Fall Term 2003
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.