## Math 211: Multivariable calculus I, Spring 2015

Section F01: MTWF 9-9:50am, Physics 122
Section F04: MTWF 12-12:50pm, Library 204

Office Hours: Tu 3-4, Wed 11-12, Th 3-4, Library 313
Math Center: SuMTWTh 7-9pm, Library 387

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

### Week 1: October 31 - September 4

• Monday: What is (multivariable) calculus? Hand out syllabus.
• Tuesday: Euclidean space, multivariable functions,
• Wednesday: Partial derivatives, the Jacobian matrix, linear functions, and the derivative.
• Friday: Best affine approximation. Download Dave Perkinson's 211 notes [pdf] and review Chapter 1.

#### Homework due Friday

• Review the syllabus.
• Fill out the office hour doodle. (You were emailed a link.)
• Use one to two pages to answer the following prompts: (1) Why are you taking this class? (2) What do you hope to learn? (3) Describe a multivariable function you would like to understand better at the end of this course. What do you know about it, and what do you want to know?

Like all our assignments, this is a writing assignment. Make sure to use sentences and paragraphs to answer the prompts.

• [Optional]: Open an account at cloud.sagemath.com. Access this sage worksheet and run it in your account. Play around with changing the functions and producing new visualizations. Share your new worksheet with me if you would like comments.

### Week 2: September 7 - 11

• Monday: Labor Day. No class.
• Tuesday: Read Colley 1.1, 1.2. Linear structure and metric structure.
• Wednesday: Read Colley 1.3, 1.4. Dot products and projection.
• Friday: Read Colley 1.5. Cauchy-Schwartz and triangle inequalities. Angles. Cross product.

#### Homework due Friday

• From Dave Perkinson's 211 notes [pdf], Chapter 1 Exercises: 2, 3, 4, 5. (Exercise 5 is really six different problems).
• Colley §1.1 Exercises: [2, 3, 4], 16, 23.
• Colley §1.2 Exercises: [1, 4, 6, 9], 13, 45(a,d), 46*.
• Colley §1.3 Exercises: 34.

Exercises in [brackets] are rote problems that do not need to be turned in, but which you should be able to do with relative ease. Starred* exercises are bonus problems; they are optional, but everyone is encouraged to attempt them.

### Week 3

• Monday: Linear subspaces (planes, hyperplanes, and all that).
• Tuesday: Read Colley §1.6. Linear functions and matrices.
• Wednesday: Read Perkinson §3.4. More linear functions and matrices.
• Friday: Review of linear structures and homework problems.

#### Homework due Friday

• Colley §1.3 Exercises: [3, 6, 7, 17, 19], 36.
• Colley §1.4 Exercises: [5], 24, 25, 26.
• Colley §1.6 Exercises: [16, 17, 18], 20, [30], 33*, 34*.
• Perkinson Chapter 3 Exercises: 11, 16, 19, [26], 32, 37, 43.

### Week 4

• Monday: Start reading Colley §2.2. Topology of Rn.
• Tuesday: Topology and homework problems.
• Wednesday: Finish reading Colley §2.2. Limits of multivariable functions. Continuity.
• Friday: Read Colley §2.3. Limits and continuity. Start the derivative.

Announcement: Our first take-home exam will be distributed during class on Friday of Week 5 (i.e. October 2). It will be due the following Monday (October 5). Week 5's homework set will be accordingly shorter in order to give you time to study and review. For the exam, you will have 3 hours to think about the problems and do scratch work, and 1 hour to write up your solutions.

#### Homework due Friday

• Perkinson Chapter 3 Exercises: 33, 34, 35, 36, 44.
• Perkinson Chapter 4 Exercises: 1, 2, 3, 4, 9(a,b,c)*, 10.
• Colley §2.2: 50, 52*, 53*.

### Week 5

• Monday: Read Colley §2.3. Uniqueness of the derivative.
• Tuesday: Read Colley §2.5. The chain rule.
• Wednesday: Exam review session.
• Friday: The chain rule, continued. Distribute first exam (covers material up to end of Week 4).

#### Homework due Friday

• Perkinson Chapter 4 Exercises: 7(c), 7(d), 9(d)*, 14, 15, 18.
• Prove part 3 of Theorem 3.2 in Perkinson Chapter 4.
• Prove part 1 of Theorem 3.4 in Perkinson Chapter 4.*

### Week 6

• Monday: Turn in your exam at the start of class. Read Perkinson §4.7. Partial derivatives.
• Tuesday: Read Colley pp.158-163. Directional derivatives.
• Wednesday: Read Perkinson §4.8. Jacobians and derivatives.
• Friday: Review and exam problems.

#### Homework due Friday

This is another relatively light assignment since you just took an exam. In all of the Colley problems, you may assume that the derivative is induced by the Jacobian.
• Colley §2.3: 43
• Colley §2.6: 2, 8
• Colley p.188: 43
• Perkinson Chapter 4 Exercises: 16, 17(a), 17(b), 17(c)

### Week 7

• Tuesday: Ascent and level activities. Week 7 sage worksheet.
• Wednesday: Read Colley pp.244-254. Taylor's Theorem I.
• Friday: Read Colley pp.255-261. Taylor's Theorem II.

