Instructor's Solution Manual for Vector Calculus, 5th edition by Susan Colley (Ch. 1-8) |All Chapters

INSTRUCTOR’S SOLUTIONS
MANUAL
Vector Calculus, 5th edition
by Susan Colley, Santiago Cañez
SC
O
R
EG
U
ID
ES

, Table of Content
1. Vectors
1.1 Vectors in Two and Three Dimensions
1.2 More About Vectors
1.3 The Dot Product
1.4 The Cross Product
1.5 Equations for Planes; Distance Problems
1.6 Some n-dimensional Geometry
SC
1.7 New Coordinate Systems
True/False Exercises for Chapter 1
Miscellaneous Exercises for Chapter 1
2. Differentiation in Several Variables
O
2.1 Functions of Several Variables; Graphing Surfaces
2.2 Limits
R
2.3 The Derivative
2.4 Properties; Higher-order Partial Derivatives
EG
2.5 The Chain Rule
2.6 Directional Derivatives and the Gradient
2.7 Newton's Method (optional)
True/False Exercises for Chapter 2
U
Miscellaneous Exercises for Chapter 2
3. Vector-Valued Functions
ID
3.1 Parametrized Curves and Kepler's Laws
3.2 Arclength and Differential Geometry
3.3 Vector Fields: An Introduction
ES
3.4 Gradient, Divergence, Curl, and the Del Operator
True/False Exercises for Chapter 3
Miscellaneous Exercises for Chapter 3
4. Maxima and Minima in Several Variables
4.1 Differentials and Taylor's Theorem
4.2 Extrema of Functions
4.3 Lagrange Multipliers
4.4 Some Applications of Extrema
True/False Exercises for Chapter 4

,Miscellaneous Exercises for Chapter 4
5. Multiple Integration
5.1 Introduction: Areas and Volumes
5.2 Double Integrals
5.3 Changing the Order of Integration
5.4 Triple Integrals
5.5 Change of Variables
5.6 Applications of Integration
SC
5.7 Numerical Approximations of Multiple Integrals (optional)
True/False Exercises for Chapter 5
Miscellaneous Exercises for Chapter 5
6. Line Integrals
O
6.1 Scalar and Vector Line Integrals
6.2 Green's Theorem
R
6.3 Conservative Vector Fields
True/False Exercises for Chapter 6
EG
Miscellaneous Exercises for Chapter 6
7. Surface Integrals and Vector Analysis
7.1 Parametrized Surfaces
7.2 Surface Integrals
7.3 Stokes's and Gauss's Theorems
U
7.4 Further Vector Analysis; Maxwell's Equations
True/False Exercises for Chapter 7
ID
Miscellaneous Exercises for Chapter 7
8. Vector Analysis in Higher Dimensions
8.1 An Introduction to Differential Forms
ES
8.2 Manifolds and Integrals of k-forms
8.3 The Generalized Stokes's Theorem
True/False Exercises for Chapter 8
Miscellaneous Exercises for Chapter 8

, Chapter 1


Vectors
SC

1.1 Vectors in Two and Three Dimensions

1. Here we just connect the point (0, 0) to the points indicated:

y
O
3
b
2.5

2
R
c
1.5

1 a
EG
0.5

x
-1 1 2 3

2. Although more difficult for students to represent this on paper, the figures should look something like the following. Note that
the origin is not at a corner of the frame box but is at the tails of the three vectors.
U
3
ID
2 a
z

1 b
c
0
ES
-2 -2 0 2
0
2
x y


In problems 3 and 4, we supply more detail than is necessary to stress to students what properties are being used:
3. (a) (3, 1) + (−1, 7) = (3 + [−1], 1 + 7) = (2, 8).
(b) −2(8, 12) = (−2 · 8, −2 · 12) = (−16, −24).
(c) (8, 9) + 3(−1, 2) = (8 + 3(−1), 9 + 3(2)) = (5, 15).
(d) (1, 1) + 5(2, 6) − 3(10, 2) = (1 + 5 · 2 − 3 · 10, 1 + 5 · 6 − 3 · 2) = (−19, 25).
(e) (8, 10) + 3((8, −2) − 2(4, 5)) = (8 + 3(8 − 2 · 4), 10 + 3(−2 − 2 · 5)) = (8, −26).
4. (a) (2, 1, 2) + (−3,  9, 7) = (2 − 3, 1 + 9, 2 + 7) = (−1,
 10, 9).
(b) 12 (8, 4, 1) + 2 5,−7, 14 = 4, 2, 12 + 10, −14, 12 = (14, −12, 1).
(c) −2 (2, 0, 1) − 6 12 , −4, 1 = −2((2, 0, 1) − (3, −24, 6)) = −2(−1, 24, −5) = (2, −48, 10).
5. We start with the two vectors a and b. We can complete the parallelogram as in the figure on the left. The vector from the
origin to this new vertex is the vector a + b. In the figure on the right we have translated vector b so that its tail is the head of
vector a. The sum a + b is the directed third side of this triangle.


c 2012 Pearson Education, Inc. 1