Maxwell's Equations: Symbols & Understanding

[FluxCapacitator]

I was curious about the famous Maxwell's equations, and decided, perhaps foolishly, to learn them myself. I know basic Electricity and Magnetics with Calculus, so I figured it was the next logical step. I understood the integral forms of Maxwell's equations, but I got completely lost when I saw the upside down deltas in the differential forms of the equations. Could someone tell me what they stand for and/or what mathematical operation they entail?

 

[StatusX]

That's called the del operator, and it is a concise way of representing a number of different operations in vector calculus. Maxwell's equations use two of these operations: divergence and curl. The divergence of a vector field is a scalar field that rougly measures how much the field is flowing into or out of each point. The curl is a vector field that measures how much it curls around each point, with the magnitude of the vector representing the magnitude of the curl and the direction representing the direction of flow (the same way the angular momentum vector works). To actually compute these quantities, the del operator can be written as \frac{d}{dx}\hat x+\frac{d}{dy}\hat y+\frac{d}{dz}\hat z, and then the appropriate operations (dot or cross products) can be performed on the field, substituting the differential operation for multiplication. This only works in cartesian coordinates, and in other systems (eg, spherical), the del operator is written differently.

 

[FluxCapacitator]

So, just to clarify, if you had the electric field as some vector function...you'd take the derivative of that vector, then dot or cross that derivative with the original electric field function, as shown here?

deloperator(E)= E' x E

Where E=electrive field (vector), E' = the derivative of the electric field.

Or have I just horribly confused it?

Also, is there any advantage to using the differential forms over integral forms?

EDIT: By derivative, I mean take the derivative of each vector component with respect to that components axis.

 

[StatusX]

No, I'm sorry, I should have been clearer. You treat the del as if it were a vector. So if the vector field has components E_x, E_y, and E_z, then dotting the del (taking the divergence) would look like this:

\nabla \cdot E= (\frac{d}{dx} \hat x + \frac{d}{dy} \hat y + \frac{d}{dz} \hat z) \cdot (E_x \hat x + E_y \hat y +E_z \hat z)

= \frac{d}{dx} E_x + \frac{d}{dy} E_y + \frac{d}{dz} E_z = \frac{dE_x}{dx} + \frac{dE_y}{dy} + \frac{dE_z}{dz}

Curl is a little more complicated, but it's the same idea. As for the usefulness of each form, it all depends on what you're trying to do. Gauss' law, for example, is usually used in its integral form for most problems. I would say that most proofs and derivations of things like EM waves and the energy and momentum stored in the fields are easier using the differential forms of the equations.

 

[FluxCapacitator]

Thanks a 3x10^8 ;) .