# Maxwell's equations with differential forms

• I
In summary, we have shown that the two forms of Maxwell's equations, written in differential forms and covariant forms, are equivalent. This provides a more "direct" derivation of the covariant form using the definition of the exterior derivative and the Hodge dual operator. I hope this helps to clarify any confusion you may have had.Best regards,[Your Name]
Hello!

I was not quite sure about posting in this category, but I think my question fits here.

I am wondering about Maxwell equations in vacuum written with differential forms, namely:
\label{pippo}
dF = 0 \qquad d \star F = 0

I know ##F## is a 2-form, and It can be written by using the 1-form ## F = d A##. I am not now interested in the derivation (even if the first is straightforward), but I want to see the equivalence with the Maxwell equation written in a covariant way:
\label{pluto}
\varepsilon^{\mu \nu \rho \sigma} \partial_\nu F_{\rho \sigma} = 0 \qquad \partial^\mu F_{\mu \nu} = 0

If I explicit everything, knowing the components of ##F_{\mu \nu}##, I can recover both \ref{pluto} equations starting from \ref{pippo}. I'm interested in a more "direct" derivation, but I cannot find any reference in textbooks.

Francesco

NB: I am not a native english speaker, sorry for that.

Hello Francesco,

Thank you for your question. It is great to see someone interested in the mathematical formulation of Maxwell's equations.

To see the equivalence between the two forms of Maxwell's equations, we can start by considering the definition of the exterior derivative of a 1-form. We know that for a 1-form ##\omega##, the exterior derivative is defined as ##d\omega = (\partial_\mu \omega_\nu - \partial_\nu \omega_\mu) dx^\mu \wedge dx^\nu##. In the case of the 2-form ##F##, we can write it as ##F = F_{\mu \nu} dx^\mu \wedge dx^\nu##.

Applying the definition of the exterior derivative, we have ##dF = (\partial_\mu F_{\nu \rho} - \partial_\nu F_{\mu \rho}) dx^\mu \wedge dx^\nu \wedge dx^\rho##. Notice that this is the same as the first equation in \ref{pluto}, with the indices ##\mu,\nu,\rho,\sigma## being replaced by ##\nu,\rho,\mu,\sigma## respectively.

To see the equivalence of the second equations, we can use the Hodge dual operator ##\star##. Recall that for a p-form ##\omega##, the Hodge dual is defined as ##\star \omega = \frac{1}{p!} \varepsilon_{\mu_1 \mu_2 ... \mu_p} \omega^{\mu_1 \mu_2 ... \mu_p}##, where ##\varepsilon_{\mu_1 \mu_2 ... \mu_p}## is the Levi-Civita symbol. Applying the Hodge dual to the first equation in \ref{pippo}, we have ##\star dF = 0##. Using the definition of the Hodge dual, we can rewrite this as ##d\star F = 0##, which is the second equation in \ref{pippo}.

Finally, we can use the fact that for a 2-form ##F##, we have ##\star F = -F##. Applying this to the second equation in \ref{pippo}, we have ##d\star F = -dF = 0##, which is the second equation in \ref{pluto}

## 1. What are Maxwell's equations with differential forms?

Maxwell's equations with differential forms are a set of four equations that describe the fundamental laws of electricity and magnetism. They are written using differential forms, which are mathematical objects that represent the properties of a physical field at a specific point in space and time.

## 2. What is the significance of using differential forms in Maxwell's equations?

The use of differential forms in Maxwell's equations allows for a more elegant and concise representation of the laws of electromagnetism. It also allows for a deeper understanding of the underlying mathematical structure of these equations.

## 3. How are Maxwell's equations with differential forms different from the traditional vector form?

In the traditional vector form, Maxwell's equations are written using vector calculus operations such as divergence and curl. In the differential form, they are written using differential forms and exterior calculus, which provides a more geometric interpretation of the equations.

## 4. What are the advantages of using Maxwell's equations with differential forms?

Using differential forms in Maxwell's equations allows for a more elegant and compact representation of the equations. It also allows for a more intuitive understanding of the physical meaning behind the equations and their geometric interpretation.

## 5. How are Maxwell's equations with differential forms used in practical applications?

Maxwell's equations with differential forms are used in many practical applications, such as designing and analyzing electromagnetic devices, predicting the behavior of electromagnetic waves, and understanding the properties of materials in the presence of electromagnetic fields.

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