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Good morning and welcome back. Today we'll turn to differential forms and that will be the last chapter of the,

the last section of the chapter on differentiable manifolds.

Differential forms are very important in physics because for instance the electromagnetic field strength

is a differential form and the question of whether you can find it like the magnetic potential

will be a question that will concern us.

But in a more general context differential forms also give rise to something called the D'Ram cohomology

of a smooth manifold and the big surprise is that although the definition of the so-called D'Ram cohomology groups

employs the differentiable structure of a smooth manifold it simply tests the underlying topology.

So it gives you a tool to construct topological invariance by using the differentiable structure.

And of course that makes it more easily computable than if you would just touch the topology first hand

and that is why it has such an important role in mathematics.

So this is physically and mathematically important.

It's 4.7 differential forms and D'Ram cohomology.

So what's a differential form? That's something extremely simple.

It's simply a totally anti-symmetric tensor definition.

If we look at the smooth manifold M then an N-form or often one doesn't say differential, a differential N-form

is a 0 N tensor field omega that is totally anti-symmetric.

That means if you consider omega you evaluate it on its N entries and because 0 N tensor field it eats N vectors

X1 to XN and the idea is you pick up a minus sign if you exchange any pair of entries

and more formally that is that if you now consider omega but you permute the vector fields by some permutation pi

so you look at the entries X pi 1 to X pi N then you pick up a minus sign depending on whether the signature of your permutation

is even or odd so this is where pi is a permutation of the first N numbers and obviously X1 to XN they're all vector fields

so they're smooth sections of the tangent bundle. Any such 0 N tensor that has this total anti-symmetry property

is called an N-form. So examples. I think we saw N-forms before because if a smooth manifold M is orientable

now what is orientable? Orientable means that you can make a certain pick of charts so you can restrict an atlas in a certain way

such that the transition functions behave in a certain way. I put this on a problem sheet and if M is orientable

then there exists a nowhere vanishing top form. I think I introduced this notion before a top form and this simply means that

I should say here that we can have 0 less or equal to N less or equal to dim M for these N-forms because obviously

if I have more vectors in here than the dimension of the manifold is then any permute that will always yield 0

so you could have a dim M plus 1 form but it necessarily would have to be 0 by anti-symmetry and it's also clear that

1 forms are not restricted by the anti-symmetry condition. It's void there and 0 form neither but the 0 forms

are the 0 0 tensors are the scalar fields. But now we have a nowhere vanishing top form and top form means that the N

is the maximum, maximal N, N equals dim M omega providing a volume. I think we talked about this before

so that's an example for a differentiable form and so this provides a volume now in every tangent space because it's a field.

B for instance the electromagnetic field strength, field strength F is a 2 form. That's the Faraday tensor.

And finally C in classical mechanics on classical mechanical phase space. Let's be more concrete.

So if Q is a smooth manifold that in classical mechanics describes your configurations of the system

then the phase space is T star Q. It's the cotangent bundle where to every point roughly speaking you attach all the momenta, all the covectors.

So if we're here in classical mechanics in the simplest case and then on T star Q there exists even a canonically defined

so you don't need to define this as extra structures already there on T star Q there exists a 2 form omega which is called the symplectic form.

Now not every 2 form omega is a symplectic form. It needs to have additional properties but I can't write them down yet.

I will only be able to write them down once we develop differential forms a little bit but one of the ideas, one of the facts is that it's a 2 form

and so you have a 2 form that means you have 2 entries. You can switch them and you get a minus sign

and that minus sign that you get is the minus sign in the Hamilton equations. You have the Hamilton equations, right?

Q dot is dh by dp and p dot is minus dh by dq and this minus, the second minus in the Hamilton equations comes from here.

Very mysterious remarks, very simple. I just want to show you this has immediate implications for physics.

Okay, so I will introduce some notation. So the set of all n forms be denoted omega n of m.

So this is first of all a set and which naturally becomes, is whatever, is if you equip it with the appropriate structures,

you inherit the structures from the sections of the tensor bundle is a C infinity m module.