I started with showing how to solve section 4.10 #2 [Transparancy 0a], involving a time-independent vector field and the flux through a square moving with constant velocity up the z-axis. We verify that the flux transport theorem is valid [Transparancy 0b].
Next, I showed how to work out the left-hand side of the flux transport theorem for section 4.10 #3. This is one of the problems on homework assignment #09. It involves a time-independent vector field and the flux through a square rotating with constant angular velocity. We show that the flux is sinusoidal with time [Transparancy 0c]. The right-hand side of the flux transport theorem is left for the student to work out.
Continuing with Wednesday's lecture on Green's formulas, we looked at a generalization of the 3rd Green formula to having the origin at a vector R instead of (0,0,0) [Transparancy 1]. The integrals are now over "primed" coordinates, and the del operator is also over the primed coordinates.
Taking the Laplacian of this expression, and using Lemma 5.3 to get rid of some terms, we end up with Lemma 5.4 [Transparancy 2].
A differential version of this lemma can be obtained by using the Dirac Delta Function, which is defined to be zero everywhere except when its argument is the zero vector, at which point it becomes infinite. However, the integral of the Dirac Delta Function over all space is 1. More about this next lecture...
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