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

#### Peter

##### Well-known member
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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 $$\displaystyle r$$ (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:

Hope that helps,

Peter

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