#### Homework due Friday

• Colley §2.6: 10, 15, 25, 37
• Perkinson Chapter 5 Exercises: 1 (fv is Perkinson's notation for ∇v f), 2 (each part graded as a full problem)
• Colley §4.1: 9, 10

### Week 8

• Monday: Taylor approximation redux (now with ∞% more Hessians).
• Tuesday: Read Colley §4.2 over the rest of this week. Optimization I: critical points.
• Wednesday: Optimization II: quadratic forms.
• Friday: Optimization III: extreme value theorem.

#### Homework due Friday

• Colley §4.1: [8, 12], 14, [16], 20, 25, 26, 42 (in order to do this problem, read about Legrange's form of the remainder on p.257 and make sure you understand Example 12).
• Colley §4.2: 4, [6], 16, 22
• Perkinson Chapter 5 exercises: 6, 16 (each part graded separately).

### Week 9

• Monday: Read Colley §4.3. Lagrange multipliers I: intuition.
• Tuesday: Read Perkinson §4.4. Lagrange multipliers II: practice.
• Wednesday: Read Colley pp.168-170. Lagrange multipliers III: proof via the implicit function theorem.
• Friday: Implicit and inverse function theorems.

#### Homework due Friday

• Colley §4.2: 36, 38
• Colley §4.3: 21, 26, 28, 38, 44
• Perkinson Chapter 5 exercises: 21
• [This and the next problem adapted from Shurman §5.3] Do the simultaneous conditions x2(y2+z2)=5 and (x-z)2+y2=2 implicitly define (y,z) as a function of x near (1,-1,2)? If so, then what is the function's best affine approximation?*
• Same question for the conditions x2+y2=4 and 2x2+y2+8z2=8 near (2,0,0).*

### Week 10

• Monday: Read Colley §4.4. Applications of Extrema: physical equilibria.
• Tuesday: Applications of Extrema: least squares approximation, Cobb-Douglas.
• Wednesday: Review session.
• Friday: Read Colley §3.1. Paths and curves. Distribute second midterm.

#### Homework due Friday

• Colley §4.3: 36, 39, 40
• Colley §4.4: 1, 5, 7, 10*, 13
• The multivariable mean value theorem has the following statement (adapted from Folland, Advanced Calculus): Let S be a region in Rn that contains points a and b as well as the line segment L joining these points. Suppose that f is a function defined on S that is continuous at each point of L and differentiable at each point of L except perhaps the endpoints a and b. Then there is a point c on L such that
f(b) - f(a) = ∇ f(c) · (b-a).
Use the single variable mean value theorem to prove the multivariable mean value theorem.
• Use the multivariable mean value theorem to prove that the function G we constructed in the Implicit Function Theorem is of class C1.*

### Week 11

• Monday: Read Colley pp.202-210. Curvature.
• Tuesday: Read Colley pp.211-214. Torsion.
• Wednesday: Twist, link, and writhe.
• Friday: Read Colley pp.214-219. The Frenet-Serret formulas and spline extrusion in 3D graphics.

#### Homework due Friday

• Prove Proposition 1.4 in §3.1 of Colley.
• Colley §3.1: 26, 32.
• Colley §3.2: 9 (you may use sage to sketch the curve), 14, 21, 22, 38.
• Think up an interesting curve in R3. (For the purposes of this exercise, line segments are not interesting. Below are pictures of curves that might be considered interesting.) Find an explicit parametrization of your curve with nonvanishing velocity. Compute the tangent, normal, and binormal vectors of your curve. Modify this sage worksheet* to create an animation of the Frenet frame of your path along with plots of its curvature and torsion. Write a qualitative description of the system (1 or 2 paragraphs), noting any interesting features. Submit your animation, plot, and description via email, but please DO NOT attach files to your email. Instead, provide public access links to the files you've saved in the sage cloud.

*You are also welcome to produce the animation by different means (Mathematica, openGL, ...). If you choose one of these alternate methods, please talk to me about how to submit the final files.

Sources: Lissajous knot, 3D Hilbert curve
• [Bonus] Prove that the torsion of a curve γ(t) is given by the formula
(γ' × γ'') · γ''' / |γ' × γ''|2

### Week 12

We are now embarking on our study of differential equations. Follow this link [pdf] for up-to-date notes on this material.
• Monday: Systems of differential equations.
• Tuesday: Matrix exponentials and operator norms, I.
• Wednesday: Matrix exponentials and operator norms, II.
• Friday: No class. Happy Thanksgiving!

### Week 13

• Monday: Solutions via exponential matrices.
• Tuesday: Properties of the exponential matrix. Examples and computations. Week 13 sage worksheet.
• Wednesday: Problem session and review.
• Friday: Similarity and exponentiation. Distribute course evaluations.

### Week 14

• Monday: Bolzano-Weierstrass revisited.
• Tuesday: No class.
• Wednesday: Norms on Rn are equivalent. Distribute final exam.

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Kyle M. Ormsby