Use the result of Exercise 27 to establish the formula for the divergence of a C1 vector
Chapter 7, Problem 28(choose chapter or problem)
Use the result of Exercise 27 to establish the formula for the divergence of a C1 vector field F = F1(x, y,z) i + F2(x, y,z) j + F3(x, y,z) k. That is, show that div F = F1 x + F2 y + F3 z in the following manner: Let P have coordinates (x0, y0,z0) and consider the (small) cube S, of edge length a, centered at P with faces parallel to the coordinate planes. Note that the volume V enclosed by S is a3. It will help to recall that if f (x, y,z) is differentiable, then f x = lim x0 f (x + x, y,z) f (x, y,z) x = lim x0 f x + x 2 , y,z f x x 2 , y,z x .
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