Students in the Physical Sciences and Engineering. Applied Math, Engineering and Physics (AMEP) majors.

Math 321 covers vector algebra, vector calculus and an introduction to complex calculus. Math 321 was redesigned to provide mathematical preparation for Physics 311 (Mechanics of points and rigid bodies) and Physics 322 (Electromagnetism) as well as further courses in Mathematics, Physics and Engineering (e.g. Transport phenomena, Fluid and Solid Mechanics, Aerodynamics, Aircraft and Satellite dynamics, Computer graphics, plasma physics, differential geometry, etc.). The course emphasizes understanding of the geometrical concepts and covers vector (Gibbs) notation, index notation (including summation convention) and linear algebra notation. See http://www.math.wisc.edu/~waleffe/M321/ for further information.

Physics 311, Physics 322, Math 322

- Vector Algebra
- Cylindrical, Spherical and Cartesian representation of vectors. Points, coordinates and position vectors.
- Bases, components, linear (in)dependence. Lines and planes.
- Dot product, Kronecker delta, orthonormal bases
- Cross product, Levi-Civita symbol
- Index notation and summation convention.
- Applications of dot and cross products. Mixed products and determinants. Rotation of vectors.
- Rotation and reflection of bases. Orthogonal matrices, Euler angles.
- (Optional) Basic matrix operations and concepts, linear systems of equations and geometrical interpretations.

- Vector Calculus
- Vector functions of a scalar variable
- Applications to motion of a particle, a system of particles, a rigid body
- Curves, Surfaces and Volumes and integrals over them. Examples from mechanics and electromagnetism.
- Coordinate transformations, curvilinear coordinates, cylindrical and spherical coordinates. Jacobian
- Grad, div, curl
- Derivation of vector identities using vector and index notation. Laplacian.
- Divergence and Stokes' Theorems
- Irrotational fields. Solenoidal fields.
- Applications to Electromagnetism and mechanics of continua.

- Complex Variables
- Complex algebra, series, radius of convergence
- Functions of a Complex Variable, branch cuts
- Conformal Mapping, Laplace's equation
- Contour Integration and residue calculus