Lecture Notes (Chapter 4.5 Gauss and Stokes Theorem )

8/19/2019 Lecture Notes (Chapter 4.5 Gauss and Stokes Theorem )

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1 Chapter 4: Vector Calculus

Example 1

Evaluate surface integral of ( , , ) 2 x y z x y z F i j k over all the surfaces of the cube

0, 1, 0, 1, 0, x x y y z and 1 z , with the oriented outward unit normal vector.

Solution

2 x y z x y z

F i j k i j k

1 1 2

4

S G

dS dV F n F

11 1

0 0 0

11

0 0

1

0

4

4

4

4

y z x

z y x

y z

z y

z

z

dxdydz

dydz

dz

Gauss’s Theorem / Divergence Theorem

Gauss’s Theorem / Divergence Theorem

Let G be a simple and closed solid region and let S be the surface of G given with positive

(outwards) orientation.

Let ( , , ) ( , , ) ( , , ) ( , , ) x y z u x y z v x y z w x y z F i j k be a vector field whose component

functions have continuous partial derivatives on an open region that contains G.

Hence, the surface integrals of the vector field F is given by:

S G

dS dV

F n F




Example 2

Use Gauss’s Theorem to evaluateS

dS F n , where ( , , ) x y z y xy z F i j k and S is the

surfaces enclosed by cylinder 2 2 4 x y , plane 0 z , and paraboloid 2 2. z x y

Solution

y xy z x y z

F i j k i j k

0 1

1

x

x

By Gauss’s Theorem,

S G

dS dV F n F

( 1)G

x dV

By cylindrical coordinates,

22 2

0 0 0

( cos 1)r z r

S r z

dS r dzrdrd

F n

22 2

2

0 0 0

2 2

2 2

0 0

2 2

4 3

0 0

22 5 4

0 0

2

0

2

0

( cos )

( cos )

( cos )

cos

5 4

32cos4

5

32sin 4

5

8

r z r

r z

r

r

r

r

r r dzdrd

r r r drd

r r drd

r r d

d

It’s projection on xy-plane




Example 3

Let S be the surfaces of the solid G enclosed by upper hemisphere 2 21 z x y , and

plane 0 z , and oriented outward. Use Gauss’s Theorem to find the surface integrals of the

vector field 3 3 3( , , ) x y z x y z F i j k over the surface S .

Solution

3 3 3 x y z x y z

F i j k i j k

2 2 2

2 2 2

3 3 3

3( )

x y z

x y z

By Gauss’s Theorem,

S G

dS dV F n F

2 2 23( )G

x y z dV

By spherical coordinates,

/2 12

2 2

0 0 0

3 sinS

dS d d d

F n

/2 2

0 0

/2

0

/2

0

3sin

5

6sin

5

6 cos5

6

5

d d

d

It’s projection on xy-plane




Stoke’s Theorem can be regarded as an extension of Green’s Theorem. Stoke’sTheorem relates a surface integral over a surface S to a line integral around the

boundary curve of S .

Example 4

EvaluateS

d F r by using Stoke’s Theorem for 2 32 z x y F i j k with C is the circle

2 2 1 x y in the xy-plane with counterclockwise orientation looking down the positive z -axis.

Solution

2

2 3

3 2 2

2

y z x y z

z x y

i j k

F i j k

For plane 0 ( , , ) z f x y z z

2

2

1 1

( 3 2 2 ) 2

f f f f

x y z

f

f

f

y z

i j k k

|| ||

n k || ||

F n i j k k

Stoke’s Theorem

Stoke’s Theorem

Let S be an orientable piecewise smooth surface, which is bounded by a simple, closed,

piecewise smooth boundatu curve C with positive orientation.

Let ( , , ) ( , , ) ( , , ) ( , , ) x y z u x y z v x y z w x y z F i j k be a vector field whose component

functions have continuous partial derivatives on an open region that contains S .

Hence,

S S

d dS F r F n




By Stoke’s Theorem,

S S

d dS F r F n

2 1

0 0

2

0

2

2

1

2

S

r

r

dA

rdrd

d

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Lecture Notes (Chapter 4.5 Gauss and Stokes Theorem )