# Maxwell Tensor Identity Explained: Deriving Formula 8.23 in Schawrtz's Book

• I
• dm4b
In summary, the conversation discusses the expansion of the square of the Maxwell tensor, as suggested by Schwartz on page 116. The formula involves the manipulation of tensors, but it is noted that achieving this may require integrating by parts.
dm4b
Hello,

In Schawrtz, Page 116, formula 8.23, he seems to suggest that the square of the Maxwell tensor can be expanded out as follows:

$$-\frac{1}{4}F_{\mu \nu}^{2}=\frac{1}{2}A_{\mu}\square A_{\mu}-\frac{1}{2}A_{\mu}\partial_{\mu}\partial_{\nu}A_{\nu}$$

where:

$$F_{\mu\nu}=\partial_{\mu} A_{\nu} - \partial_{\nu}A_{\mu}$$

For the life of me, I can't seem to derive this. I get close, but always with an extra unwanted term, or two.

Anyone have a hint on the best way to proceed?

Thanks!

... keeping it under the integral (of S) and differentiating by parts works out here. However, is there a way to achieve this with just tensor manipulation? I thought so, but I may not be remembering correctly.

dm4b said:
However, is there a way to achieve this with just tensor manipulation?

No, you need to integrate by parts. The equality sign there is a bit misleading.

bhobba
The equality is modulo a total derivative.

Demystifier

## 1. What is the Maxwell Tensor Identity?

The Maxwell Tensor Identity, also known as the Bianchi Identity, is a mathematical relationship between the derivatives of the electromagnetic field tensor. It is a fundamental equation in electromagnetism that is derived from Maxwell's equations.

## 2. Why is the Maxwell Tensor Identity important?

The Maxwell Tensor Identity is important because it allows us to understand the behavior of electromagnetic fields and their relationship to each other. It also helps us to solve complex problems in electromagnetism by providing a mathematical framework for analyzing the electromagnetic field tensor.

## 3. How is Formula 8.23 derived in Schwartz's book?

Formula 8.23 in Schwartz's book is derived using vector calculus and the properties of the electromagnetic field tensor. It involves taking the curl of the divergence of the electromagnetic field tensor and simplifying the resulting equations.

## 4. What is the significance of Formula 8.23 in electromagnetism?

Formula 8.23 is significant because it is a direct consequence of the Maxwell Tensor Identity and helps to explain the behavior of electromagnetic fields. It is also used in many applications, such as in the study of electromagnetic waves and in the development of electromagnetic field theories.

## 5. How is the Maxwell Tensor Identity related to Maxwell's equations?

The Maxwell Tensor Identity is derived from Maxwell's equations, which are a set of four fundamental equations that describe the behavior of electric and magnetic fields. The identity is a mathematical consequence of these equations and provides a deeper understanding of their relationship to each other.

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