Inverse and implicit functions theorems, rank of a map. Regular and critical values. Sard's theorem. Differentiable manifolds, submanifolds, embeddings, immersions and diffeomorphisms. Tangent space and bundle, differential of a map. Partitions of unity, orientation, transversality embed-dings in Rn. Vector fields, local flows. Lie bracket, Frobenius theorem. Lie groups, matrix Lie groups, left invariant vector fields, tensor fields, differential form, integration, closed and exact forms. DeRham's cohomology, vector bundles, connections, curvature.

One-semester review of manipulative algebra, introduction to functions, some topics in analytic geometry, and that portion of trigonometry needed for calculus. Prerequisite: MATH 011 or Placement Exam Part A score above 15. Students with a weak background should take the two-semester sequence MATH 101-102. (Gen.Ed. R1)

Calculus of several variables, Jacobians, implicit functions, inverse functions; multiple integrals, line and surface integrals, divergence theorem, Stokes' theorem.

Basic concepts of linear algebra. Matrices, determinants, systems of linear equations, vector spaces, linear transformations, and eigenvalues. Prerequisite or corequisite: MATH 132, or 136, or consent of instructor. (Gen.Ed. R2)

[Note: Because this course presupposes knowledge of basic math skills, it will satisfy the R1 requirement upon successful completion.]

Basic concepts of linear algebra. Matrices, determinants, systems of linear equations, vector spaces, linear transformations, and eigenvalues. Prerequisite or corequisite: MATH 132, or 136, or consent of instructor. (Gen.Ed. R2)

[Note: Because this course presupposes knowledge of basic math skills, it will satisfy the R1 requirement upon successful completion.]

Basic concepts of linear algebra. Matrices, determinants, systems of linear equations, vector spaces, linear transformations, and eigenvalues. Prerequisite or corequisite: MATH 132, or 136, or consent of instructor. (Gen.Ed. R2)

[Note: Because this course presupposes knowledge of basic math skills, it will satisfy the R1 requirement upon successful completion.]

Maximum credit, 6.

Maximum credit, 6.

Continuation of Math 623. Introduction to functional analysis; elementary theory of Hilbert and Banach spaces; functional analytic properties of Lp-spaces, appli-cations to Fourier series and integrals; interplay between topology, and measure, Stone-Weierstrass theorem, Riesz representation theorem. Further topics depending on instructor.