1. Vector-valued functions. Smooth curves and parameterization.
Parameterization by arc length.
2. Functions of several real variables. Continuity.
Partial derivatives. Taylor expansion.
3 Implicit functions. Finding constrained extrema of functions through
the method of Lagrange multipliers.
4. Multiple integrals. Change of coordinates in integrals. Polar coordinates.
cylindrical coordinates, spherical coordinates. Applications
of multiple integrals to the calculation
of areas, volumes and mean values. Examples of improper integrals.
5. Differential forms . Line integrals and surface integrals.
Divergence Theorem, Green's Theorem, Stokes' Theorem.
Exact differential forms. Closed differential forms.
6. Fourier series. Fourier transform. Plancherel's Theorem.
Applications to differential equations and to some
partial differential equations.