Math 14

Fall Term 2003

Syllabus

Math 14 is a course in Vector Calculus. It assumes you have a knowledge of representations of lines in two and three space, representations of planes in three space, dot and cross products, partial derivatives, directional derivatives, gradients, the chain rule as it applies to all of these, extreme values of multivariable functions, and Lagrange Multipliers. It assumes you have at least an intuitive idea of limits and continuity of functions of two and three. The course reviews these topics very quickly as they arise.

The course begins with a discussion of the geometry of *n*-dimensional space and an elementary introduction to
linear transformations of *n*-dimensional
space and matrices. It then takes
up limits, continuity, and differentiability of functions from *m*-dimensional space to *n*-dimensional space. It covers the general chain rule and its applications to the
implicit function theorem and the inverse function theorem. It takes up multiple integrals, such
coordinate systems as spherical and cylindrical coordinates, change of variable
in multiple integrals and the relations of these integrals to concept such as
volume and mass. It takes up line
integrals and their relationship with the physical concept of work, Green’s
Theorem, and path independence of line integrals and its relationship to
gradient fields. The course closes
with a discussion of surface integrals and their relationship to multiple
integrals, including a thorough treatment of Stokes’s and Gauss’s
Theorems and the intuitive meaning of the divergence and curl of a vector
field.