {(1, 1, 1), (1, 0, 2), (0, 1, 1)}

Use the graph of f to determine its domain and range:

Draw a contour map for:

Functions of Two or More Variables

Limits of Functions of Two or More Variables

Limits Along Paths

Limits Along Paths

Limits Along Paths

Limits Along Paths

f (x, y) = lim f (x, g (x))

Limits Along Paths

f (x, y) = lim f (x, g (x))

In the case that C has equation x = g (y), then

f (x, y) = lim f (g (y), y)

lim+ f (x) = lim f (x)

For functions of two variables:

Intuitive Notion of Limits

Intuitive Notion of Limits

Intuitive Notion of Limits

Intuitive Notion of Limits

Intuitive Notion of Limits

If f (x, y) is a polyomial function, then

Theorem (Properties of Limits)

Take home SW 3-1 due on February 5, Wednesday (will be uploaded

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