# Curl or maxwell equations in higher dimensions

#### snowstorm69

Anyone know what topic, branch of math, book, or subject I should look up in order to find a formulation for Maxwell's equations in higher spatial dimensions? I don't mean having time as a 4rth dimension. I mean a 4rth (and more) spatial dimension. This would require the maxwell exquations involving Curl to be represented in higher dimensions, which would require that the curl itself be represented in higher dimensions. Does the curl (and do the 2 maxwell's equations involving curl) only apply to 3-D or is it extendable to higher dimensions? Where can I read about this?
Thanks!

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#### snowstorm69

Dear Vid,
Thank you so much! Can't wait to read this, and very cool that he was your high school physics teacher! Meanwhile I hope this doesn't discourage others from posting a response but... this response does look excellent! My heartfelt thanks for your help.

#### haushofer

The book of Zwiebach on string theory covers this topic quite early, somewhere in the first chapters. Maybe interesting for you.

#### snowstorm69

Thanks, haushofer,
Looks like a great book; I think I'm going to get it. Also... what about the Helmholtz equation that you get from the Maxwell's equations; any idea where to read about that in higher spatial dimensions? I may post that as a separate question but am very interested in an answer whether from this thread or another. Thanks guys.

#### Phrak

To do it 'right' you first might want to redefine the 4-vector potential as a 5-vector.

But it doen't partition as nicely as it does in spacetime. In the usual spacetime you get the electric and magnetic fields that both appear to be vectors (but don't really transform as vectors with a change in inertial frame).

In 5 dimensions you get something you might call an 'electric field' that 4 components--one that can be associated with each spacial dimension. But the other part that generalizes the magnetic field has 6 elements. (Each element is associated with two of your spacial dimensions rather than one-on-one.)

On top of all that, you get magnetic monopoles popping out of it, after taking another derivative.

This all has to do with the way the generalization of the crossproduct that utilizes the completely antisymmetric field tensor behaves in higher dimensions.

There's nothing wrong with investigating this, of course, it just might not turn out to be as you expect.
-deCraig

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