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Vector Calculus - Maxwell's Equation

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I'm having trouble figuring out how to solve this. I think I figured out some of it, but I don't understand it in general. I attached the problem as a jpeg. I'd appreciate seeing how to show the answer. You can ignore the pencil markings, they are notes to myself in trying to figure out the problem. Thanks.

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The expert examines vector calculus for Maxwell's equations.

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Look at the vector identity for del of the cross product of two vectors with E and B the two vectors. the curl of B has the time derivitive of E and ...

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