Question

If `u=r^2cos2theta, v=r^2sin2theta. "find"(del(u,v))/(del(r,theta))`

Answer 1

`u=r^2cos2theta`     `v=r^2sin2theta`

Diff. u and v w.r.t r and 𝜽 partially to apply it in jacobian

`(del(u,v))/(del(r,theta))=|(u_r,u_theta),(v_r,v_theta)|=|(2rcos2theta,-2r^2sin2theta),(2rsin2theta,2r^2cos2theta)|`

`=4r^3cos^2 2theta+4r^3sin^2 2theta`

`=4r^3(cos^2 2theta+sin^2 2theta)`

`(del(u,v))/(del(r,theta))=4r^3`