I’ve spent a little bit of time over the last few weeks casually filling in some of the gaps unresolved by the Cambridge tripos syllabus (much of which is excellent, but does tend to be taught within what I am beginning to believe is a generally quite old-fashioned way).

In “applied” lectures in the first year, one is introduced to the curious trio Grad, Curl and Div, who perform various magical tricks which then prove to be incredibly handy for solving problems in electromagnetism and fluid mechanics (or anywhere else where there is some kinds of generic ‘stuff’ flowing around in 3-dimensional space). At no point does anyone explain to the audience how the tricks are done, and this may be to some extent because the mechanisms are quite complicated. In this series of posts, while avoiding going into too much gory detail, I will summarise a few of their more spectacular illusions and hint at the reality that lies beneath (much of the surface of which is scratched in any course on “Complex Analysis”).

**Trick 1: Being defined at all**

I suspect I was not alone in suspecting that Curl was a rather strange member of this family of derivatives. The idea of defining this weird antisymmetrised thing definitely felt quite idiosyncratic and not much to do with the other two operators. It was also obvious how the other two operators would behave in higher dimensions, while generalising the idea of curl to 4-dimensions doesn’t seem so easy.

A more general way to define them is to use the language of *differential forms*. These are really abstract objects which should be thought of “things you want to integrate”. In fact, if we are in some n-dimensional space , for any we define a p-form to be something we will integrate over a ‘p-dimensional subset’ of the space. We assume the 0-forms are just the smooth functions on (with ‘0-dimensional subsets’ being ‘finite sets of points’ and ‘integration’ being ‘evaluation at’). In fact, if we have some co-ordinates ( for points in ) we can define the 1-forms over the space as simply the objects

where are smooth functions of space.

Then for any curve , the integral is defined in the natural way

.

When we go to higher dimensions, p-forms tend to pick up an antisymmetric character (as is familiar from the fact that *volume forms* are alternating, and when we learn how to integrate over more than one dimension we frequently multiply by a Jacobian, which is the determinant of a matrix). This motivates the definition of the (non-commutative) **wedge product:**

If are a p-form and a q-form respectively, then is a (p+q)-form, and we insist the wedge product is bilinear and associative.

This sounds a little abstract, but it can be seen to be well-defined and sensible by looking at a concrete example. Suppose we are in 2-dimensional space, and . Then applying the above axioms gives , exactly the determinant-like expression we would expect when passing to an area. Note also that if we just multiply through by a smooth function (a 0-form) this can be interpreted as wedge multiplication.

Now comes the clever part (again an abstract definition that turns out to extend to a very well-defined concept). The **exterior derivative **of a p-form is a (p+1)-form with the derivation operation satisfying the properties:

- d is linear.
- If is a p-form, any differential form, (d is “wedge-Leibniz”).
- (or ) (d is “cohomological”).

Though d is defined in this axiomatic way, in concrete situations it really is distinct maps, one for each increase in dimension, and they might take on apparently different concrete characters, but are ultimately instances of the same thing: differentiation, all the time with a view to later integration.

Let us just examine the case where is 3-dimensional space, and see what the three exterior derivatives are (you should be able to see where this is going…).

Firstly, taking exterior derivative of the zero forms turns out simply to be the map

.

In other words, it takes a scalar function and gives us a covector of partial derivatives. Up to worrying about whether or not it was originally an element of the dual space, this is precisely Grad.

What now happens at the next stage? Here the ‘cohomological’ property comes into play, along with the others (since is of course the 1-form obtained by taking the exterior derivative of a “co-ordinate” function).

which expands to

This is just the Curl we know and love. Since the spaces of 1-forms and 2-forms both have dimension 3 here, it can be identified with a vector in the space again, but we should bear in mind it is actually in some sense really just an object we want to integrate over an area.

Finally, what happens when we pass from “things to be integrated over areas” to “things to be integrated over volumes”? You’ve probably guessed correctly, but it’s worth checking anyway:

(since all the other terms are eliminated by the alternating property of wedge products of 1-forms, or by the cohomological property of d)

These results are not only interesting because we have managed to abstractify in total generality a few things of which we previously only knew special cases(we can now easily derive the 57-dimensional analogues of the 3 vector calculus derivatives of 3-dimensions if we were so minded). We now know that our perspective on them was previously incomplete, in that whenever we were taking a curl we inadvertently were interpreting our original object as something which exists as integrable in 1-dimension and turning it into something which exists as integrable in 2-dimensions. The deeper implications of this, including Stokes’ theorem (the more general form of which an astute reader might be able to hazard a guess at now) will be explored in future posts.

Two more remarks. The notion of integration with differential forms entirely removes (or abstracts) the mysterious Jacobian from the calculations. In converting between co-ordinates (changing the basis of our p-forms), it will be automatically encorporated in the resulting recasting of the differential form being integrated as a straightforward change-of-basis matrix. Secondly, the identities are revealed here to be special cases of precisely what I have called the “cohomological condition” on the exterior derivative, and not actually particularly connected with the antisymmetries of the wedge product (contrary to what one might assume after proving these identities using suffix notation in the 3-dimensional case).

With this closing act, we shall take a break, and come back with two far more daring magic tricks after the intermission.

## 4 comments

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August 3, 2010 at 9:27 pm

ThomasHi Tom,

thanks for this post! I never got around studying differential forms in detail so far – and I haven’t really needed them, doing functional analysis mostly.

But say, does the rule for deriving 0-forms follow from the axioms, or is it another definition?

Cheers,

Thomas

August 7, 2010 at 1:56 am

GauravThanks for posting this!

August 25, 2010 at 11:22 am

tloveringI believe that provided everything is smooth, and we are working on a manifold, the rule for deriving 0-forms is determined by the axioms (i.e. there is a unique derivation operation satisfying the axioms – the obvious one).

A google search revealed http://www.springerlink.com/content/j6283766681450x7/fulltext.pdf as a reasonable reference.

January 21, 2011 at 4:39 pm

DZSo it wasn’t just me who learnt differential forms over the summer.

Strange they don’t teach it, the only hard part to prove (that I remember) is change of variables in multiple dimensions.