Gradient, Directional Derivatives, Divergence & Curl
Gradient, Directional Derivatives, Divergence & Curl
Gradient, Directional Derivatives, Divergence & Curl
A A
OA or
a
O O
A(u1 , u2 , , un ) a x (u1 , u 2 , , u n ) i
~ ~
a y (u1 , u2 , , un ) j
~
a z (u1 , u2 , , u n ) k
~
5
Partial derivative of A with respect to
~
u1 is
given by
A a x a y a z
~
i j k,
u1 u1 ~ u1 ~ u1 ~
2 A a 2
2
a 2
az
~
x
i y
j k
u1u2 u1u2 ~ u1u2 ~ u1u2 ~
e.t.c.
6
Example
If F 3uv2 i (2u 2 v) j (u 3 v 2 ) k
~ ~ ~ ~
then
F
~
3v 2 i 4u j 3u 2 k ,
u ~ ~ ~
F 2 F
~
6uv i j 2v k , ~
4 j 6u k ,
v ~ ~ ~ u 2
~ ~
2 F 2 F 2 F
~
6u i 2 k , ~
~
6v i
v 2 ~ ~ uv vu ~
7
Del Operator Or Nabla (Symbol )
i j k .
x ~ y ~ z ~
8
Grad (Gradient of Scalar Functions)
9
Example
If x 2 yz3 xy 2 z 2 , determine grad at P (1,3,2).
Solution
Given x 2 yz3 xy 2 z 2 , hence
2 xyz3 y 2 z 2
x
x 2 z 3 2 xyz2
y
3 x 2 yz 2 2 xy 2 z
z 10
Therefore,
i j k
x ~ y ~ z ~
(2 xyz3 y 2 z 2 ) i ( x 2 z 3 2 xyz2 ) j
~ ~
(3x 2 yz 2 2 xy 2 z ) k .
~
At P (1,3,2), we have
(2(1)(3)(2) 3 (3) 2 (2) 2 ) i ((1) 2 (2)3 2(1)(3)(2) 2 ) j
~ ~
84 i 32 j 72 k .
~ ~ ~
11
Grad Properties
1) ( A B) A B
2) ( AB ) A(B) B(A)
12
Directional Derivative
Directional derivative of in the direction of a is
~
d
a . grad
ds ~
dr
where a ~ ,
~
dr
~
13
PHYSICAL INTERPRETATION
In mathematics, the directional derivative of
a multivariate differentiable function along a given
vector v at a given point x intuitively represents the
instantaneous rate of change of the function,
moving through x with a velocity specified by v. It
therefore generalizes the notion of a partial
derivative, in which the rate of change is taken
along one of the coordinate curves, all other
coordinates being constant.
14
Example
Compute the directional derivative of x 2 z 2 xy 2 yz 2
at the point (1,2,1) in the direction of the vector
A 2i 3 j 4k.
~ ~ ~ ~
15
Solution
Directional derivative of in the direction of a
~
d
a . grad
ds ~
A
where grad i j k and a ~ .
x ~ y ~ z ~ ~
A
~
Given x 2 z 2 xy 2 yz 2 , hence
(2 xz 2 y 2 ) i (4 xy z 2 ) j ( x 2 2 yz) k .
~ ~ ~
16
At (1,2,-1),
(2(1)(1) 2(2) 2 ) i ( 4(1)(2)
~
6 i 9 j 3k .
~ ~ ~
A 2 2 32 (4) 2
~
29.
17
A 2 3 4
Therefore, a ~
i j k.
~
A 29 ~ 29 ~ 29 ~
~
d
Then, a .
ds ~
2 3 4
i j k .(6 i 9 j 3 k )
29 ~ 29 ~ 29 ~ ~ ~ ~
2 3 4
( 6) (9) (3)
29 29 29
51
9.470462.
29
18
Unit Normal Vector
Equation (x, y, z) constant is a surface equation.
Since (x, y, z) constant, the derivative of is
zero; i.e.
d d r .grad 0
~
d r grad cos 0
~
cos 0
90.
19
This shows that when (x, y, z) constant,
grad d r .
~
y grad
ds
n .
~
Example
At (-1,1,1), (1 1) i (2 1) j (2 1) k
~ ~ ~
2 i j k
~ ~ ~
and 4 1 1 6 .
The unit normal vector is
2 i j k
1
n (2 i j k )
~ ~ ~
~ 6 6 ~ ~ ~
22
Divergence of a Vector
If A a x i a y j a z k , the divergence of A is
~ ~ ~ ~ ~
defined as
div A . A
~ ~
i j k .( a x i a y j a z k )
x ~ y ~ z ~ ~ ~ ~
a x a y a z
div A . A .
~ ~ x y z
23
PHYSICAL INTERPRETATION
The physical significance of the divergence of
a vector field is the rate at which "density" exits a
given region of space.
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Example
If A x 2 y i xyz j yz 2 k ,
~ ~ ~ ~
Answer
ax a y az
div A . A
~ ~ x y z
2 xy xz 2 yz.
At point (1,2,3),
div A 2(1)(2) (1)(3) 2(2)(3)
~
13. 25
Remarks
A is a vector function, but div A is a scalar function.
~ ~
26
Curl of a Vector
If A a x i a y j a z k , the curl of A is defined by
~ ~ ~ ~ ~
curl A A
~ ~
i j k (a x i a y j a z k )
x ~ y ~ z ~ ~ ~ ~
i j k
~ ~ ~
curl A A .
~ ~ x y z
ax ay az
27
PHYSICAL INTERPRETATION
The physical significance of the curl of a
vector field is the amount of "rotation" or angular
momentum of the contents of given region of space.
It arises in fluid mechanics and elasticity theory.
28
Example
If A ( y 4 x 2 z 2 ) i ( x 2 y 2 ) j x 2 yz k ,
~ ~ ~ ~
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Solution i j k
~ ~ ~
curl A A
~ ~ x y z
y4 x2 z 2 x2 y2 x 2 yz
2 2
( x yz) ( x y ) i
2
y z ~
( x 2 yz) ( y 4 x 2 z 2 ) j
x z ~
2 4 2 2
( x y ) ( y x z ) k
2
x y ~
x 2 z i (2 xyz 2 x 2 z ) j (2 x 4 y 3 ) k . 30
~ ~ ~
At (1,3,-2),
curl A (1) 2 (2) i (2(1)(3)(2) 2(1) 2 (2)) j
~ ~ ~
(2(1) 4(3)3 ) k
~
2 i 8 j 106 k .
~ ~ ~
Exercise
If A ( xy3 y 2 z 2 ) i ( x 2 z 2 ) j x 2 yz 2 k ,
~ ~ ~ ~
(2 x 3xy 2 2 yz 2 ) k .
~
At (1,2,3), curl A 15 i 12 j 26 k .
~ ~ ~ ~
Remark
A is a vector function and
~