


Darin Stephenson, Math 231232 Book
This page will contain details about a textbook I am writing for our sophomore sequence, Math 231232.
As I learn about errors and omissions in the text, I will post corrections here.
Table of Contents
 The Geometry of R^{n}
 Matrices and Linear Systems
 Vector Spaces
 Linear Transformations
 Eigenvalues and Eigenvectors
 First Order Differential Equations
 Linear Differential Equations
 Systems of Differential Equations
 Functions of Several Variables
 Derivatives of Multivariable Functions
 Multiple Integrals
 Calculus on Vector Fields
Errata for the 201213 version (Version 3):
 I'll add errata for the new version as I learn about them. Please email
me at stephenson@hope.edu if you notice any errors.
Errata for the 201112 version (Version 2.3):
 In the definition of "boundary point" (in the box near the bottom
of page 439), both occurrences of
the phrase "(other than a itself)" should be omitted.
Errata for the 201011 version (Version 2.2):
 In Section 1.4, Theorem 1.19 statement: It should be 0 is less than or equal to theta less than or
equal to pi. In the book, it says "theta less than pi".
 In Section 1.4, Example 1.22: It says we are finding the projection of y on x. What is actually computed
is the projection of x on y.
 In Exercise 26 on page 58, for clarity, it should say that "the length of the cross product
of two vectors gives the area of the parallelogram that they define."
 Exercise 19 from Section 4.1 should say "f(x,y)" instead of "f(x)" and "g(x,y)" instead of "g(x)".
 The formula given in the book for the kth order Taylor method (formula 6.5.6 on page 312) is incorrect. The correct formula is
 In the definition of "boundary point," both occurrences of
the phrase "(other than a itself)" should be omitted.
 In the Section 11.3 exercises, p. 498, exercises 12 and 13 should be in Section 11.4. These
exercises involve change of variables.
 In the boxed formulas on pages 504 and 505, the Jacobian matrices as computed are incorrect.
The positions of sin(theta) and r*sin(theta) are interchanged. The determinants are still the same.
Errata for the 200910 version (Version 2.1):
 In Section 1.4, Theorem 1.19 statement: It should be 0 is less than or equal to theta less than or
equal to pi. In the book, it says "theta less than pi".
 In Section 1.4, Example 1.22: It says we are finding the projection of y on x. What is actually computed
is the projection of x on y.
 In Section 1.1, Exercise 9: Before issuing these commands, you will need to type in
the command "with(plots):" and press enter, just as described in Exercise 10.
 In Section 3.3, Theorem 3.38: The sentence "Let B of vectors in S corresponding
to leading columns in rref(A)." should say "Let B be the set of vectors in S corresponding
to leading columns in rref(A)."
 Theorem 4.9: The notation and proof of parts 2 through 4 implicitly require that V is finite
dimensional. However, this is not required, and the proofs can be rewritten so that a finite
basis is not needed. Finite dimensionality of V is needed for part 5, however.
 The formula given in the book for the kth order Taylor method (formula 6.5.6 in Section 6.5) is incorrect. The correct formula is
 There are several typos in the Section 8.2 exercises. A corrected version of the exercise
set is here.
 There is a typo in the function j(x,y) given in Example 9.5 part 3, on page 419. The function
should be (x^{2}+4y^{2})^{(1/4)}.
 The hint given for Exercise 11 of Section 9.3 is incorrect. (It leads to an incorrect
answer.) With a corrected hint, the problem becomes very complicated, and so will eventually
be replaced with a different exercise.
 In Section 10.1, the text says that there will be an exercise at the end of the section
related to the open ball of radius a in R, R^{2}, and R^{3}. This exercise
was inadvertently omitted.
 In the statement of Theorem 10.10, the coordinate functions f_{i} should
map into R, not R^{n}.
 In Section 10.1, Example 10.18: B_{1}00(0) should instead be
B_{100}(0).
 In the definition of "boundary point," both occurrences of
the phrase "(other than a itself)" should be omitted.
 In the definition of derivative (p. 442),
the matrix f '(a) should be defined as an n x m matrix, not an m x n matrix.
 The function at the end of Example 10.26 that is listed as sqrt(xy) should be
the square root of xy (with a square root symbol).
 In Exercise 2(d) of Section 10.3, the pointy brackets were left off the vector
1,1,1.
 In Exercise 7 of Section 10.4, the function should be
z=xe^{(x^2/8y^2)}.
 In Exercise 10 of Section 10.5, the function should read
f(x,y) =x+y^{2}, as in the previous exercise.
 In the statement of Theorem 11.40 on page 506, as well as the definition
of the line integral on page 507, the phrase "the curve defined by the image
of f" should say "the curve defined by the image of g".
 The answer to Exercise 5
in Section 1.6 is listed in
Appendix A as 30, but the answer is actually 12.
The answer to Exercise 6 from Section 1.6 is 30.
 The answers to the following exercises are incorrect in the book:
8.1 #6, 8.1 #11, 8.2 #5, and 8.3 #1. Corrected answers will be posted
on the course Moodle site.

