What is differential about differential forms?

[pellman]

TL;DR Summary

Why are n-forms called differential forms? What is differential about them?

Why are n-forms called differential forms? What is differential about them? And why was the dx notation adopted for them? It must have something to do with the differential dx in calculus. But dx in calculus is an infinitesimal quantity. I don't see what n-forms have to do with infinitesimal quantities.

We find in both real calculus and tensor calculus expressions similar to df = f' dx , but it seems to me the quantities in each case are very different things. Aren't they?

 

[fresh_42]

Summary:: Why are n-forms called differential forms? What is differential about them?

What is differential about them?

It is still about tangent spaces, cotangent vectors and differential forms are integrable, only in higher dimensions. See the example in
https://www.physicsforums.com/threads/why-the-terms-exterior-closed-exact.871875/#post-5474443
or the more detailed description here
https://www.physicsforums.com/insights/pantheon-derivatives-part-iii/

 

[Orodruin]

cotangent vectors and differential forms are integrable, only in higher dimensions

Note to self, commas are important. I missed the one in this sentence on first reading ...

 

[martinbn]

What would you prefer to call them?

 

[haushofer]

I guess this question boils down to "what is the exact relation between the basis vectors dx we use for n-forms and the dx we use as infinitesimal quantities in derivatives".

If so: I've never understood that either.

 

[fresh_42]

I guess this question boils down to "what is the exact relation between the basis vectors dx we use for n-forms and the dx we use as infinitesimal quantities in derivatives".

If so: I've never understood that either.

I think there is no difference. We are just not used to calling the infinitesimal quantity dx a one form or Pfaffian form, that's it. I once decided to make a list
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/ (end of section -1)
of all viewpoints on a derivative. And I didn't even use the words slope or limit of secants.

My personal dream would be to start lecturing infinitesimal calculus by the (co-)boundary operators of simplicial complexes!

 

[fresh_42]

Note to self, commas are important. I missed the one in this sentence on first reading ...

It is especially hard for me. I observed that the comma laws in English and in German are basically complementary. And I do not like SPO very much. I find it far more pleasant to start a sentence with the most important part, not with the unimportant subject.

 

[pellman]

I think there is no difference. We are just not used to calling the infinitesimal quantity dx a one form or Pfaffian form, that's it. I once decided to make a list
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/ (end of section -1)
of all viewpoints on a derivative. And I didn't even use the words slope or limit of secants.

My personal dream would be to start lecturing infinitesimal calculus by the (co-)boundary operators of simplicial complexes!

Right but I don't see it. Hence the question.

 

[fresh_42]

Right but I don't see it. Hence the question.

How about the following point of view:

Consider three vector fields $H,X,Y$ and the one form $dx$ eating them and spitting out a function in $C^\infty (\mathbb{R})$: $H\,dx = 2x\, , \,X\,dx =x^2\, , \,Y\,dx=-1.$ This is simply three differential operators using the infinitesimal $dx$
$$
H=2x \cdot \frac{d}{dx} \, , \, X=x^2\cdot \frac{d}{dx}\, , \, Y=- \frac{d}{dx}
$$
written in a different way. $\operatorname{span}\{H,X,Y\} \cong \mathfrak{sl}_2 \cong T_1(\operatorname{SL(2)}).$

 

[pellman]

If H is a vector field and dx is one from, what is H dx?

But I don't know that it matters to my question. What I am looking for is a conceptual bridge between the two types of mathematical objects. What was it about differential forms in the early days of the subject which led mathematicians to adopt the dx notation?

Suppose I am a former calculus student that is used to using df , dx notation for "differentials", e.g., the expression for a total differential. And I don't know anything about differential forms. How would one introduce the concept of forms in way where the conceptual continuity from the differentials in calculus is clear, and the use of the dx symbol is obviously appropriate?

 

[fresh_42]

It all started in 1866 with Riemann's habilitation lecture:
http://lymcanada.org/wp-content/uploads/sites/2/2015/01/Riemann_Habilitation-Dissertation.pdf

I can give you a modern version of how you get from $T^*(V)$ to $C^k-$differential forms. It basically starts with $p \longmapsto d_pf$ from $M \longrightarrow T^*(M)$ called the differential of $f.$

 

[martinbn]

If H is a vector field and dx is one from, what is H dx?

It is an unusual notation, but if i am not wrong it means $dx(H)$.

But I don't know that it matters to my question. What I am looking for is a conceptual bridge between the two types of mathematical objects. What was it about differential forms in the early days of the subject which led mathematicians to adopt the dx notation?

Suppose I am a former calculus student that is used to using df , dx notation for "differentials", e.g., the expression for a total differential. And I don't know anything about differential forms. How would one introduce the concept of forms in way where the conceptual continuity from the differentials in calculus is clear, and the use of the dx symbol is obviously appropriate?

What is the definition if differential you know? I am asking so that we can build on that in order to answer your question.

 

[Mozibur Rahman Ullah]

Instead of thinking them of as 'differential forms', think of calling them smooth forms. We call them this because they are defined form smooth manifolds as opposed to say, topological manifolds. Then recall differentials are usded to analyse smoothness. Hence, also differential forms.

