## Syllabus

### This is an advanced course in multivariate calculus. This course covers the same subjects (roughly) as MA 18 and MA 20, but is on a higher level, and is more difficult.

Topics will include:

**Differentiation**

- Brief survey of Euclidean geometry, scalar and vector products.
- Multivariate functions: graphical representation (surfaces), continuity.
- Differentiation in two and three dimensions: partial derivatives, directional derivatives.
- Gradients, tangent lines and planes.
- Extremal problems.
- Lagrange multipliers and constraints.
- Higher order derivatives and Taylor's theorem.
- The implicit function theorem, the inverse function theorem.

**Integration**

- Brief survey of one dimensional integration.
- Integration in two dimensions: Cartesian, polar.
- Fubini's theorem.
- Integration in three dimensions: Cartesian, cylindrical, spherical.
- Change of variables: the Jacobian.
- Geometrical applications: solid volumes, surface area, center of mass.

**Vector analysis**

- Vector valued functions.
- The divergence and the curl of a vector field.
- Line integrals in two and three dimensions.
- Green's theorem (in two dimensions).
- Surface integrals.
- Divergence theorem (Gauss' theorem).
- Stokes' theorem.