Mathematics and Statistics

## Syllabus for Multivariable Calculus

Elementary vector analysis:
Scalar or dot product
Cross or vector product
Lines and planes
Differentiation of vector-valued functions
Tangent vectors
Velocity and acceleration

Definitions involving functions of several variables:
Limit
Continuity
Differentiability
Partial derivative
Directional derivative
Tangent plane to a surface

Computations using the usual rules of differentiation:
Partial derivatives
Directional derivatives
Tangent planes to surfaces

Computations and proofs using the definitions:
Prove continuity (or lack thereof) of a function at a point
Compute partial derivatives and directional derivatives from the definition

Maxima and minima of functions of several variables:
Finding critical points
The second derivative test for local maxima/minima and saddle points
The method of Lagrange multipliers

Double integrals:
Cartesian and polar coordinates
Finding area, surface area and volume

Triple integrals:
Cartesian, cylindrical and spherical coordinates
Finding volume and mass

Line integrals:
Basic properties
Green's Theorem