partial derivatives and gradient

Question

For c）,can anyone provide some hints? I can only think of showing the cross product equals 0 but don't know how

Answer 1

Hint :

• What is the explicit function of $x(\phi, \theta)$ ? $y(\phi, \theta)$ ? $z(\phi, \theta)$ ?
• Compute $\partial_\theta F=(\partial_\theta x, \partial_\theta y, \partial_\theta z)$, and $\partial_\phi F=(\partial_\phi x, \partial_\phi y, \partial_\phi z)$
• Compute $\partial_\theta F \times \partial_\phi F $
• What is $\nabla(x^2+y^2+z^2)$ in terms of $(x,y,z)$ ? Replace with $(x(\phi, \theta), y(\phi, \theta), z(\phi, \theta))$
• Can you notice the relation of proportionality ? If not compute the cross product $(\partial_\theta F \times \partial_\phi F ) \times \nabla(x^2+y^2+z^2)$