INTRODUCTION TO MANIFOLDS

MATH 761

Differentiable manifolds, vector bundles, implicit function theorem, submersions and immersions, vector fields and flows, foliations and Frobenius theorem, differential forms and exterior calculus, integration and Stokes' theorem, De Rham theory, Riemannian metrics. Familiarity with basic undergraduate analysis coursesMATH 521and522orMATH 621) is strongly recommended.

MATH 375

Vector spaces and linear transformations, differential calculus of scalar and vector fields, determinants, eigenvalues and eigenvectors, multiple integrals, line integrals, and surface integrals. Freshmen students are invited to enroll by the Department of Mathematics.

MATH 321

Vector analysis: algebra and geometry of vectors, index notation, 3D rotations, Euler angles; vector differential and integral calculus, theorems of Green, Gauss, and Stokes; grad, div, curl, and Laplacian in cylindrical and spherical coordinates; Complex analysis: analytic functions, Cauchy-Riemann equations, conformal mapping, complex integrals, residue theorem.

MATH 222

Techniques of integration, improper integrals, first order ordinary differential equations, sequences and series, Taylor series, vector geometry in two and three dimensions.

MATH 856

The theory of differential manifolds such as differential forms and de Rham theorem, cobordism groups, Lie groups, homogeneous spaces, fiber bundles, characteristic classes.