So after taking MVC, we’ve all been through those tedious proofs that and . Here, we give a unified way to view these identities.

We start by giving the notion of a tangent space of a point in , which is a vector at that point. Denote this with , which also has the convenient notation . We can then take the union of all these tangent spaces and call it the tangent bundle .

We can view tangent vectors and vector fields as differential operators for a function. The directional derivative of a function at a point in direction is , where . After some crunching, we arrive at

We then naturally extend this to vector fields, namely

Now we can talk about differential forms. Recall that the total differential of a function is

We now define a differential 1-form to be a function such that when we restrict to a certain point and consider , is a linear functional. It turns out that we can equip 1-forms with function coefficients. If is a 1-form, then . We also let the standard set of differential 1-forms to be the set of functions , so in , reads out the 2nd coordinate of . Then the total differential becomes a very natural 1-form.

If we have 1-forms, what about what about 0-forms and 2-forms? We can naively let 0-forms be the set of functions, and higher dimensional forms to be some kind of product of lower dimensional forms, with some sort of derivative operation that moves from one form space to the next. It turns out that when we do this, we get a lot of nice things. We can define a new product of differential forms, called the wedge product, such that , and takes in more tangent vectors and linear in each component when you fix a point. Then if is a 1-form, we can write , and define analogously for higher dimensional spaces (here we took ) and forms. If we crunch out the products, we learn that , or if is a differential form, then . From this, we can construct an interesting diagram.

where

are isomorphisms and denotes the space of -forms.

Construct a similar chain of morphisms to get good at -dimensional calculus where .