Cartesian and polar coordinates are related through the transformation equations
{x = r cos Theta or {r^2= x^2 + y^2
{Y = r sin theta { tan theta = y/x

A. Evaluate the partial derivatives Xr, Yr,Xtheta, and Ytheta.
B. Evaluate the partial derivatives Rx, Ry, Theta x, Theta y.
C. For a function z = f(x,y), find z, and Ztheta, where x and y are expressed in terms of r and theta.
D. for a function z = g(r,theta), find Zx, and Zy, where r and theta are expressed in terms of x and y.
E. show that (∂z/∂x)^2 + (∂z/∂y)^2 = (∂z/∂r)^2 + 1/r^2 (∂z/∂theta)^2

Question

Cartesian and polar coordinates are related through the transformation equations
{x = r cos Theta     or         {r^2= x^2 + y^2
{Y = r sin theta                   { tan theta = y/x

A. Evaluate the partial derivatives Xr, Yr,Xtheta, and Ytheta.
B. Evaluate the partial derivatives Rx, Ry, Theta x, Theta y.
C. For a function z = f(x,y), find z, and Ztheta, where x and y are expressed in terms of r and theta.
D. for a function z = g(r,theta), find Zx, and Zy, where r and theta are expressed in terms of x and y.
E. show that (∂z/∂x)^2 + (∂z/∂y)^2 = (∂z/∂r)^2 + 1/r^2 (∂z/∂theta)^2