Question

Answer with chain rule

Z = cos (x² + y²), x = u cos v, y = u sin v

Answer 1

$Z_u=Z_xx_u+Z_yy_u$

$Z_u=-2xsin(x² + y²)cosv-2ysin(x² + y²)sinv=$

$=-2ucos^2vsin(u^2)-2usin^2vsin(u^2)=-2usin(u^2)$

$Z_v=Z_xx_v+Z_yy_v$

$Z_v=2xusin(x² + y²)sinv-2yusin(x² + y²)cosv=$

$=u^2sin2vsinu^2-u^2sin2vsinu^2=0$