Clark University
                                        Math 131, Multivariate Calculus
Spring 2014
Prof. D. Joyce, BP 322, 793-7421
Department of Mathematics and Computer Science
Clark University

Gauss-Weber Monument
   
Gauss Weber Statue
   
click for source Wilhelm Weber (1804–1891) and Carl Friedrich Gauss (1777–1855)
Gauss-Weber Monument in Göttingen
Date 1899, by Carl Ferdinand Hartzer


Please bookmark this page, http://aleph0.clarku.edu/~djoyce/ma131/, so you can readily access it.

General information


Syllabus

Not all of the topics listed below will be covered in the same depth. Some are fundamental and will be covered in detail; others indicate further directions of study and will be treated as surveys. The only concepts of physics we will study in depth are velocity, acceleration, angular velocity, and angular acceleration, but a few others will be mentioned such as force.

There are probably more topics than we can discuss in one semester, so some will have to be eliminated because of time. Likely candidates are those in brackets.

The listed exercises are tentative. They may be changed as the course progresses.

1. Vectors       4 meetings

2. Differentiation in several variables       8 meetings

Conic helix illustrating curvature

3. Vector-valued functions       5 meetings

4. Maxima and minima in several variables       4 meetings

A solid illustrating a double integral

5. Multiple integration       6 meetings

Helicoid

6. Line integrals       5 meetings

7. Surface integrals       5 meetings

Class notes, quizzes, tests, homework assignments

The dates for the discussion topics and the assignments are tentative. They will change as the course progresses.
Pants

  1. Monday, 13 Jan 2014. Welcome to the class! Outline of the course.
    Things you need to know about linear algebra
    Preview. We’ll study functions Rn → Rm where not both n and m are 1. Three important kinds of these.

  2. Wednesday, 15 Jan.
    Notes on Curves and paths. Velocity, speed, and acceleration
    Quick review of important topics in linear algebra.
    Discuss exercises from sections 1.1: Exercises: 1–11 odd, 15, 23, 25, and 1.2. Exercises: 1–7 odd, 13, 15, 17, 35, 44.

  3. Friday, 17 Jan. More topics from linear algebra including dot products and cross products.
    Dot and cross products
    Discuss exercises from sections 1.3: Exercises: 1–13 odd, 17, 21, 29, 30.
    Review Sets and set notation when we have time.

    Monday, 20 Jan. Martin Luther King day. No classes.

  4. Wednesday, 22 Jan. Discuss functions of several variables including concepts of domain, codomain, range; onto (surjective) and one-to-one (injective) functions, one-to-one correspondences (bijective functions) and their inverses; scalar-valued functions (also called scalar fields), vector-valued functions and their component functions.
    Notes on Functions.
    Homework due from sections 1.4: Exercises: 1, 3, 5, 11–19 odd, 25.
    Discuss section 1.5: Exercises: 1, 3, 5, 13, 23, 25, 27, 31.

    A lyre of Ur

  5. Friday, 24 Jan.
    Graphs of functions as surfaces and described by level curves; surfaces and hypersurfaces
    Limits of functions of several variables including the intuitive concept and formal definition of limits for multivariate functions; topological concepts of open and closed subsets, boundaries of subsets, neighborhoods of points
    A gallery of surfaces
    Notes on Limits.
    Homework due from sections 1.5 (see above).
    Discuss section 1.6: Exercises: 3, 5, 7, 16–19, 21–23, 25, 27, 28a, 30a.

  6. Monday, 27 Jan.
    Properties of limits; multivariate polynomials; continuous functions.
    Partial derivatives
    Notes on Partial derivatives
    Exercises due from section 1.6 (see above).

  7. Wednesday, 29 Jan.
    Tangent planes, total derivatives, gradients for scalar fields Rn → R
    Notes on Gradients
    Exercises due from section 2.1: 1–7, 10, 15–21 odd, 31, 39.

  8. Friday, 31 Jan.
    Notes on total derivatives
    Derivative matrix for vector-valued functions Rn → Rm, rules of differentiation
    Exercises due from section 2.2: 7–13, 29, 30, 38–43, 47, 48.

  9. Monday, 3 Feb.
    Quiz on dot and cross products. Answers
    Higher order derivatives
    Discuss Exercises from section 2.3: 1–10, 15–17, 21, 22, 26–28 due Wed.

  10. Wednesday, 5 Feb. Snow day. No classes.

  11. Friday, 7 Feb.
    Newton’s method
    Chain rule, part 1
    Exercises from section 2.3 due, see above.
    Discuss Exercises due from section 2.4: 1, 2, 9–11, 16, 20, 21a due Monday.

  12. Monday, 10 Feb.
    Chain rule, part 2
    Exercises from section 2.4 due, see above.
    Discuss Exercises from section 2.5: 1, 2, 5, 8, 9, 11, 15–17

    A heart

  13. Wednesday, 12 Feb.
    Directional derivatives, steepest ascent, tangent planes
    Exercises from section 2.5 due, see above.
    Discuss Exercises due Friday from section 2.6: 2, 3, 12, 13, 15, 16

  14. Friday, 14 Feb.
    Speed, velocity, acceleration
    Exercises from section 2.6 due, see above.

  15. Monday, 17 Feb. Review.
    Discuss Exercises from section 3.1: 1–4, 7, 8, 15, 16 due Monday

  16. Wednesday, 19 Feb. First midterm on chapter 2.
    First test. Answers.

  17. Friday, 21 Feb.
    Arclength, the arclength parameter, and parameterizing curves by their arclength
    Kepler’s laws
    The arc length parameter s

  18. Monday, 24 Feb.
    The unit tangent vector and curvature
    Exercises from section 3.1: due, see above

  19. Wednesday, 26 Feb.
    Vector fields, gradient fields, potential functions, equipotential sets, flow lines
    Notes on Vector fields
    Exercises due from section 3.2: 1, 2, 4, 8, 12, 18a.

