Vector, Covector and Tensor Fields
Last time I said that our motivation for defining the tangent bundle was so that we'd be able to define smooth vector fields on a manifold, but I didn't quite get there! Let's
differential geometry
A collection of 5 posts
The Tangent Bundle
In multivariable calculus, we have the concept of a smooth vector field, which is an assignment of a vector to each point in $\R^n$ in such a way that the vectors vary smoothly as we move around in space. We would like to do the same on smooth manifolds, but we immediately run into some problems.
Cotangent Spaces and the Pullback
Last time, we defined the tangent space to a smooth manifold at a point. This turned out to be a vector space with the same dimension as the manifold, and tangent vectors were
Tangent Spaces and the Pushforward
In calculus, you learn how to construct tangent lines to differentiable curves at a point. In multivariable calculus, you learn how to construct tangent planes to differentiable surfaces at a point. In differential
Smooth Manifolds
Smooth manifolds are the primary object of study in differential geometry, and are an essential ingredient in general relativity (spacetime is assumed to be a smooth manifold) and other branches of physics.