Math 2240: Multivariable calculus


Prof. Allen Knutson


Final exam: Sunday 5/12/2019 2:00 PM Hollister Hall 110 Cumulative, with one double-sided page of notes.

Final with answers. Grade boundaries: D 30 C 50 B 70 A 100

Class times: TTh 11:40-12:55 in 253 Malott.

Midterm (here with answers) was class Thursday March 14. Despite our having only 30+ students, there were grades in every decile (0-10,10-20,...,90-100). 45-100 is A, 30-45 is B.

Text: Vector Calculus, Linear Algebra, and Differential Forms, Hubbard & Hubbard 5th edition (aka [HH]).

Prof. Knutson's office hours: after class on Tuesday, and Monday 1-2, in 515 Malott.

Special office hours Thursday May 9 from 10-2pm, Friday May 10 from 9:30-11am.

TA Max Lipton's office hours: Wednesdays 5-7 pm in the tutor lab. Starting May, they will be Thursday 1:45-3:45.

Syllabus: chapters 4-6 of the book [HH].

Homework is due at the beginning of class on Tuesdays Thursdays -- just march up to the front of class.

HW #1 (due Thurs Jan 31): [HH] 4.1.5, 4.1.8, 4.1.11, 4.1.12, 4.1.15 (see def 4.1.17 on p411), 4.1.21

HW #2 (due Thurs Feb 7): [HH] 4.2.4, 4.3.1, 4.3.3, 4.3.4

HW #3 (due Thurs Feb 14): [HH] 4.4.1, 4.4.3, 4.4.7.
#4. Let f(x,y) = 0 unless (x,y) is of the form (a/2^m, b/2^n) in lowest terms (both x,y in [0,1]), with m < n.
Try to integrate f by doing iterated integration (over x then y, or vice versa.) Show that one way gives 0 and the other way doesn't.

HW #4 (due Thurs Feb 21):

  • 1. Let S in R^n be defined by the inequalities each x_i nonnegative, total of the x_i is at most 1. Do iterated integrals to find its volume. Figure out, once you know the answer, why it should have been obvious.
  • 2. Let S in R^2 be the unit disc, intersected with "x+y >= 0". The area is obvious, but compute it by doing iterated integrals.
  • 3. Read the Fubini appendix and figure out why the statement there is stronger than the one in 4.5. (Don't turn anything in for this.)
  • 4. We proved that if f,g are integrable so is fg. Give examples where neither is integrable, but fg is:
  • where f,g are nonnegative
  • where f,g are never 0 on [0,1] (and always 0 off).
    • (That's two separate questions, unless you find an f,g that do both!)

    • HW #5 (due Thurs Feb 28): [HH] 4.8 #2,4,10 (note: "cycle form" is not one-line notation; it's explained on p470 after 4.8.38), 11

    HW #6 (due Thurs March 7):

  • [HH] 4.8.11 oops
  • [HH] 4.8.23ab
  • Find two matrices of determinant zero whose sum has determinant one.
  • Find two matrices of determinant one whose sum has determinant zero.
  • Let M's characteristic polynomial \( \chi_M(t) \) factor as \( \prod_{i=1}^n (t-\lambda_i) \) with all roots \(\lambda_i\) distinct. Let \( \vec v_j, \vec v_k \neq \vec 0 \) be eigenvectors of \( M\) with eigenvalues \( \lambda_j, \lambda_k \). Let \( T = \prod_{i\neq j} (\lambda_i {\bf 1} - M) \). Show that \( \vec v_k \) is an eigenvector of \( T \), compute its \(T\)-eigenvalue, and show that eigenvalue is nonzero iff \( j = k \).

    • HW #7 (due Thurs March 28):

  • [HH] 4.10.5a, 4.10.13, 4.10.17
  • [HH] 4.11.1, 4.11.8ac, 4.11.11

    • HW #7 (due Thurs April 11):

  • [HH] 5.1.2, 5.1.5, 5.2.2, 5.2.3
  • Recall an orthogonal matrix M is one with orthonormal columns. Show that it also has orthonormal rows, and determine the possible determinants. Finally, parametrize the set of 2x2 orthogonal matrices.

    • HW #8 (due Tues April 23 Thurs Apr 25):

  • Let \( V \) be a real vector space, and \( \vec v_1,\ldots,\vec v_m \) be a list L of vectors in \( V \). Define a linear map \( T_L:\ \mathbb R^m \to V\) by \( (c_1,\ldots,c_m) \mapsto \sum_{i=1}^m c_i \vec v_i \). Figure out the exact conditions on \( L \) under which \( T_L \) is invertible.
  • Let \( V \) have a basis \( \vec v_1,\ldots,\vec v_m \). For each \( i=1\ldots m\), define an element \( w_i \in V^* \), i.e. in the dual space, such that \( w_i\left( \sum_j c_j \vec v_j \right) = c_i \). Show that \( w_1,\ldots,w_m \) is a basis for \( V^* \), called the dual basis.
  • Let \( V\) be the space of size \(n \) column vectors \(\vec v\), and think of \( V^*\) as the space of size \(n \) row vectors \(w\), using matrix multiplication then trace \(Tr(w \vec v\)) to compute the application of \(w \in V^*\) to \(\vec v\in V\).

    Let \(M\) be an \(n\times n\) matrix whose columns form a basis for \( V \). Compute the dual basis.

  • Let \( \alpha = \sum_{i=1}^{n} dx_{2i-1} \wedge dx_{2i} \), a 2-form on \(\mathbb R^{m}\) where \(m\geq 2n\). Compute \( \alpha \wedge \cdots \wedge \alpha \), the n-fold wedge power, expressed in elementary 2n-forms. (Maybe start with n=1,2,3 to get an idea.)

    • Final HW #9 (due Tues May 7):

  • [HH] 6.2.5, 6.3.2, 6.3.15, 6.4.2, 6.4.4, 6.7.1, 6.7.10
  • Let \( S := \{ (x \neq 0, y) \} \subset \mathbb R^2 \), \( T := \{ (x, y \neq 0) \} \subset \mathbb R^2 \), \( U := S \cup T \), the punctured plane. On \(S\) define a function \(\theta_1(x,y) := \tan^{-1}(y/x) \in (-\pi/2,\pi/2) \), and on \(T\) define a function \(\theta_2(x,y) := \pi/2 - \tan^{-1}(x/y) \).

    Finally, define a 1-form on \(U\) (and therefore on the open subsets \(S,T\)), \(\alpha = (y\, dx - x\ dy)/(x^2+y^2) \).

  • Show that \(d\theta_1 = \alpha\) on \(S\), and \(d\theta_2 = \alpha\) on \(T\).
  • Show that \(d\alpha = 0\).
  • Why is there no function \(f\) on \(U\) such that \(df = \alpha\)?