Spring 2009

## Syllabus

The following is a tentative syllabus for Math 14, Spring 2009. The actual topics covered will depend upon interest and pace.
The rough outline of the course is as follows. Multivariable functions and their derivatives, vector fields, gradient, divergence, curl, double and triple integrals, parametrized curves, flows, line integrals, Green's theorem, and flux integrals. (The topics in Math 61, the half-course that continues Math 14 in the sophomore year, will cover optimization, Taylor's theorem, Lagrange multipliers, Stokes' Theorem, and the Divergence Theorem.)

## Lecture topics

Week 1
1. Introduction, overview, scalars/vectors, distance, lines, dot product review
Colley 1.1-1.3
2. dot products: alg/geom defns, application: rhombus diags projections, cross products: alg/geom defns, area of triangle, applications to work, torque, eqn of planes, dist between planes
Colley 1.4-1.5
Week 2
3. multivariable functions, examples, how to visualize: graphs/sections/level sets, examples of surfaces: paraboloids, hyperboloids, etc.
Colley 2.1
4. functions as mappings, meaning of limits of m-v functions, examples where limits do not exist, what continuity means
Colley 2.2
5. partial derivative defn, examples, meaning, tangent plane, when does tangent plane exist, "differentiable" means good approximation exists, the derivative matrix
Colley 2.3
Week 3
6. derivative matrices and meaning of differentiability, example, meaning: best linear approximation in a small nbhd, 2nd partials and meaning
Colley 2.4
7. mixed partials, when and why equal, chain rule: tree diagrams, matrix multiplication
Colley 2.5
Colley 2.6
Week 4
9. param paths, curves, velocity, accel, circles, ellipses, cycloids, arclength
Colley 3.1 [minus Kepler], 3.2 [only arclength]
Take Home Exam handed out
10. vector fields, flow lines,
Colley 3.3
11. grad, div, curl, del operator: examples, defn/meaning
Colley 3.4
Week 5
12. double integrals, meaning/properties, iterated integrals, Cavalieri
Colley 5.1-5.2
13. more examples, Fubini, switching the order of integration, type I/II/III regions
Colley 5.3
14. triple integrals, meaning, examples, choosing an order of integration
Colley 5.4
Week 6
15. polar, cylindrical, spherical integration, volume elements for each
Colley 1.7, 5.5 [only the special cases of cylinders/spheres!]
16. more cylindrical/spherical examples, which to use?
Colley 5.5 [only the special cases!]
17. line integrals: scalar/vector, mass of wire, area of fence, work
Colley 6.1
Week 7
18. Green's Theorem
Colley 6.2
19. conservative vector fields, fundamental theorem of line integrals,
Colley 6.3