Review of multivariable calculus: Differentiability of functions of several variables, Jacobian, inverse

function theorem, implicit function theorem.

Topological manifolds, differentiable manifolds, examples of differentiable manifolds, manifolds with

boundary.

Differentiable functions on manifolds, partition of unity, tangent vectors and tangent space, differential

of a smooth map.

Rank of differentiable functions, immersion, submersion, embedding, submanifolds.

Vector fields, flows, exponential maps, Lie groups, Lie algebras.

Multilinear algebra, symmetric and alternating tensors, tensors and tensor fields on manifolds, the

exterior derivative, Lie derivatives.

Orientations of manifolds, the geometry of volume measurement, integration of differential forms,

Stokes’s theorem.