Multivariable Calculus

Lecture Notes

The notes below represent summaries of the lectures as written by Professor Auroux to the recitation instructors.

LEC #	TOPICS	LECTURE NOTES
I. Vectors and matrices
0 1 2	Vectors Dot product Determinants; cross product	Week 1 summary (PDF)
3 4 5	Matrices; inverse matrices Square systems; equations of planes Parametric equations for lines and curves	Week 2 summary (PDF)
6	Velocity, acceleration Kepler’s second law	Week 3 summary (PDF)
7	Review
II. Partial derivatives
8	Level curves; partial derivatives; tangent plane approximation	Week 4 summary (PDF)
9	Max-min problems; least squares
10	Second derivative test; boundaries and infinity
11	Differentials; chain rule	Week 5 summary (PDF)
12	Gradient; directional derivative; tangent plane
13	Lagrange multipliers
14	Non-independent variables	Week 6 summary (PDF)
15	Partial differential equations; review
III. Double integrals and line integrals in the plane
16	Double integrals	Week 7 summary (PDF)
17	Double integrals in polar coordinates; applications
18	Change of variables	Week 8 summary (PDF)
19	Vector fields and line integrals in the plane
20	Path independence and conservative fields
21	Gradient fields and potential functions	Week 9 summary (PDF)
22	Green’s theorem
23	Flux; normal form of Green’s theorem
24	Simply connected regions; review	Week 10 summary (PDF)
IV. Triple integrals and surface integrals in 3-space
25	Triple integrals in rectangular and cylindrical coordinates	Week 10 summary (PDF)
26	Spherical coordinates; surface area	Week 11 summary (PDF)
27	Vector fields in 3D; surface integrals and flux
28	Divergence theorem
29	Divergence theorem (cont.): applications and proof	Week 12 summary (PDF)
30	Line integrals in space, curl, exactness and potentials	Week 13 summary (PDF)
31	Stokes’ theorem
32	Stokes’ theorem (cont.); review
33	Topological considerations Maxwell’s equations	Week 14 summary (PDF)
34	Final review
35	Final review (cont.)