1 - 4 of 4 results for: MATH 61CM: Modern Mathematics: Continuous Methods

This is the first part of a theoretical (i.e., proof-based) sequence in multivariable calculus and linear algebra, providing a unified treatment of these topics. Covers general vector spaces, linear maps and duality, eigenvalues, inner product spaces, spectral theorem, metric spaces, differentiation in Euclidean space, submanifolds of Euclidean space as local graphs, integration on Euclidean space, and many examples. The linear algebra content is covered jointly with Math 61DM. Students should know 1-variable calculus and have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on the BC-level Advanced Placement calculus exam, or consent of the instructor. This series provides the necessary mathematical background for majors in all Computer Science, Economics, Mathematics, Mathematical and Computational Science, Natural Sciences, and Engineering.

A proof-based introduction to manifolds and the general Stokes' theorem. This includes a treatment of multilinear algebra, further study of submanifolds of Euclidean space (with many examples), differential forms and their geometric interpretations, integration of differential forms, Stokes' theorem, and some applications to topology. Prerequisites: Math 61CM.

A proof-based course on ordinary differential equations. Topics include the inverse and implicit function theorems, implicitly-defined submanifolds of Euclidean space, linear systems of differential equations and necessary tools from linear algebra, stability and asymptotic properties of solutions to linear systems, existence and uniqueness theorems for nonlinear differential equations, behavior of solutions near an equilibrium point, and Sturm-Liouville theory. Prerequisite: Math 61CM or Math 61DM.