# 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 $$df = \alpha$$?