18.02 | Fall 2007 | Undergraduate

Multivariable Calculus

Lecture Notes

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

I. Vectors and matrices
0 1 2 Vectors

Dot product

Determinants; cross product

Week 1 summary (PDF)
3 4


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.)

Course Info

As Taught In
Fall 2007
Learning Resource Types
Lecture Videos
Problem Sets
Exams with Solutions
Lecture Notes