Vector-valued differential equations are often a topic felt to be too advanced to be mentioned in elementary differential equations courses. Part of the problem, of course, is that many of the equations actually involve things like differential forms and other vector-like quantities that are best explained in terms of differential geometry, which takes an introductory course too far afield. The following picture is the magnetic field of current flowing through a solid cube (it is a constant current coming approximately out of the plane of the picture) requiring it to be tangent to the boundary. The method of solution involves special kinds of finite elements specifically for vector-valued functions (really, differential forms), and as talked about in

Anil Hirani's talk. I'll be working with these types of elements for a while, so expect to see more posts on this!

where is the magnetic field coming from? Is it already magnetic, like a block of magnetized iron? Or is it a cubic electromagnet?

ReplyDeleteThere is a wire running through the center moving in the vertical direction (the magnetic field encircles it). The block is not made of iron, but of modestly conducting material...

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