Mathematical Foundations I: Manifolds and Tensor Calculus

Define a smooth manifold using charts and atlases, and provide examples such as the n-sphere.

Define the tangent space at a point and construct vector fields as derivations.

Define the cotangent space (dual space) and construct one-forms.

Define a general (r,s) tensor as a multilinear map and explain the tensor product.

Define the metric tensor and its signature, and use it to raise and lower indices.

Define the Lie derivative of a tensor field with respect to a vector field.

Define differential forms and the exterior derivative operator.

Define integration on manifolds using volume forms.