**Course** Description: | Inverse function theorem and manifolds, integration in several
variables, the change of variables theorem. Differential forms,
line integrals, surface integrals, and Stokes's Theorem;
applications to physics. Eigenvalues, eigenvectors, spectral
theorem, and applications. |

**Course Objectives:** | In addition to learning the customary computational skills in calculus and
linear algebra, the student will be exposed to mathematics as
mathematicians view it. Students will learn to write proofs and think more
rigorously about mathematics, and will come to grips with challenging
concepts and problems. In that regard, this course is an excellent preparation
for students considering a career in law, medicine, or the sciences. |

**Topical Outline:** | 1. Solving nonlinear problems: contraction mapping principle, the inverse and
implicit function theorems, manifolds.
2. Integration: multiple and iterated integrals and Fubini's Theorem; polar,
cylindrical, and spherical coordinates; physical applications.
3. Determinants, n-dimensional volume, and the Change of Variables
Theorem.
4. Differential forms and integration on manifolds: differential forms, line
integrals, Green's Theorem, surface integrals and flux, Stokes' Theorem,
applications to physics and topology.
5. Eigenvalues, eigenvectors, and applications: change of basis formula,
diagonalizability, difference equations, differential equations, and the spectral
theorem for symmetric matrices. |