Vector Calculus 4th Edition — by Susan Colley — Complete Instructor's Solution Manual (Ch. 1-8)

INSTRUCTOR’S SOLUTIONS
MANUAL
Vector Calculus, 4th edition
by Susan Colley
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, 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
1.7 New Coordinate Systems
True/False Exercises for Chapter 1
Miscellaneous Exercises for Chapter 1
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2. Differentiation in Several Variables
2.1 Functions of Several Variables; Graphing Surfaces
2.2 Limits
2.3 The Derivative
2.4 Properties; Higher-order Partial Derivatives
2.5 The Chain Rule
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2.6 Directional Derivatives and the Gradient
2.7 Newton's Method (optional)
True/False Exercises for Chapter 2
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Miscellaneous Exercises for Chapter 2
3. Vector-Valued Functions
3.1 Parametrized Curves and Kepler's Laws
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3.2 Arclength and Differential Geometry
3.3 Vector Fields: An Introduction
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
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4.1 Differentials and Taylor's Theorem
4.2 Extrema of Functions
4.3 Lagrange Multipliers
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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
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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
5.7 Numerical Approximations of Multiple Integrals (optional)
True/False Exercises for Chapter 5
Miscellaneous Exercises for Chapter 5
6. Line Integrals
6.1 Scalar and Vector Line Integrals
6.2 Green's Theorem
6.3 Conservative Vector Fields
True/False Exercises for Chapter 6

,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
7.4 Further Vector Analysis; Maxwell's Equations
True/False Exercises for Chapter 7
Miscellaneous Exercises for Chapter 7
8. Vector Analysis in Higher Dimensions
8.1 An Introduction to Differential Forms
8.2 Manifolds and Integrals of k-forms
8.3 The Generalized Stokes's Theorem
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True/False Exercises for Chapter 8
Miscellaneous Exercises for Chapter 8
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, Chapter 1


Vectors
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1.1 Vectors in Two and Three Dimensions

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

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3
b
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1 a
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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.
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2 a
z

1 b
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-2 -2 0 2
0
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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.


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