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

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In summary, Browder defines a differential form as a map from a set into the wedge product of the dual space of the vector space, while Weintraub's example illustrates this definition by providing a specific calculation for a ##2##-form. The two notations may differ, but they are equivalent in terms of the definition of a differential form.
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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 ##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

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Last edited:
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.

Orodruin said:
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

Here is an example with an actual calculation:

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.

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Math Amateur

## 1. What are differential forms?

Differential forms are mathematical objects used in multivariable calculus and differential geometry to describe and analyze geometric and physical quantities that vary smoothly from point to point. They are represented as a combination of basis elements and can be used to represent vectors, scalars, and higher-order tensors.

## 2. What are tensor fields?

A tensor field is a mathematical object that assigns a tensor to each point in a given space. Tensors are used to represent geometric and physical quantities that vary in multiple directions and can be thought of as a generalization of vectors and matrices. Tensor fields are important in the study of differential geometry, general relativity, and other areas of mathematics and physics.

## 3. How are differential forms and tensor fields related?

Differential forms and tensor fields are closely related, as differential forms can be used to describe and manipulate tensor fields. In particular, differential forms can be used to define operations such as differentiation and integration on tensor fields, making them a powerful tool in the study of differential geometry and other fields.

## 4. What is the significance of Section 13.1 in Browder's book?

Section 13.1 in Browder's book is significant because it introduces the concept of differential forms and tensor fields and their applications in differential geometry. It provides a foundation for understanding more advanced topics in differential geometry and other areas of mathematics and physics.

## 5. How are differential forms and tensor fields used in real-world applications?

Differential forms and tensor fields have numerous real-world applications, particularly in physics and engineering. They are used to model and analyze physical systems in fields such as fluid dynamics, electromagnetism, and general relativity. They are also used in computer graphics and computer vision for geometric modeling and image processing.

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