Advanced Calculus fi..

Chapter 3 Vector Differential Calculus 181a) Find this matrix for the functions w = x3y - y3z and w = x: + 2x1x2 + 5x1x3 +2 x 2 + ~ 4x22 ~ + x2xg + 5 ~ ~ + x x3x2 1 + 2x32.b) Show, under appropriate assumptions, that H is symmetric.c) If z = f (x, y), show that V,Vgz = [cos ,b sin ,b] H[cos a sina]' (see Section 2.21).3.4 THE DIVERGENCE OF A VECTOR FIELDGiven a vector field v in a domain D of space, one has (for a given coordinate system)three scalar functions v, , v,, v,. If these possess first partial derivatives in D, we canform in all nine partial derivatives, which we arrange to form a matrix:From three of these the scalar div v, the divergence of v, is constructed by theformula:It will be noted that the derivatives used form a diagonal (principal diagonal) of thematrix and the divergence is the trace (Problem 6 following Section 1.11).For example, if v = x2i - xyj + xyzk, thenFormula (3.15) can be written in the symbolic form:for, treating V as a vector, one hasdiv v = V v; (3.16)~he.definition of the divergence at first appears to be quite arbitrary and todepend on the choice of axes in space. It will be seen in Section 3.8 that this isnot the case. The divergence has in fact a definite physical significance. In fluiddynamics it appears as a measure of the rate of decrease of density at a point. Moreprecisely, let u = u(x, y, z, t) denote the velocity vector of a fluid motion and letp = p(x, y, z, t) denote the density. Then v = pu is a vector whose divergencesatisfies the equationa~div v = --. at