﻿In one lecture you have tensors.
In another lecture you have tangent spaces.
In a third lecture you have bundles.


First is the idea of a tensor. A p,q-tensor is a map that takes a cartesian product of p copies of a covector space, and q copies of a vector space, and bilinearly maps them to a scalar. There is a whole theory around tensors, and it can pretty much be considered as the extension of linear algebra into multiple dimensions: multilinear algebra.


Second idea is tangent spaces. Pretty much you have a manifold. And using curves on that manifold going at different speeds, you consider the directional derivative operator along that curve. Basically a thing that’s looking for a function, and when it gets a function, it calculates the directional derivative at that point, for that function, along that curve. It does this by making everything into a nice function from R->R. If you consider all the possible curves that pass a point p, and you collect all the created directional derivative operators, you can notice that this collection has a structure of a vector space. This vector space of directional derivative operators is called the tangent space.


Third idea is bundles. First we need to talk about the theory of bundles for manifolds. Basically, products of manifolds are very useful. You can take the product of two circles and get a torus, and other interesting things. But some manifolds can’t be expressed as products. Take for example the mobius ring. So instead what we do is develop a theory of bundles: where you can be more careful about describing properties of the global object, and what it means to attach something at every point of the manifold. There’s multiple different levels of the theory of bundles, where you consider a topological bundle, a manifold bundle, and higher or lower levels of bundles, but I’ll focus on manifold bundles. Basically you need three things for a bundle: a base space M, a total space E, and a map down from the total space to the base space, pi. The kind of bundle dependents on restrictions on the map. If the map must be continuous, then it’s a topological bundle. It must be smooth, then it’s a smooth bundle. Bundles are basically smarter ways to attach something at every point. If you attach a vector at every point then its a vector bundle.


If you combine all three of the previous ideas together, then you can get fields. More specifically tensor fields that are tangent at every point of the manifold.


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Tensors. Bundles. Tangent spaces. All three come together beautifully in mathematical physics to describe matter and energy and other things in spacetime. They’re tensor fields. To understand them, we need to understand each individually.


What is a tensor? It’s a multilinear map that takes p covectors and q vectors multi linearly to a scalar in the underlying field.


Ironically enough, people usually say that tensors are the generalization of vectors. But in reality it is vectors that are the generalization of tensors. I mean it’s already questionable to talk about tensors without saying p,q-tensors. And if you do, it dispels a lot of the confusion. Because a p,q-tensor can’t be confused at the generalization of anything, whereas a “tensor” can be confused as the generalization of specific p,q-tensors.


Vectors are not arrows in space. Vectors are elements of a vector space. In fact it’s the same thing with tensors. Tensors don’t exist on their own. Tensors exist as elements of a vector space of p,q-tensors.


So that’s a p,q-tensor. What other theory do you have about tensors? Well to truly form them, you need to understand covectors. Covectors are elements of the dual vector space. Every vector space has a dual vector space. In fact the best way to understand dual vectors is to more generally study linear maps on vector spaces, where you realize that the linear maps from the vectors to the underlying scalars have their own vector space structure, and we call those the dual vectors, in other words the functionals. The linear functions that take in a vector and spit out a number linearly. Physical examples can help with intuition.


If you have bananas that are vectors for example. Say 1 banana, 2 bananas, 3 bananas. Then the covectors in that system are linear functionals that say 2$/banana, or say 3$/banana, or maybe 10$/banana. The space of all possible exchange rates is the dual space. A specific choice of exchange rate is an example of a dual vector. 2$/banana is a dual vector because it is a linear functional.


Imagine you consider all the linear maps from V to W and call that set of linear maps Hom(V,W). We can easily define and + and * pointwise, which allows us to make Hom(V,W) into a vector space itself. Now this is the general theory. But we can still develop interesting specific cases, in fact most of the interesting things happen in the specific. For example End(V), which means Hom(V,V), is clearly valuable. As is Aut(V), which is the automorphisms on the vector space, which basically just requires that the linear map is invertible.


But the most interesting case of all is when you have Hom(V, K), where K is the underlying field, and this is the case of dual vectors in the dual vector space.


But let’s talk more about tensors. When you have a specific p,q-tensor, you can analyze the tensor through the lense of its components, since a tensor is uniquely defined by the play on its components. And if you align it’s components in a specific way, you get the classical understanding of vectors and linear transformations and covectors as column vectors, matrices, and row vectors.


So to be clear, all the simplified versions of linear algebra all depend on the choice of basis, which is sometimes necessary, although usually makes proving things easier.


But yeah you have a 1,1-tensor for example, which is a bilinear map that takes a vector and a covector to a scalar bilinearly.


This map can also be interpreted as an endomorphism (a map that takes a vector to a vector).


Double duals: how do you go from covectors to scalars? You apply the covector to a vector.


How does a vector become a double dual vector? It goes out to beg for a covector.