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.
Athena Title
Multivariable Mathematics II
Equivalent Courses
Not open to students with credit in MATH 3510H
Prerequisite
MATH 3500 or MATH 3500H
Semester Course Offered
Offered spring
Grading System
A - F (Traditional)
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.
Syllabus