How Can Differential Forms Be Used to Compute Areas and Volumes?

[zpconn]

I've just begun investigating differential forms. I have no experience in this field and no formal, university level training in mathematics, so please bear with me.

I understand that a differential form may be thought of as a family of linear functionals; more precisely, it is a function that assigns to each point of a manifold a linear functional (i.e., a member of the cotangent space at that point) mapping a tuple of k tangent vectors to a real number.

This is all good for me. My confusion arises with regard to what sorts of quantities can actually be computed using forms.

For example, I understand that, if we take our manifold to be the xy-plane in ordinary Euclidean space, a two-form, say A dx dy where A is constant, can be evaluated on a geometrical figure in this plane to give its oriented area up to the factor A (and by "evaluated" I really mean integrated over the figure). But what if I wanted to compute the surface area of a surface in general xyz-space, say the upper-hemisphere of the unit sphere? This is straightforward in vector calculus using surface integrals, but I can't see how this could be accomplished using differential forms.

To make my difficulty plain, consider an analogous but simpler problem. To find the arc length of a curve in the xy-plane, we would need to integrate the quantity $$\sqrt{dx^2 + dy^2}$$ over the curve in the language of differential forms. Clearly the product here cannot be the Grassmann (wedge) product, since those terms would then vanish. But even if a suitable definition were provided for the products, the result doesn't seem like it'd be a differential form.

My problem is that it seems to me that differential forms naturally compute areas, volumes, etc, within a tangent space, but for computing such quantities over a portion of a manifold forms are not sufficient--the use of some more general object becomes necessary.

I'd be very grateful for anyone who can help to correct any misunderstandings I might have. I'm learning this subject on my own, just for the love of it. This isn't for a class or anything.

 

[zpconn]

I've given this some more thought. I was thinking that maybe if the curve were parameterized by t, say $$x = x(t), y = y(t)$$, then the arc length could be obtained by integrating $$\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2} dt$$ over the curve. This looks like a one-form to me as far as I can tell, where the dx/dt and dy/dt are just the formal symbols for derivative and are not related to forms in any way. In other words, the integral has been redefined as an integral over the parameter space, and in this space it seems that it may be possible to find a form that can be used to calculate lengths such as this.

But then intuitively this form should be the pullback of some other form under the parameterizing map which gives arc length when directly integrated over the curve. But no such form can exist as far as I can tell, since length does not depend linearly on the components of the tangent vectors to the curve (and forms are special types of variable multilinear maps).

I would be so very thankful to anyone who could help me work through this confusion. Thanks much in advance.

 

[Phrak]

Hello, zp. None of the smart guys has yet responded to you, so I'll do what I can. I'm about at the same level of understanding of differential forms as yourself. Can you tell me what text you're reading?

I've just begun investigating differential forms. I have no experience in this field and no formal, university level training in mathematics, so please bear with me.

I understand that a differential form may be thought of as a family of linear functionals; more precisely, it is a function that assigns to each point of a manifold a linear functional (i.e., a member of the cotangent space at that point) mapping a tuple of k tangent vectors to a real number.

If I understand this correctly, the same could be said of all type(0,k) tensors, rather than the subset in k-forms. I've prefer to think of k-forms as antisymmetric tensors with lower indices. In notation, $$A_{[ijk...z]}=A_{ijk...z}$$

This is all good for me. My confusion arises with regard to what sorts of quantities can actually be computed using forms.

For example, I understand that, if we take our manifold to be the xy-plane in ordinary Euclidean space, a two-form, say A dx dy where A is constant, can be evaluated on a geometrical figure in this plane to give its oriented area up to the factor A (and by "evaluated" I really mean integrated over the figure). But what if I wanted to compute the surface area of a surface in general xyz-space, say the upper-hemisphere of the unit sphere? This is straightforward in vector calculus using surface integrals, but I can't see how this could be accomplished using differential forms.

There's a generalization of Stoke's Theorem in forms that preports to do just this. It's an object of study I intend to get to. If you recall, Stoke's theorem as normally presented equates a boundry integral to the area integral of the cross-product.

You may have seen the notation. It's presented as
$$\int_{\partial M}A = \int_{M}dA$$

A marginal introduction to this is found here, http://en.wikipedia.org/wiki/Discrete_exterior_calculus" . It's the best I could come up with.

To make my difficulty plain, consider an analogous but simpler problem. To find the arc length of a curve in the xy-plane, we would need to integrate the quantity $$\sqrt{dx^2 + dy^2}$$ over the curve in the language of differential forms. Clearly the product here cannot be the Grassmann (wedge) product, since those terms would then vanish. But even if a suitable definition were provided for the products, the result doesn't seem like it'd be a differential form.

My problem is that it seems to me that differential forms naturally compute areas, volumes, etc, within a tangent space, but for computing such quantities over a portion of a manifold forms are not sufficient--the use of some more general object becomes necessary.

