# The Dual Space and Differential Forms .... ....

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In summary, Peter is trying to understand what the result of $\phi(v) = a_{1}dx^{1}+ \cdots + a_{n}dx^{n}$ is and why it is equal to a_{1}v^{1}+\cdots+a_{n}v^{n}.
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I am reading the book: Multivariable Mathematics by Theodore Shifrin ... and am focused on Chapter 8, Section 2, Differential Forms ...

I need some help in order to fully understand some statements of Shifrin at the start of Chapter 8, Section 2 on the dual space ...

The relevant text from Shifrin reads as follows:
View attachment 8791In the above text from Shifrin we read the following:

" ... ... Then $$\displaystyle \phi = a_1 dx_1 + \ ... \ ... \ + a_n dx_n$$ ... ... "
Can someone please demonstrate and explain how/why $$\displaystyle \phi = a_1 dx_1 + \ ... \ ... \ + a_n dx_n$$ ... ...
Help will be much appreciated ... ...

Peter

#### Attachments

• Shifrin - Start of Ch. 8, Section 2 ... Differential Forms ... .png
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Hi Peter,

To establish the result, let $v=\sum_{k=1}^{n}v^{k}e_{k}$ be a vector in $\mathbb{R}^{n}$. Then, using linearity of $\phi$ judiciously, $$\phi(v) = \sum_{k=1}^{n}v^{k}\phi\left(e_{k}\right)=\sum_{k=1}^{n}a_{k}v^{k}.$$ On the other hand, using linearity of the $dx^{k}$,
\begin{align*}
\left(a_{1}dx^{1}+\cdots + a_{n}dx^{n}\right)\left(v\right) &= \left(a_{1}dx^{1}+\cdots + a_{n}dx^{n}\right)\left(\sum_{k=1}^{n}v^{k}e_{k}\right)\\
&= a_{1}dx^{1}\left(\sum_{k=1}^{n}v^{k}e_{k}\right) +\cdots + a_{n}dx^{n}\left(\sum_{k=1}^{n}v^{k}e_{k}\right)\\
&=a_{1}v^{1}+\cdots +a_{n}v^{n}.
\end{align*}
Since the two are equal and $v$ was chosen arbitrarily, we have $\phi=a_{1}dx^{1}+ \cdots + a_{n}dx^{n}.$

GJA said:
Hi Peter,

To establish the result, let $v=\sum_{k=1}^{n}v^{k}e_{k}$ be a vector in $\mathbb{R}^{n}$. Then, using linearity of $\phi$ judiciously, $$\phi(v) = \sum_{k=1}^{n}v^{k}\phi\left(e_{k}\right)=\sum_{k=1}^{n}a_{k}v^{k}.$$ On the other hand, using linearity of the $dx^{k}$,
\begin{align*}
\left(a_{1}dx^{1}+\cdots + a_{n}dx^{n}\right)\left(v\right) &= \left(a_{1}dx^{1}+\cdots + a_{n}dx^{n}\right)\left(\sum_{k=1}^{n}v^{k}e_{k}\right)\\
&= a_{1}dx^{1}\left(\sum_{k=1}^{n}v^{k}e_{k}\right) +\cdots + a_{n}dx^{n}\left(\sum_{k=1}^{n}v^{k}e_{k}\right)\\
&=a_{1}v^{1}+\cdots +a_{n}v^{n}.
\end{align*}
Since the two are equal and $v$ was chosen arbitrarily, we have $\phi=a_{1}dx^{1}+ \cdots + a_{n}dx^{n}.$
Thanks GJA ... most helpful ...

Appreciate your help ...

Peter

## What is the concept of dual space in mathematics?

The dual space is a mathematical concept that refers to the set of all linear functionals on a vector space. It is the space of all linear maps from the vector space to its underlying field of scalars. In simpler terms, it is a space of linear functions that can take a vector as an input and produce a scalar as an output.

## How is dual space related to differential forms?

In differential geometry, differential forms are mathematical objects that can be used to describe the properties of a space at each point. The dual space is closely related to differential forms, as the coefficients of a differential form are elements of the dual space. This allows for a more geometric interpretation of differential forms and their operations.

## What is the significance of the exterior derivative in differential forms?

The exterior derivative is a fundamental operation in differential forms that maps a differential form of degree k to a differential form of degree k+1. It is significant because it allows for the calculation of important geometric quantities, such as the volume and surface area of a space, as well as the integration of differential forms over a space.

## How are differential forms used in physics?

Differential forms are used extensively in physics, particularly in the field of classical mechanics. They are used to describe physical quantities such as position, velocity, and acceleration, as well as to formulate laws and equations in a geometrically invariant manner. They also play a crucial role in the study of electromagnetism and general relativity.

## What are some real-world applications of differential forms?

Differential forms have a wide range of applications in various fields, including engineering, computer graphics, and computer vision. They are used to model and analyze complex systems, such as fluid flow and deformable objects. They also have applications in data analysis, as they can be used to extract meaningful information from large datasets.

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