I'm having kind of a problem on calculating the normal vector to a sphere using a parameterization. Consider a unit-radius sphere centered at the origin.
One can parameterize it using the following:
$$P(\phi, \theta)=(\sin(\phi)\cos(\theta),\,\sin(\phi)\sin(\theta),\,\cos(\phi)) $$
My Vector Calculus book says that the Vector Product between the two partial derivates of the parameterized surface gives a Normal Vector to the surface.
I found that
$$\frac {\partial P(\phi, \theta)}{\partial \phi} \times \frac {\partial P(\phi, \theta)}{\partial \theta} = (\sin^2(\phi)\cos(\theta))\hat i + (\sin^2(\phi)\sin(\theta))\hat j + (\sin(\phi)\cos(\phi))\hat k$$
Which, as one can easily verify, yields $(0, 0,0)$ for $(\phi,\theta) = (0,0)$ which means that the sphere is not regular at the point $(0,0,1)$. That is obviously wrong, so I would like to know where is my mistake.
Best Answer
This is a typical example of singular point(s) arising from the parametrization, they are called the artificial singularities.
For a parametrized surface: $S: \mathbf{r} = \mathbf{r}(u,v)$, like you said: $$ \mathrm{If } \;\; \mathbf{r}_u \times \mathbf{r}_v \neq 0, $$ then it indeed is a normal vector (un-normalized) to the surface $S$. Now: $$ \frac {\partial P(\phi, \theta)}{\partial \phi} \times \frac {\partial P(\phi, \theta)}{\partial \theta} = \begin{vmatrix} \mathbf{i}& \mathbf{j} &\mathbf{k} \\ \cos\phi \cos\theta & \cos\phi \sin\theta & -\sin\phi \\ -\sin\phi \sin\theta & \sin\phi \cos\theta & 0 \end{vmatrix}, $$ notice the cross product is 0 for $\phi = 0,\pi$, this happens due to the choice of parametrization. For example, if you choose the angle between $y$-axis as the polar angle, then this singular point will be gone (other arises though), and this gives you a heuristics of one of the reasons why we wanna cut a manifold into pieces and establish a local coordinate system...