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De Rham Theory

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Intuitive cohomology

Now we can marry the two lines of argument and arrive at one
of the most beautiful and useful advanced mathematical area, de Rham
cohomology. I cannot cover this in only one post, and there will be some back
to physics posts (one or more guest posts) to prevent the math topics to become
too dry. The end goal of the math series is to be able to talk about Yang-Mills
theory and the Standard Model, so there is light (physics) at the end of the
tunnel.

Let us start gentle into the topic and consider the real 3
dimensional space. Let us talk about differential forms (the stuff that you **pull back**). We have 0-forms, 1-forms,
2-forms, and 3-forms. On R^{3} there are no 4 or higher forms? Why? Read
on…

What 0-forms might be? They are simply the usual functions.
Let’s call them *f.*

Take the differential of a function:

d*f* =∂*f*/∂x dx + ∂*f*/∂y dy + ∂*f*/∂z dz

and you get the

**gradient
**which is a 1-form. 1-forms are isomorphic with vector fields X:

*f*_{1} dx +
*f*_{2} dy + *f*_{3 }dz ~ X = (*f*_{1}, *f*_{2} , *f*_{3})

Take the differential of 1-form and you get 2-forms (the

**curl**):

d ( *f*_{1} dx
+ *f*_{2} dy + *f*_{3 }dz ) =

+(∂*f*_{3}/∂y - ∂*f*_{2}/∂z)
dy dz

– (∂*f*_{1}/∂z - ∂*f*_{3}/∂x)
dx dz

+ (∂*f*_{2}/∂x - ∂*f*_{1}/∂y)
dx dy

Then take the differential of 2-forms and you get a 3-form
(the

**divergence**)

d ( *f*_{1} dy
dz - *f*_{2} dx dz + *f*_{3 }dx dy ) = (∂*f*_{1}/∂x + ∂*f*_{2}/∂y + ∂*f*_{3}/∂z)
dx dy dz

The gradient, curl, and divergence are the bread and butter
of Maxwell’s equations in college.

When working with differentials remember 2 rules:

- (dx dx)
= 0
- dx dy
= - dydx

Now
the key idea is the d d = 0 and we are getting somewhere interesting. **We will be able to study topology by
investigating partial differentials equations. **Mathematically this is very
surprising because it reveals a bridge between two very different domains. In
topology you deal with accumulation points, closed and open sets, while in differential
equations you encounter say Schrodinger equation. What can Schrodinger equation
tell you about open sets?

Physically
on the other hand it is not surprising at all!!! You know that a violin sounds
like a violin, and a drum like a drum. This is because the solution to the wave
equation in a cavity depends on the shape of the cavity. By the way, this is
why I like physics: it provides an easy context for intractable mathematical
abstractions.

So from above we have:

- d(0-forms)
= gradient
- d(1-forms)
= curl
- d(2-forms)
= divergence

What is the curl of a gradient? What is the divergence of a
curl? They are zero because dd = 0 (recall how long it took to prove those
things in college?)

Now suppose we have a vector field **v**

and we ask if it is a gradient of some potential:

**v**= grad (A)

We know that curl (grad (A)) = 0 and *locally* this is always obeyed, but not *globally*. If we have a loop (recall homotopy) the integral of **v** along the loop __depends only on the
homotopy class__ and the following vector space:

**v **for which curl
(**v**) = 0 BUT **v** is not grad (A) for some A.

is the same as the homotopy class!

In general we can define the following p-th **de Rham cohomology **(this is a vector
space)**:**

** α | d α = 0 **

H^{p}_{dR} =
--------------------------------

** α | α = dβ**

where **α **is a
p-form and **β **is a (p-1)-form.

In other words: __closed
forms which are not exact.__ This is the same Ker/Image pattern we encountered
in homology. To be continued…