Useful Formulas for the Divergence. Prove

(a) div (kv) = k div v (k constant)

(b) div (fv) = f div v + v • ∇f

(c) div (f∇g) = f ∇²g + ∇f • ∇g

(d) div (f∇g) - div (g∇f) = f∇²g - g∇²f

Verify (b) for f = e^xyz and v = axi + byj + czk. Obtain the answer to Prob. 6 from (b). Verify (c) for f = x² - y² and g = e^x+y. Give examples of your own for which (a)–(d) are advantageous.

Data from Prob. 6

Find div v and its value at P.

v = (x² + y² + z²)^-3/2[x, y, z]