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Differential Forms and Tensor Fields ... Browder, Section 13.1 ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,882
Hobart, Tasmania
Andrew Browder in his book: "Mathematical Analysis: An Introduction" ... ... defines a differential form in Section 13.1 which reads as follows:



Browder - Defn of a differential form .png


In the above text from Browder we read the following:

" ... ... A differential form of degree \(\displaystyle r\) (or briefly an \(\displaystyle r\)-form) in \(\displaystyle U\) is a map \(\displaystyle \omega\) of \(\displaystyle U\) into \(\displaystyle { \bigwedge}^r V ( \mathbb{R}^{ n \ast } )\) ... ... "


In other words if \(\displaystyle \omega\) is a differential form of degree \(\displaystyle r\) in \(\displaystyle U\), then we have

\(\displaystyle \omega : p \to \omega_p ( v_1, \cdot \cdot \cdot , v_r )\)



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

" ... ...

\(\displaystyle \phi = xyz \text{ dy } \text{ dz } + x e^y \text{ dz } \text{ dx } + 2 \text{ dx } \text{ dy }\)




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





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



Weintraub - Differential Forms ,,, Ch. 1, page 6 ... .png



Hope that helps,

Peter
 
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