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
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.
Permutations and Parity
In my next post, I would like to introduce a very special type of tensor whose properties are invaluable in many fields, most notably differential geometry. Although it's possible to understand antisymmetric tensors
Tensor Products and Multilinear Maps
If you're the sort of person who cowers in fear whenever the word "tensor" is mentioned, this post is for you. We'll pick up right where we left off last time in our discussion of the dual space, and discover how tensor products are a natural extension of the ideas developed in that post.
Dual Spaces
Since I haven't posted for a while, I decided to break up my rants about homology with some posts on linear (and multilinear) algebra. In this post, we will (as usual) deal only with finite dimensional vector spaces.
Simplicial Homology
Homology groups are topological invariants which, informally, give information about the types of holes in a topological space. They are not the only such invariant in algebraic topology, but they are particularly nice to work with since they are always abelian and easy to compute.
Simplicial Complexes and Boundary Maps
The idea behind our definitions is that lots of topological spaces can be "triangularized" in such a way that they look sort of like a bunch of "triangles" glued together.
Matrices for Linear Maps
If you already knew some linear algebra before reading my posts, you might be wondering where the heck all the matrices are. The goal of this post is to connect the theory of linear maps and vector spaces to the theory of matrices and computation.
Linear Maps
Just as we decided to study continuous functions between topological spaces and homomorphisms between groups, much of linear algebra is dedicated to the study of linear maps between vector spaces.