This exam covers topics at the undergraduate level, most of which might be encountered in courses here such as Math 233, 235, 425, 523, 545. Faculty members who teach these courses can recommend texts for review purposes. The emphasis is on understanding basic concepts, rather than performing routine computations. But exam questions often center on concrete examples of matrices, functions, series, etc.
- Vector spaces: subspaces, linear independence, basis, dimension.
- Linear transformations and matrices: kernel and image, rank and nullity, transpose.
- Linear operators: change of basis and similarity, trace and determinant, eigenvalues and eigenvectors, characteristic polynomial, diagonalizable operators.
- Inner product spaces: orthonormal basis, orthogonal complements and projections, orthogonal matrices, diagonalization of symmetric matrices.
- Functions of one real variable: continuity and uniform continuity, derivative and Mean Value Theorem, Riemann integral, improper integrals, Fundamental Theorem of Calculus.
- Sequences and series of numbers or functions: pointwise, uniform, absolute convergence; term-by-term differentiation and integration of series; Taylor’s Theorem with remainder.
- Functions of several variables: continuity, partial and directional derivatives, differentiabilitity, maps from Rn to Rm, Jacobian, implicit and inverse function theorems, chain rule.
- Extrema of functions of several variables: constrained extrema, Lagrange multipliers.
- Multiple and iterated integrals, change of variables formula.
- Vector calculus: gradient, divergence, curl; line and surface integrals; theorems of Green, Gauss, Stokes; conservative vector fields.