  20. Friday, 28 Feb.
    Critical points, maxes, mins, saddle points,
    Notes on Critical points
    Exercises due from section 3.3: 1, 4, 9, 10, 19–21, 24, 26. Lorenz attractor

    Monday–Friday, 3–7 Mar. Spring Break

  21. Monday, 10 Mar.
    The del operator, divergence of vector fields, incompressible vector fields, curl of a vector field, irrotational vector fields
    Notes on Gradient, divergence, and curl
    A Lorenz attractor
    Discuss Exercises due Wednesday from section 3.4: 1--4, 7--10, 13, 28a.

  22. Wednesday, 12 Mar.
    Taylor's theorem for a single variable, Taylor polynomials, remainder term; the first-order formula for the multivariate Taylor's theorem, total differentials; the second-order formula and the Hessian.
    Notes on Differentials and Taylor polynomials
    Exercises from section 3.4 due, see above
    Discuss Exercises due Friday from section 4.1: 1, 2, 8, 9, 11, 19, 20, 24

  23. Friday, 14 Mar.
    Local minima and maxima for scalar fields, critical points and the Hessian criterion; quadratic forms, positive definite form; the second derivative test
    Compact sets, the Extreme Value Theorem (EVT)
    Maxima and minima of scalar fields
    Exercises from section 4.1 due, see above.

  24. Monday, 17 Mar.
    Quiz on sections 3.2–3.4.
    Discuss Exercises due Wednesday from section 4.2: 3--6, 13--16
    Lagrange multipliers

  25. Wednesday, 19 Mar.
    An application of Lagrange multipliers to pyramids
    Exercises from section 4.2 due, see above.
    Discuss Exercises due Friday from section 4.3: 3, 4, 5, 9.

  26. Friday, 21 Mar.
    Double integrals. Volumes as integrals, integration over rectangles and other regions, Riemann sums, integrability.
    Exercises due from section 4.3 due, see above. Answers
    Discuss Exercises due Monday from section 5.1 and 5.2

  27. Monday, 24 Mar.
    Fubini’s theorem, linearity of integrals, other basic properties. Double integrals over general regions. Changing order of integration. Triple integrals over boxes.
    Exercises due from section 5.1: 1, 2, 3, 6, 9,
    and from section 5.2: 5, 6, 7, 10, 16.

  28. Wednesday, 26 Mar.
    Jacobians for change of variables. Transformations of the plane R2 → R2 and their expansion factors. Change of variables and the Jacobian for double and triple integrals
    Double integrals in polar coordinates
    Exercises due from section 5.3: 3–6, 15, 17,

  29. Friday, 28 Mar.
    Exercises due from section 5.4: 1, 2, 5, 6, 11, 13, 17 Answers

  30. Monday, 31 Mar. Review.

  31. Wednesday, 2 Apr. Second midterm on chapters 3–5.
    Second test. Answers.

  32. Friday, 4 Apr.
    Introduction to line integrals: scalar line integrals, vector line integrals, and differential forms.
    Line integrals

    Green's theorem

  33. Monday, 7 Apr.
    Green's theorem as a generalization of the fundamental theorem of calculus, Stokes' theorem and the divergence theorem in the plane.
    Green’s theorem

  34. Wednesday, 9 Apr.

  35. Friday, 11 Apr.
    Proof of Green’s theorem
    Exercises from section 5.5: 1, 3, 9, 13, 17 due Answers

  36. Monday, 14 Apr.
    Irrotational vector fields with simply connected domains are conservative.
    Exercises from section 6.1.: 1–3, 9, 16, 17, 34 due Answers

  37. Wednesday, 16 Apr.
    Conservative vector fields: as gradient fields, as vector fields having path independent line integrals, as fields with 0 integrals over closed paths.
    Exercises from section 6.2: 1–3, 7, 9, 15, 17 due

  38. Friday, 18 Apr.
    Surfaces in space, their parameterizations and tangent planes.
    Exercises from section 6.3: 3–6 due

  39. Monday, 21 Apr.
    Scalar surface integrals
    Vector surface integrals
    Exercises from section 7.1: 1, 3, 24, 26 due

  40. Wednesday, 23 Apr. Academic Spree Day.
    More on Stokes' theorem, orientation of surfaces, the statement of the divergence theorem, a.k.a. Gauss's theorem.
    Exercises from section 7.2: 1, 3, 7, 11 due

  41. Friday, 25 Apr.
    Gauss’s theorem
    Exercises from section 7.3: 1, 3, 7, 9 due

  42. Monday, 28 Apr.
Past tests

rule line

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http://aleph0.clarku.edu/~djoyce/ma131/

David E. Joyce,