The Space of 2-Forms .... Fortney, Darling and Weintraub ....

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In summary, the definitions of 2-forms and the space of two forms are different, but the underlying concept is the same.
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I have a question regarding the nature of 2-forms (indeed k-forms ... but I'll focus the question on two forms ... ) ... Defining 2-forms and the space \(\displaystyle {\bigwedge}^2 ( \mathbb{R}^3 )\) , Jon Fortney in his book: A Visual Introduction to Differential Forms and Calculus on Manifolds, writes the following:View attachment 8783R. W. R. Darling in his book defines \(\displaystyle { \bigwedge}^2 V \) similarly when he writes the following:View attachment 8784... BUT ...Steven Weintraub in his book: Differential Forms: Theory and Practice defines k-forms (and hence 2-forms ...) as follows:View attachment 8785(NOTE: Weintraub omits the wedge notation writing the wedge product \(\displaystyle dx \wedge dy\) as just \(\displaystyle dx dy\) ... ... )
My question is ... how do we reconcile the differences between Fortney and Darling's definition of a 2-form and the space of two forms ... with Weintraub's definition ...

Essentially ... Fortney defines a 2-form as

\(\displaystyle a dx_1 \wedge dx_2 + b dx_2 \wedge dx_3 + c dx_1 \wedge dx_3 \)

where \(\displaystyle a,b, c \in \mathbb{R} \)... while Weintraub defines a 2-form as \(\displaystyle f_1 dx_1 \wedge dx_2 + f_2 dx_2 \wedge dx_3 + f_3 dx_1 \wedge dx_3 \)

where \(\displaystyle f_1, f_2,\) and \(\displaystyle f_3\) are smooth functions ...
Can someone please help reconcile the differences between the definitions of Fortney and Weintraub ...?Help will be appreciated ...

Peter
EDIT ... reflection ... the definitions could be very similar ( ... the same ...? ... ) since the smooth functions are (I think) real-valued functions ... and so when evaluated at a point are real numbers ... does that make sense of the above?Peter
 

Attachments

  • Fortney - Two-forms ... Ch. 3, page 78 ... .png
    Fortney - Two-forms ... Ch. 3, page 78 ... .png
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  • Darling ...Second  Exterior Power ... Ch. 1, page 1 .png
    Darling ...Second Exterior Power ... Ch. 1, page 1 .png
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  • Weintraub - Differential Forms ,,, Ch. 1, page 6 ... .png
    Weintraub - Differential Forms ,,, Ch. 1, page 6 ... .png
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Hi Peter,

Weintraub is defining a differential $k$-form, which is one where the smooth function $f$ is used to allow the coefficients to vary from point to point (as opposed to the fixed values of $a$, $b$, and $c$ used by Fortney). Also, you are correct that $f$ is a smooth, real-valued function.
 
  • #3
GJA said:
Hi Peter,

Weintraub is defining a differential $k$-form, which is one where the smooth function $f$ is used to allow the coefficients to vary from point to point (as opposed to the fixed values of $a$, $b$, and $c$ used by Fortney). Also, you are correct that $f$ is a smooth, real-valued function.
Thanks for the help, GJA ...

Peter
 

FAQ: The Space of 2-Forms .... Fortney, Darling and Weintraub ....

1. What is a 2-form in the context of space?

A 2-form is a mathematical object that describes how a vector field changes over a surface in three-dimensional space. It is often used in the study of differential geometry and is an important tool in understanding the curvature of space.

2. Who are Fortney, Darling, and Weintraub?

Fortney, Darling, and Weintraub are three scientists who have made significant contributions to the field of differential geometry, specifically in the study of 2-forms in space. They have published several papers together on the topic and are considered experts in the field.

3. Why is the study of 2-forms important?

The study of 2-forms is important because it allows us to understand the curvature and geometry of space, which has implications in fields such as physics and engineering. It also has applications in computer graphics and computer vision.

4. What are some real-world examples of 2-forms?

Some real-world examples of 2-forms include the magnetic field around a wire, the flow of a fluid over a surface, and the curvature of spacetime in general relativity. These examples demonstrate how 2-forms can be used to describe physical phenomena in our world.

5. How are 2-forms related to other mathematical concepts?

2-forms are closely related to other mathematical concepts such as vectors, tensors, and differential forms. They are also used in conjunction with other mathematical tools, such as the exterior derivative and the wedge product, to solve problems in differential geometry and other fields of mathematics and science.

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