That's very insightful. The more general object you're looking for is one side, or the other, of the above integral equation. M is a region of a k-dimensional manifold, partial M is it's boundry. A is a differential k-form, and dA is the first exterior derivative of A.

Apparently, we're on similar quests. I could use differential forms to obtain Stoke's Theorm in cylindrical coordinates in order substantiate Faraday's Law, where the fields are themselves differential forms.

 

[Hurkyl]

One main thing to remember is the analogy

forms : functions :: tangent vectors : curves​

and since functions and curves behave in a dual manner, you will see the same thing happen with forms and vectors.


The fact you have embedded your curve in a larger ambient space means that you can also consider vectors along your curve as if they were vectors in the ambient space. The converse fails: a vector in the ambient space usually cannot be considered as a vector on the curve.

Functions work in the opposite direction -- if you have a function on the ambient space, you can restrict it to a function on your curve, but you cannot go in the opposite direction. And therefore the same is true of forms: if you have a form on the ambient space, you can restrict it to the curve. But you usually cannot take a form on your curve and extend it to the ambient space.



As it turns out, I believe you can often find some form on the ambient space that will restrict to the form you have on your curve... but you cannot expect it to have any good properties, aside from the fact it has the correct restriction.

For example, if dy/dt never vanishes on your curve, you have the following identity on your curve:

$$\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2} dt = \sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2} \frac{1}{dy/dt} dy$$

If you can manage to express the coefficient as a function of x and y (by eliminating t), then you have a differential form on your ambient space whose line integral along your particular curve really does give length. (Of course, it will usually not be usable to compute length along any other curve).

 

[zpconn]

Thank you both for the responses.

I'm reading the book Advanced Calculus: A Differential Forms Approach by Harold Edwards.

I'm actually already aware of the generalized Stokes' theorem. But the Stokes theorem is really just the old theorems of vector calculus recast in a unified and more concise notation and generalized to a certain class of manifolds. In this sense surface area/arc length can be computed using differential forms, but the methods are ultimately "isomorphic" to the methods used in vector calculus, if I may abuse some terminology.

I was wondering about a differential form that could be directly integrated over the curve (or surface) to give arc length (or surface area) without having to appeal to theorems like the Stokes' theorem.

If I understand this correctly, the same could be said of all type(0,k) tensors, rather than the subset in k-forms. I've prefer to think of k-forms as antisymmetric tensors with lower indices.

Thanks for the heads-up. My understanding of tensors is, at this point, very limited since I haven't seriously sat down to study them, having instead just gleaned a few important tidbits from various passive glances at Wikipedia and through other books. However, I will certainly keep this in mind, especially when I do (as I plan to do) study tensors.

For example, if dy/dt never vanishes on your curve, you have the following identity on your curve:

$$\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2} dt = \sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2} \frac{1}{dy/dt} dy$$

If you can manage to express the coefficient as a function of x and y (by eliminating t), then you have a differential form on your ambient space whose line integral along your particular curve really does give length. (Of course, it will usually not be usable to compute length along any other curve).

So if I understand correctly, it is, in principle at least, sometimes possible (depending on the exact properties of the curve) to find a differential form given a curve that, when integrated over the curve, will give the length, but this form will *not* work when integrated over other curves in general? If so, then my original suspicion seems right that there is no general form that could accomplish this task for all curves.

Once again, thanks for the very helpful replies. I'm certainly getting closer to getting an intuitive feel for all this differential form business.

 

[Phrak]

"This is all good for me. My confusion arises with regard to what sorts of quantities can actually be computed using forms."

This particular question of yours has been nagging at me for some time. Not to dismiss integration of differential forms--how else could you construct a lagrangian using forms? But there at least one example of what can be done with operations restricted to the wedge product and the duality transform that takes k-forms to (n-k)-forms.

If any of my notation or jargon is wrong, maybe someone can help clean it up.

The four equations of Maxwell can be reduced to zero equations: 1) postulate a 1-form field having real valued entries covering Minkowski space (or more generally a pseudo-Riemann manifold), and 2) make associations between various elements of various derivatives of the 1-form with physically quantities.

So that we have some (hopefully) common notation, the exterior derivative of the form A, is defined as the wedge product of the partial derivative with A.

$$dA = \partial\wedge A$$ or in a coordinate basis,

$$(dA)_{[\mu_{1}\mu_{2}\ldots\mu_{p+1}]}=(p+1)\partial_{[\mu_{1}}A_{\mu_{2}\mu_{3}\ldots\mu_{p+1}]}$$ .

For example, for the one-form A,

$$dA = 2(\partial_{\mu}A_{\nu} - \partial_{\nu}A_\mu})$$ .

The only other operator needed is the Hodge (*) star operator that uniquely maps a k-form to its (n-k)-form.

$$(*A)_{ \mu_{1}\mu_{2}\ldots\mu_{q} } = (1/p!)\epsilon_{ \mu_{1}\mu_{2}\ldots\mu_{q} } \ ^{ \nu_{1}\nu_{2}\ldots\nu_{p} } A _{\nu_{1}\nu_{2}\ldots\nu_{p} }$$ , where p+q=n.

