A review of continuity and differentiability in more than one variable. The inverse, implicit, and constant rank theorems. Definitions and examples of manifolds, maps between manifolds, regular and critical values, partition of unity, Sard’s theorem and applications. Tangent spaces and the tangent/cotangent bundles, definition of general vector bundles, vector fields and flows, Frobenius’ theorem. Tensors, differential forms, Lie derivative and the exterior derivative, integration on manifolds, Stokes’ theorem. Introduction to de Rham cohomology.