Let \(F(\varphi,\theta) =
(x(\varphi,\theta),y(\varphi,\theta),z(\varphi,\theta))\) be the spherical
polar coordinate system on the unit sphere \(S^{2} =
\left\{(x,y,z) : x^2+y^2+z^2=1\right\}\).
- Sketch the coordinate curves
\(\theta = \pi/4\) and \(\varphi = \pi/2\)
- Compute the derivative to the
coordinate curves from part (1) at the point \((\varphi,\theta) =
(\pi/2,\pi/4)\). Add these arrows to your plot.
- Prove that
\(\partial_{\varphi}F \times \partial_{\theta}F\) is parallel to
\(\nabla(x^2+y^2+z^2)\).