Differential Forms & Tensor Fiekds .... Browder, Section 13.1

[Math Amateur]

Andrew Browder in his book: "Mathematical Analysis: An Introduction" ... ... defines a differential form in Section 13.1 which reads as follows:

In the above text from Browder we read the following:

" ... ... A differential form of degree $r$ (or briefly an $r$-form) in $U$ is a map $\omega$ of $U$ into ${ \bigwedge}^r V ( \mathbb{R}^{ n \ast } )$ ... ... "In other words if $\omega$ is a differential form of degree $r$ in $U$, then we have

$\omega : p \to \omega_p ( v_1, \cdot \cdot \cdot , v_r )$Browder, unfortunately, gives no example of a differential form ... but I found an introductory example of a $2$-form on page 6 of Steven Weintraub's book: Differential Forms: Theory and Practice ... the example (Example 1.1.3 (2) ) reads as follows:

" ... ...

$\phi = xyz \text{ dy } \text{ dz } + x e^y \text{ dz } \text{ dx } + 2 \text{ dx } \text{ dy }$
My question is as follows:

How do we interpret Weintraub's example in terms of Browder's definition of a differential form ... and further, how do we translate Weintraub's example into an example in Browder's notation/definition ... ...Help will be much appreciated ... ...

Peter=========================================================================================

So that readers can access Weintraub's definition and notation I am providing the relevant text ... as follows:

Hope that helps,

Peter

 

[Orodruin]

The differential of a function $f$ is the 1-form $df$ such that $df(v) = df/dt$, where the derivative of $f$ is taken with respect to the curve parameter $t$ such that $v$ is the tangent vector of the curve. A complete basis for 1-forms is given by the differentials of the coordinate functions. These are the $dx$, $dy$, and $dz$ in the expression. The products of the differentials is intended as the antisymmetric wedge product, which makes them 2-forms. Since you have three dimensions, there are three [n(n-1)/2] independent such products and the general 2-form is a linear combination of those.

 

[Math Amateur]

The differential of a function $f$ is the 1-form $df$ such that $df(v) = df/dt$, where the derivative of $f$ is taken with respect to the curve parameter $t$ such that $v$ is the tangent vector of the curve. A complete basis for 1-forms is given by the differentials of the coordinate functions. These are the $dx$, $dy$, and $dz$ in the expression. The products of the differentials is intended as the antisymmetric wedge product, which makes them 2-forms. Since you have three dimensions, there are three [n(n-1)/2] independent such products and the general 2-form is a linear combination of those.

Thanks Orodruin ... appreciate your help ...

However I am still struggling to understand the explicit link/equivalence between Browder's definition of $\omega$ and Weintraub's example ...

Peter

 

[fresh_42]

Here is an example with an actual calculation:
https://www.physicsforums.com/threads/why-the-terms-exterior-closed-exact.871875/#post-5474443

Weintraub does not write the wedges and only uses notations up to three, which is a bit confusing, as he does not mention (in what you copied) that differential forms are alternating.

Say we have a differential (alternating) $2-$form $\omega = xyz (\text{dy} \wedge \text{dz}) + x e^y (\text{dz}\wedge \text{dx}) + 2 (\text{dx} \wedge \text{dy})$. Then Weintraub writes it as $\varphi = xyz \text{ dy } \text{ dz } + x e^y \text{ dz } \text{ dx } + 2 \text{ dx } \text{ dy }$, but they are the same thing.

I would quote this section in Wikipedia https://de.wikipedia.org/wiki/Differentialform#Differentialform but I'm too lazy to translate it (and the translator in Chrome needs some additional work - see below). You don't need to understand sections and cuts for now, and they say "smooth" which is $C^\infty$, which can be substituted by $C^k$. The English page is less explicit.