Let f(z)=u+iv be an analytic function, phi(u,v) any function with second order partial derivatives and g(u,v) any function with first order partial derivatives.
a) Let L_x,y be the Laplace operator in x,y coordinates and L_u,v be the Laplace operator in u,v coordinates. Show that L_x,y(phi o f)=L_u,v |f'(z)|^2
b)Let G_u,v be the gradient vector in u,v, and G_x,y be the gradient vector in x,y coordinates. Show that |G_x,y(g o f)|^2=|G_u,v(g)|^2 |f'(z)|^2 where the | | is the euclidean norm in C.
Laplace Operators and Gradient Vectors are investigated. The solution is detailed and well presented.