The document is a chapter from a calculus textbook covering vector calculus. It includes figures and examples of vector fields, grid curves, and computer-generated plots related to concepts in vector calculus like line integrals, Green's theorem, and Stokes' theorem. Exercises at the end of sections apply these concepts to specific vector fields and vector calculus problems.

1. Calculus 5e
Early Transcendentals Multivariable
James Stewart
Chapter 16
Vector Calculus

2. © 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Thomson Learning™ is a trademark used herein under license.
2
Follow the link to the slide.
Then click on the figure to play the animation.
Animations
A
Figure 16.3.9
Figure 16.6.5
Figure 16.6.10

3. © 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Thomson Learning™ is a trademark used herein under license.
3
Section 16.1  Figure 1
Air velocity vectors indicating wind speed and direction
(a) 12:00 P.M., June 11, 2002 (b) 4:00 P.M., June 30, 2002

4. © 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Thomson Learning™ is a trademark used herein under license.
4
Section 16.1  Figures 10-12
Examples of computer-generated vector fields
kjiF xzyzyx ++=),,( kjiF xyzyx +−= 2),,(
kjiF
4
),,(
z
z
x
z
y
zyx +−=

5. © 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Thomson Learning™ is a trademark used herein under license.
5
Section 16.1  Exercises 11-14
I II
xyyx ,),( =F
ysyx in,1),( =F
1,2),( +−= xxyxF
xyyx /1,),( =F
11
.
12
.
13
.
14
.
III IV

6. © 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Thomson Learning™ is a trademark used herein under license.
6
Section 16.1  Exercises 15-18
I II
III IV
kjiF 32),,( ++=zyx
kjiF zzyx ++= 2),,(
kjiF 3),,( ++= yxzyx
kjiF zyxzyx ++=),,(
15
.
16
.
17
.
18
.

7. © 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Thomson Learning™ is a trademark used herein under license.
7
A
Section 16.3  Figure 9
The vector field in Example 3

8. © 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Thomson Learning™ is a trademark used herein under license.
8
A
Section 16.6  Figure 5
Grid curves for
Animate u
constant
Animate v
constant
A

9. © 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Thomson Learning™ is a trademark used herein under license.
9
A
Section 16.6  Figure 10
Computer-generated plot of the graph in Example 8
sinsincossin θθ xzxyxx ===

10. © 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Thomson Learning™ is a trademark used herein under license.
10
Section 16.6  Exercises 11-16
I II III
IV V VI
kjir11. uvvvu ++= sincos),( kjir12. uvuvuvu ++= sincos),(
kjir13. vvuvuvu ++= sincos),( vuzvuyux cos,sin,3
===14.
( ) ,cossin vuux −=15. ( )( ) ,4coscos31 uvux π+−=16.
( ) uzvuy =−= ,sincos1
uz =
( )( ) ,4sincos31 uvuy π+−=
( ) vuuz sin13 −+=

11. © 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Thomson Learning™ is a trademark used herein under license.
11
Section 16.10  Summary Box
Summary: Higher-dimensional versions of the Fundamental Theorem
Fundamental Theorem
of Calculus
Fundamental Theorem
for Line Integrals
Green’s Theorem
Stokes’ Theorem
Divergence Theorem
)()()( aFbFdxxF
b
a
−=′∫
dyQdxPdA
y
P
x
Q
C
D
+=





∂
∂
−
∂
∂
∫∫∫
))(())(( afbfdf
C
rrr −=∇∫ ⋅
SFF ddV
SE
∫∫∫∫∫ =div ⋅
rFSF dd
C
S
∫∫∫ =curl ⋅ ⋅