 

[pellman]

It all started in 1866 with Riemann's habilitation lecture:
http://lymcanada.org/wp-content/uploads/sites/2/2015/01/Riemann_Habilitation-Dissertation.pdf

I can give you a modern version of how you get from $T^*(V)$ to $C^k-$differential forms. It basically starts with $p \longmapsto d_pf$ from $M \longrightarrow T^*(M)$ called the differential of $f.$

Thanks for that link.

 

[pellman]

It is an unusual notation, but if i am not wrong it means $dx(H)$.

What is the definition if differential you know? I am asking so that we can build on that in order to answer your question.

My understanding of it is not very deep. Just an infinitesimally small change that results in a linear change in the dependent function.

 

[pellman]

I think my trouble stems from the definition of one-forms as linear maps from a tangent space to R, which seems to have nothing to do with infinitesimal changes of a real variable. Is there another equivalent definition of one-forms?

 

[fresh_42]

I think my trouble stems from the definition of one-forms as linear maps from a tangent space to R, which seems to have nothing to do with infinitesimal changes of a real variable. Is there another equivalent definition of one-forms?

This is not true. E.g. write the derivative in my favorite form, Weierstarß's notation for $f\, : \,\mathbb{R}^n\longrightarrow \mathbb{R}$
$$
f(x_{0} +v )=f(x_{0})+J(v)+r(v)\, \text{ with } \,\lim_{v \to 0}\dfrac{r(v)}{\|v\|}=0
$$
What is the derivative here? It's the Jacobian matrix $J,$ a vector in this case. And what is it? It is a linear function that eats a direction $v$ and spits out a number. The direction is the tangent direction, the number is the slope, and all are evaluated at $x_0,$ and for $f$, which means that we get different matrices at different points or for different functions. But once fixed a function and a point we get a one-form $J$.

The same holds true in the school case $n=1.$ The derivative is the linear function, that multiplies a direction (we have only one in that case, i.e. we have a number), by the slope in the direction. E.g. let's take $f(x):=x^3.$ Then
\begin{align*}
f(x_0+\delta )&=x_0^3+3x_0^2\delta +3x_0\delta^2+\delta^3 =\underbrace{x_0^3}_{=:f(x_0)}+\underbrace{(3x_0^2)\cdot \delta}_{=:J(\delta)} +\underbrace{\delta^2(3x_0+\delta )}_{=:r(\delta )}
\end{align*}
where $\lim_{\delta \to 0} \dfrac{r(\delta )}{|\delta| }= \lim_{\delta \to 0}\left(\delta \cdot (3x_0+\delta)\right)=0.$

You see, the derivative $f' = J\, : \,\delta \longmapsto (3x_0^2)\cdot \delta$ is a one-form, multiplication by the slope at $x_0.$

 

[WWGD]

Differential forms are , by many accounts, the " right" objects to be integrated on manifolds. If f is differentiable from Reals to Reals, then f'(x)dx is a 1-form. It approximates the change of f locally , using the change along the tangent line to approximate the actual change f( x+h)-f(x). As Fresh pointed out, this genetalizes to a linear map in n>2 dimensions, represented by a matrix, which is the Jacobian.

 

[fresh_42]

One can tell the difference with Weierstraß, too:
\begin{align*}
f'_{x_0}\cdot v &= f(x_0+v) - f(x_0) - r(v) \quad \longrightarrow \text{ linear function multiplying by the slope vector, one-form}\\[12pt]
\left. \dfrac{df}{dv}\right|_{x_0}&=\dfrac{f(x_0+v)-f(x_0)}{\|v\|}-\dfrac{r(v)}{\|v\|} \quad \longrightarrow \left. \dfrac{df}{dv}\right|_{x_0}=\lim_{v \to 0}\dfrac{f(x_0+v)-f(x_0)}{\|v\|}\quad \text{ infinitesimal }
\end{align*}

 

[Mozibur Rahman Ullah]

I had trouble with that definition too. Locally, given a function $f: M \rightarrow \mathbb{R}$ on a manifold $M$, the exterior derivative of $f$ is:

$$df = \partial f / \partial x^i .dx^i$$

This is a1-form. Not all 1-forms are of this form locally. The ones that are are called closed or locally exact. This is like a locally conservative field where I mean by this that a local potential for the field can be found. Usually, fields are sections of vector bundles and tangent fields are sections of tangent bundles. Dually, we can call cofields sections of covector bundles and so cotangent fields are sections of cotangent bundles. These are exactly the differential 1-forms. Here the 'differential' and 'infinitesimal change' is hidden inside the definition of tangent bundles.

I hope this helps.

 

[WWGD]

To summarize*, differential is used to mean related to derivatives, and this is taken to stand for the usual every day calculus. Differential geometry is the application of calculus-related methods to geometry or the study of differential manifolds. A differential form is the standard object that is " integrated against" in a manifold.

* I remember someone spelling it as " Two Sammurais".

 

[pellman]

Thanks, guys. I will think it through some more.