## Math 212: Multivariable calculus II, Spring 2016

MTWF 12-12:50pm, Library 204

Office Hours: Mon 11-12, Wed 2-3, and by appointment in Library 313
Math Center: SuMTWTh 7-9pm, Library 387

Textbook: Vector Calculus, 4th ed., by Susan Colley.

### Week 1: January 25-29

• Monday: Course overview. Hand out syllabus.
• Tuesday: Read 5.1. Area, volume, and iterated integrals.
• Wednesday: Read pp. 314 - middle of 321. Entry quiz. Double integrals over rectangles.
• Friday: Read pp. middle of 321 - middle of 328. HW due. Double integrals over general regions in the plane.

#### Homework due Friday

• Review the syllabus.
• 5.1: 2, 4, 8, 16
• 5.2: 2, 10

### Week 2: February 1-5

• Monday: Review pp.321-328. Entry quiz. More double integrals over general regions.
• Tuesday: Read the proof of Theorem 2.6 (starting on p. 328) and 5.3. HW due. Fubini's theorem and changing the order of integration.
• Wednesday: Read 5.4. Entry quiz. Triple integrals.
• Friday: HW due. Problem review. More triple integrals.

#### Homework due Tuesday

• 5.2: 28, 34, 40

#### Homework due Friday

• 5.2: 24, 25, 41 (counts as two problems)
• 5.3: 2[c], 8[c]
• 5.4: 22[c]

Note: Problems marked with a [c] are [computational] in nature and do not require full sentence/paragraph-formatted answers. You must still provide sufficient work, diagrams, and explanations so that your work can be easily followed. These problems will be graded on a 5-point scale. Problems without a [c] require full explanations for full credit (5 points for mathematical insight, 3 points for writing) even if they appear computational in nature.

### Week 3: February 8-12

• Monday: Review 5.4, especially the second half. Practice with triple integrals.
• Tuesday: Read these notes [pdf]. HW due. Integration in $\mathbb{R}^n$.
• Wednesday: Read pp.30 - top of 34 and bottom of 56 - 58. Entry quiz. Determinants.
• Friday: Read pp.349 - middle of 355. HW due. Determinants and coordinate transforms.

#### Homework due Tuesday

• 5.3: 12
• 5.4: 8, 26[c]

#### Homework due Friday

• 1.4: 2[c], 4[c]
• 1.6: 25, 26, 28, 29
• 5.5: 4

### Week 4: February 15-19

• Monday: Read pp.355-364. Change of variables for double integrals.
• Tuesday: No class.
• Wednesday: More change of variables. Review.
• Friday: First in-class exam.

### Week 5: February 22-26

• Monday: Read pp.364-371. Change of variables redux. Exam review.
• Tuesday: HW due. Common changes of variable: polar, cylindrical, and spherical coordinates.
• Wednesday: Optional reading: Shurman notes [pdf] §6.8. Entry quiz. Proof of COV: concepts and overview.
• Friday: Optional reading: Shurman notes[pdf] §6.9. HW due. Proof of COV: some details.

#### Homework due Tuesday

• 5.5: 8, 20, 25

#### Homework due Friday

• 5.5: 7, 30
• Shurman: 6.7.11, 6.7.12 (count (a-b), (c), and (d-e) as separate problems)

### Week 6: February 29 - March 4

• Monday: Read Shurman [pdf] §6.7. Entry quiz. Applications of COV.
• Tuesday: Read Colley pp.408-421. HW due. Scalar and vector line integrals.
• Wednesday: Entry quiz. More line integrals and physical applications.
• Friday: Read Colley §6.2. HW due. Green's theorem.

#### Homework due Friday

• 5.6: 5, 22
• 6.1: 12, 22, 37, 39

### Week 7: March 7-11

• Monday: Entry quiz. Green's theorem - finish proof, examples, curl and divergence formulations.
• Tuesday: HW due. Exam review. Distribute take-home exam.
• Wednesday: Read Colley §6.3. Entry quiz. Conservative vector fields.
• Friday: Take-home exam due. Finding scalar potentials.

