Clark University
        Math 131, Multivariate Calculus
Spring 2010
Prof. D. Joyce
BP 322, 793-7421
Department of Mathematics and Computer Science
Clark University

[This course web page is obsolete.  I'll prepare a new one next time I teach 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.

1. Vectors       4 meetings

2. Differentiation in several variables       8 meetings

3. Vector-valued functions       5 meetings

4. Maxima and minima in several variables       4 meetings

5. Multiple integration       6 meetings

6. Line integrals       5 meetings

7. Surface integrals       5 meetings

Sample tests

Class notes, quizzes, tests, homework assignments

  1. Wednesday, 20 Jan 2010. Welcome to the class! Outline of the course.
    Things you need to know about linear algebra
    Diagnostic test. Answers
  2. Friday, 22 Jan. Quick review of important topics in linear algebra.
    Discuss exercises from sections 1.1–1.2.
  3. Monday, 25 Jan. More topics from linear algebra including dot products and cross products.
    Discuss exercises from sections 1.3–1.4.
  4. Wednesday, 27 Jan. Notes.
    Discuss functions of several variables including concepts of domain, codomain, range; onto (surjective) and one-to-one (injective) functions, one-to-one correspondences (bijective fuctions) and their inverses; scalar-valued functions (also called scalar fields), vector-valued functions and their component functions.
    Discuss exercises from sections 1.5–1.6.
  5. Friday, 29 Jan. Notes.
    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
    Discuss exercises from section 1.7.
  6. Monday, 1 Feb. Notes.
    Properties of limits; multivariate polynomials; continuous functions.
    Partial derivatives
    Exercises due from section 2.1: 1–8, 11–17 odd, 27, 35.
  7. Wednesday, 3 Feb. Notes.
    Tangent planes, total derivatives, gradients for scalar fields Rn → R
    Exercises due from section 2.2: 7–13, 29, 30, 34–39, 43, 44.
  8. Friday, 5 Feb. Notes.
    Derivative matrix for vector-valued functions Rn → Rm, rules of differentiation
    Quiz on sections 2.1 and 2.2. Answers
  9. Monday, 8 Feb. Notes.
    Higher-order partial derivatives. Newton's method. The chain rule.
    Newton Basin in C at http://aleph0.clarku.edu/~djoyce/newton/newton.html
    Exercises due from section 2.3: 1–10, 15–17, 21, 22, 26–28
  10. Wednesday, 10 Feb. Notes.
    More on the chain rule. Polar/rectangular conversions for partial derivatives.
    Exercises due from section 2.4: 1, 2, 9–11, 16, 20, 21a
  11. Friday, 12 Feb. Notes.
    Directional derivatives, steepest ascent/descent, tangent planes.
  12. Monday, 15 Feb. Notes.
    Paths, velocity, speed, acceleration; tangent line, tangents on a circular path; vector product differentiation rules; Kepler's laws of planetary motion.
    Exercises due from section 2.5: 1, 2, 5, 8, 9, 11, 15–17
  13. Wednesday, 17 Feb. Notes.
    More on Kepler.
    Exercises due from section 2.6: 2, 3, 12, 13, 15, 16
  14. Friday, 19 Feb. Arclength and the arclength parameter s. (See notes from Wednesday.)
  15. Monday, 22 Feb. Review.
    Exercises due from section 3.1: 1–4, 7, 8, 15, 16.
  16. Wednesday, 24 Feb.
    First test. Answers.
  17. Friday, 26 Feb. Notes.
    Reparametrizing a path by its arclength, the unit tangent vector T of a path, and the curvature of a curve.
  18. Monday, 1 Mar. Notes.
    Introduction of vector fields F : Rn → Rn, the gradient field (a vector field) associated to a potential function (a scalar field), equipotential sets, and flow lines of vector fields
    Exercises due from section 3.2: 1, 2, 4, 8, 12, 18a.
  19. Wednesday, 3 Mar. Notes.
    The del operators, gradient, divergence, and curl. Gradient fields are irrotational, that is, curl(grad f) = 0. Div(curl F) = 0.
  20. Friday, 5 Mar. Notes. The Lorenz attractor as an example of a vector field with chaotic flow lines.
    Exercises due from section 3.3: 1, 4, 9, 10, 19–21, 24, 26.

    Monday–Friday, 8–12 Mar. Spring Break

  21. Monday, 15 Mar. Notes.
    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.
    Exercises due from section 3.4: 1–4, 7–10, 13, 28a.
  22. Wednesday, 17 Mar. Notes.
    Local minima and maxima for scalar fields, critical points and the Hessian criterion; quadratic forms, positive definite form; the second derivative test
  23. Friday, 19 Mar. Notes.
    Compact sets, the Extreme Value Theorem (EVT)
    Quiz on sections 3.2–3.4. Answers
    Exercises due from section 4.1: 1, 2, 8, 9, 11, 15, 16, 18.
  24. Monday, 22 Mar. Notes.
    Lagrange multipliers
    Exercises due from section 4.2: 3–6, 13–16.
  25. Wednesday, 24 Mar. Notes.
    Volumes as integrals, integration over rectangles and other regions
  26. Friday, 26 Mar. Notes.
    Double and triple integrals, parameterizing domains of integration.
    Exercises due from section 4.3: 3, 4, 5, 7.
  27. Monday, 29 Mar. Notes.
    Change of variables, the Jacobian.
    Exercises due from section 5.1: 1, 2, 3, 6, 9, ; and from section 5.2: 3, 4, 5, 6, 12.
  28. Wednesday, 1 Apr. Notes.
    Exercises due from section 5.3: –6, 15, 17, ; and from section 5.4: 1, 2, 5, 6, 11, 13, 17.
  29. Friday, 3 Apr. Review.
    Exercises from section 5.5: 1, 3, 9, 13, 17.
  30. Monday, 5 Apr.
    Second test. Answers.
  31. Wednesday, 7 Apr. Notes.
    Introduction to line integrals: scalar line integrals, vector line integrals, and differential forms.
  32. Friday, 9 Apr. Notes.
    Green's theorem as a generalization of the fundamental theorem of calculus, Stokes' theorem and the divergence theorem in the plane.
  33. Monday, 12 Apr. Notes.
    Proof of Green's theorem.
    Exercises from section 6.1: 1–3, 7, 10, 11, 20.
  34. Wednesday, 14 Apr. Notes.
    Conservative vector fields: as gradient fields, as vector fields having path independent line integrals, as fields with 0 integrals over closed paths.
  35. Friday, 16 Apr. Notes.
    Irrotational vector fields with simply connected domains are conservative.
    Exercises from section 6.2: 1, 2, 3, 5, 7..
  36. Monday, 19 Apr.
    Surfaces in space, their parameterizations and tangent planes.
    Exercises from section 6.3: 3–6.
  37. Wednesday, 21 Apr. Notes.
    Surface differentials, surface areas, and scalar surface integrals.
  38. Friday, 23 Apr. Notes.
    Exercises from section 7.1: 1, 3, 20.
  39. Monday, 26 Apr. Notes.
    More on Stokes' theorem, orientation of surfaces, the statement of the divergence theorem, a.k.a. Gauss's theorem.
    Quiz on sections 6.1 –6.2. Answers
  40. Wednesday, 28 Apr. Examples of surface integrals.
    Exercises from section 7.2: 1, 3, 7, 11.
  41. Friday, 30 Apr. Notes.
    More on Gauss's theorem.
  42. Monday, 3 May. Review.
    Exercises from section 7.3: 1, 3, 7, 9.

This page is located on the web at

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

David E. Joyce,