Eric Shapiro

38 posts
Feb 23, 2020

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

Feb 22, 2020

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.

Feb 21, 2020

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

Feb 12, 2020

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

Feb 10, 2020

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.

Nov 12, 2019

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

Aug 13, 2019

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.

Aug 8, 2019

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.

Apr 14, 2019

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.

Mar 21, 2019

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.

Mar 20, 2019

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.