By Stone M.
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75) We can check this by taking the wedge product. We find F 1 F = (Fµν F µν )σ = (Bx2 + By2 + Bz2 − Ex2 − Ey2 − Ez2 )dtdxdydz. 78) we compute and check that J J = (Jµ J µ )σ = (−ρ2 + jx2 + jy2 + jz2 )dtdxdydz. 4. 80) expresses the charge conservation law. Writing out the terms explicitly shows that the source-containing Maxwell equations reduce to d F = J. All four Maxwell equations are therefore very compactly expressed as dF = 0, d F = J. Observe that current conservation, d J = 0, follows from the second Maxwell equation as a consequence of d2 = 0.
Any reasonable definition of I1 should end up with the answer we would immediately write down if we saw an expression like I1 in an elementary calculus class. That is, I1 = Γ df = f (P1 ) − f (P0 ). 2) We will therefore accept this. Notice that no notion of metric was needed. There is however a geometric picture of what we have done. We draw in our space the surfaces . . , f (x) = −1, f (x) = 0, f (x) = 1, . , and perhaps fill in intermediate values if necessary. We then start at P0 and travel from there to P1 , keeping 53 54 CHAPTER 3.
2 Lie Derivative Another derivative we can define is the Lie derivative along a vector field X. 27) 34 CHAPTER 2. 28) and on anything else by requiring it to be a derivation, meaning that it obeys Leibniz’ rule. For example let us compute the Lie derivative of a covector F . We first introduce an arbitrary vector field Y and plug it into F to get the function F (Y ). 29) and since F (Y ) is a function and Y a vector, both of whose derivatives we know how to compute, we know two of the three terms in this equation.