I hope this has neither insulted your intelligence, nor lost you. From here on, it's all about forms.

F=dA, are associated with the coordinate components of the electric and magnetic fields.

G=*F, J=-*dG, J is a one-form. Elements of J are associated with charge intensity and current intensity. (*J would contain the charge and current densities)

dF=ddA=0 says that there are no magnetic monopoles.

$$A'=A + d\phi$$, says that A can be regauged leaving higher derivatives invariant.

The charge continuity equation is expressed as d*J=0. This is a direct consequence of A being a 1-form on a manifold with Minkowski metric.

The vacuum wave equation is *d*F=0, where again, this equality is zero as a consequence of the metric, and the vacuum constraints.

There may be simple forms for the energy density and other stuff. I dunno.

In addition to above, identities in the subspaces R^3 and R^1 of Minkowski space, can be developed as equations in differential forms that look more like Maxwell's equations. One can go the otherway, introducing forms in higher dimensions, that are a result of other regauging, where the original equations are reacquired in the 4D subspace, under some constraints.

 

[zpconn]

Very interesting stuff, Phrak. I just also found in the back of this book a derivation of the famous $$E = mc^2$$ using a rather elegant approach based on forms. They seem particularly well-suited to naturally describing electromagnetism, as you indicated.

 

[Phrak]

Very interesting stuff, Phrak. I just also found in the back of this book a derivation of the famous $$E = mc^2$$ using a rather elegant approach based on forms. They seem particularly well-suited to naturally describing electromagnetism, as you indicated.

Interesting as well, zpconn. I get this feeling that either the notation I am using is not commonly use, or there's not a great deal of direct interest in form in this forum. I've decided to by a differential geometry text. Perhaps the one you are using.

 

[Phrak]

Does you text show how to evaluate any integrals in forms at all?

 

[zpconn]

Several exercises ask you to evaluate integrals of forms, and brief solutions to all exercises are in the back (usually just with the final answer, or a brief explanation). But the text itself is fairly lacking in concrete examples of how to do traditional calculus problems using forms. It's much more concerned with developing the subject rigorously (although somewhat informally). Much of the book is spent building up to a rigorous proof of Stokes' theorem without much emphasis on actually using it; a lot of other space is spent on the implicit function theorem and the algebra of forms (discussed in a more concrete, down-to-earth manner, not the "alternating covariant tensor field" interpretation which might be more common).

That said, there is an entire chapter devoted to applications, a few pages of which indicate the correspondence between forms and traditional vector calculus (including a page of exercises). The rest of the chapter discusses much meatier applications (including a proof of some of the most interesting theorems of complex analysis, the aforementioned derivation of E = mc^2, integrability conditions, homology theory, and then some mathematical physics, including Maxwell's equations, flows, etc).

Overall, it's probably one of those books that you'd need to thumb through at a book store to decide if it will be worth your time. Unfortunately you probably won't be able to find it in a bookstore.

When I got it, I ultimately had to make a somewhat blind choice because there aren't many good introductory texts on differential forms that are at my level (most seem to target graduate students, which I am far from).

 

[Phrak]

Re blind choice. One of the more difficult exercises is selecting a book on mathematics. It's often true that you don't really know the name of the kind of mathematics you want to know about, until after the fact.

From what you've told me, it sounds to be good to start with. An undergraduate level is what I need, as well, as nearly anything presented in abstract algebra means nothing to me.

Thanks for the very detailed account! I think it would make for an excellent review at any of the online outlets.

By the way, some things don't look obviously like they have equivalents as differential forms. A good example of this is the Laplacian (or del Alembertian). The del Alembertian is a perfectly good differential form dispite appearances and, I presume, the Laplacian as well.

 

[shrubber]

It's been three years but I will answer this question anyway for future reference, as I feel it has not been answered satisfactorily.

What you need is a Riemannian volume form. Area does not make sense on a general manifold because the length of a tangent vector is undefined. However, if you have an orientable Riemannian manifold, you can use its metric to define the Riemannian volume form. It is the unique differential form that yields 1 on all orthonormal frames. You can then compute the area (volume, length) of the manifold by integrating the volume form.

Unfortunatelly, the volume form cannot be inherited from an ambient manifold. The good news is you can express the volume form in terms of the Riemannian metric, which can be inherited. So in the case of a surface in R3 you do the following: Take the Eucleidian metric dx^2 + dy^2 + dz^2 (which is a symmetric 2-tensor) and restrict it to the 2-dimensional submanifold using a smooth chart u,v. Express the Riemannian metric as a matrix G in terms of u,v. The Riemannian volume form can then be expressed as sqrt(det G) du/\dv. It works similarly in all dimensions. If you have an unoriented Riemannian manifold, you have to use density instead of a volume form.