“\ [“: > In other words, if \( \{x^1,\,\dots\,,x^n\} \) are the standard orthogonal coordinates on \( \mathbb{R}^n \), then \[ \mathrm{d}x^1_p\left(\left.\frac{\partial}{\partial x^1}\right|_p\right)=1 \]

“$ $”: > and $$ \mathrm{d}x^1_p\left(\left.\frac{\partial}{\partial x^i}\right|_p\right)=0 $$ for $i\neq 1$.

I’m curious to whether any of these forms will work…

In other words, if $\{x^1,\,\dots\,,x^n\}$ are the standard orthogonal coordinates on $\mathbb{R}^n$, then $\mathrm{d}x^1_p\left(\left.\frac{\partial}{\partial x^1}\right|_p\right)=1$ and $\mathrm{d}x^1_p\left(\left.\frac{\partial}{\partial x^i}\right|_p\right)=0$ for $i\neq 1$.

Again, I would like to thank Thomas Klimpel for pointing out both the error in the post and the error in the hasty correction I was going to replace it with. The people who can edit the blog posts will probably fix the problem at their earliest convenience.

Clifford algebra (and its associated calculus) appeals to me as a physicist because I almost exclusively work in a setting with some metric or pseudo-metric. The calculus treats the exterior derivative and interior one on the same footing, and in a setting without a metric, it’s easy enough to forbid any metrical operations (common enough in projective geometry, for instance). To me, this is somewhat cleaner than dealing exclusively with forms and then resorting to duality when needed, but I understand it’s a matter of experience and taste.

Still, with respect to thi article, I make a point about the meaning of dx here as a result of more than one argument on the matter even on MSE. To me, the geometric calculus picture of integrating an n-form is very appealing: the form eats the tangent n-vector, producing a scalar function, which is then integrated according to established multivariable techniques. But, I do recognize that this merely changes the burden of how one defines the integration of a form: in forms notation, we just skip one step compared to geometric calculus, but GC still has to define that integration on an oriented manifold involves multiplying by that manifolds tangent n-vector.

I certainly agree that differential geometry provides the language to express integrals of absolutely continuous (with respect to volume form) over orientable varieties, and it is surely the right way to formalise the otherwise seemingly nonsensical $u=f(t)$ leading to $du= f'(t) dt$ squiggles. On the other hand, introducing differential forms to freshmen would likely only serve to confuse them. So yeah.

]]> http://math.blogoverflow.com/2014/11/03/more-than-infinitesimal-what-is-dx/#comment-1285 Wed, 05 Nov 2014 10:14:50 +0000 http://math.blogoverflow.com/?p=570#comment-1285 In my opinion, the apparatus of modern differential geometry is not needed to actually do calculations with differential forms — introductory calculus classes already sort of teach it, even while saying that’s not what’s really going on.

I think there is actually a more fundamental problem. There are two versions of calculus: calculus of functions, and calculus of scalars. While both versions have the same content, they have very different notational flavor and teach you to think in somewhat different ways.

Applications and calculations are, at least in my experience, predominantly in the form of the calculus of scalars. e.g. you have dependent variables x and y satisfying x^2 + y^2 = 1, and you’re relating variations in y to variations in x.

However, in introductory classes, the theory is almost exclusively presented in terms of the calculus of functions, which teaches you to think of x and y as being very different kinds of things; e.g. you pick x to be the variable, and then treat y as an abuse of notation being used as shorthand for sqrt(1-x^2).

In my opinion, treating dx and dy as objects in their own right only makes sense when you’re doing calculus of scalars, which means such ideas are inaccessible until students unless they either have a good intuition for these things, or until they finally advance to a subject that can’t get away with the calculus of functions approach and finally has to teach the calculus of scalars version. Which, I believe, is typically their first differential geometry course or their first algebraic geometry course.

Muphrid: In full sincerity, I never really looked at geometric calculus before; for some reason I’ve always assumed that it was a synonym for multiplicative calculus. At a glance, it looks very interesting, and I’ll be sure to look further into it when I find time.

That being said, I still believe that differential forms are the the natural way to go, since they make many of the ideas of calculus arise of their own accord. This opinion might simply be because I am an abecedarian when it comes to math in general, though.

mvw: Perhaps you missed the point. This post was designed to encourage exploration of differential geometry, and explain that the ideas of infinitesimals might be better explained by differential forms.

Engineers and physicists are also not, strictly speaking, mathematicians, so they can enjoy the freedoms of empirical science. If you are doing physics, and you find “numbers smaller than any real number” intuitive, then you can make that choice. However, I know for a fact that forms are well-accepted in upper tier physics.

This being said, of course they are going to stay. They haven’t been allowed to die for over 300 years, so why would that stop now?

Learning the apparatus of modern differential geometry would require extra effort and probably add not much to the problem solving skills.

So it will probably stay.

I’m not arguing against the use of oriented regions; I’m saying differential forms relies choosing orientations for regions based on arbitrary criteria (most often, merely by how the basis is ordered) and treats such choices as implicit. This in itself is not a bad thing, but it contributes to the ongoing notion that basis covectors are “the same as” or “substantially related to” the differentials that appear inside integrals.

I argue this is misleading because in geometric calculus, the differentials that appear in an integral do not–in any way, shape, or form–come from the differential form being integrated at all. Geometric calculus replicates the whole of the substance of differential forms–they do not disagree in any way as far as the result of a computation–so the meaning of notational concepts like the differential inside an integral should be compatible in both.

In my experience, both applications and methods of computation tend rely crucially on the fact you’re integrating over oriented regions. Although in top degree in Euclidean space, there is some pedagogical value in avoiding differential geometry and faking an orientation with an unoriented integral by preordaining an orientation, and making sure you always respect the orientation (e.g. by putting absolute values around the Jacobian when making a change of variable, which has the effect of flipping the sign of the integrand whenever the transformation reverses the orientation).

And it is clear that differential forms really are the right notion of integrand when you’re integrating over oriented regions; no matter which parametrization of the surface you take, you always plug the same differential form into the formulas. e.g. in the usual notation, $\int_S \omega = \int_{S’} \omega$.

Although, I’m sure there are contexts where unoriented versions of integration are useful and even important, in which case different concerns become important and the above may not apply.