#### Homework due Tuesday

• 6.2: 4[c], 10, 17
• Invent a region $D$ to which you can apply the result from 17 above. Compute the area of $D$ by computing one of $\oint_{\partial D} x\,dy$ or $-\oint_{\partial D} y\,dx$.

### Week 8: March 14-18

• Monday: Review §6.3. Entry quiz. Finding scalar potentials. Exam discussion.
• Tuesday: HW due. Proofs of Theorems 3.3 and 3.5.
• Wednesday: Entry quiz. Planimeter activity.
• Friday: HW due. Winding number revisited.

#### Homework due Tuesday

• 6.3: 18, 19, 20

#### Homework due Friday

• 6.3: 6[c], 8[c], 37
• p.453: 28, 30, 31, 32, 33

### Week 9: March 28 - April 1

• Monday: Read Colley §7.1. Intro to $k$-surface integrals and parametrized surfaces.
• Tuesday: Read §7.2. HW due. Surface integrals.
• Wednesday: Entry quiz. More surface integrals.
• Friday: Read §7.3. HW due. Stokes's theorem.

• p.452: 14

#### Homework due Friday

• p.452: 15
• 7.1: 2, 4, 10, 24, 30

### Week 10: April 4-8

• Monday: Entry quiz. Stokes's theorem: examples and proof.
• Tuesday: HW due. Practice with Stokes's theorem. Intro to Gauss's theorem.
• Wednesday: Entry quiz. Proof of Gauss's theorem.
• Friday: HW due. Practice with Gauss's theorem.

• 7.2: 6, 8
• 7.3: 2

#### Homework due Friday

• 7.2: 17
• 7.3: 10, 11, 12, 23

### Week 11: April 11-15

• Monday: Read §8.1. Intro to differential forms.
• Tuesday: Algebra of forms.
• Wednesday: Exam review.
• Friday: In-class exam. You may bring one two-sided 8.5"x11" page of notes. Emphasis on material covered since Exam 2.

#### Homework due Tuesday

• 7.3: 18, 20
• A friend makes the following argument to you: If $S$ is a closed surface (so that $\partial S = \varnothing$) and $F$ is a $C^1$ vector field on $\mathbb{R}^3$, then by Stokes's theorem, $\iint_S (\nabla \times F)\cdot dS = \oint_\varnothing F\cdot ds = 0$. Evaluate your friend's argument, and then justify their assertion by using Gauss's theorem.

### Week 12: April 18-22

• Monday: Read Shurman §§9.6-9.7. Entry quiz. Review of differential forms; algebra of forms.
• Tuesday: Read Shurman §9.8. HW due. Differentiation of forms.
• Wednesday: Read Shurman §9.9. Entry quiz. Pullback of forms.
• Friday: Read Shurman §9.10. HW due. Change of variables for forms.

#### Homework due Tuesday

• Shurman 9.3: 1, 2
• Shurman 9.5: 4
• Read through Shurman exercise 9.5.5 and convince yourself that you could execute all the steps. No need to turn anything in. (Note that Shurman uses $T'$ to denote $JT$.)

#### Homework due Friday

• Shurman 9.7: 1, 2, 4
• Shurman 9.8: 1, 2, 4, 5 (Note that Shurman uses $D_i$ to denote the $i$-th partial derivative $\partial/\partial x_i$. Further, $\langle x,y\rangle$ is used to denote the dot product $x\cdot y$.)

### Week 13: April 25-29

• Monday: Read Shurman 9.12-9.13. Entry quiz. Singular cubes, chains, manifolds, and boundaries.
• Tuesday: Read Shurman 9.14-9.15. HW due. Generalized Stokes's theorem. Classical theorems revisited.
• Wednesday: Review, synthesis, examples.
• Friday: Review, synthesis, examples. Distribute take-home final. Due May 6 by 5pm.

• 9.9.3(a)
• 9.10.1

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