# Vector fields on $\Bbb S^2$ [closed]

I have question about vector fields:(pls help me)

Consider 2-sphere $$\mathbb{S}^2= \left\{(x_1,x_2,x_3) \in R^3: x_1^2+x_2^2+x_3^2=1\right\}$$

Let $P \in \mathbb{S}^2$ has spherical coordinates$(1,\phi,\theta)$.Give a base of tangent space $T_p{S}^2$.For which values ​​of theta doesnot work?

Give some example of vector fields on ${\Bbb S}^2$ using above (who so ever touching on ${\Bbb S}^2$)Count for each example the number of points where the vector field equals $0$.

-

## closed as off-topic by Normal Human, Tunk-Fey, studiosus, Claude Leibovici, JimmyK4542Sep 6 '14 at 4:39

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Normal Human, Tunk-Fey, studiosus, Claude Leibovici, JimmyK4542
If this question can be reworded to fit the rules in the help center, please edit the question.

Could you make the question more clear. –  viplov_jain Feb 18 '14 at 20:25

The parametrizarion is $$x(\theta,\phi)=(\cos\theta\sin\phi,\sin\theta\sin\phi,\cos\phi)$$ and so the basis of the tangent space are $$x_\theta(\theta,\phi)=(-\sin\theta\sin\phi,\cos\theta\sin\phi,0) \\ x_\phi(\theta,\phi)=(\cos\theta\cos\phi,\sin\theta\cos\phi,-\sin\phi)$$ For example, at $(\pi/4,\pi/4)$ we $$x_\theta(\pi/4,\pi/4)=(-1/2,1/2,0) \\ x_\phi(\pi/4,\pi/4)=(1/2,1/2,-1/\sqrt2)$$
And how can I construct a vector field on $\mathbb{S}^2$ with just one zero –  marmar Feb 19 '14 at 21:51
The tangent space of $S^2$ at $p\in S^2$ is nothing but the normal hyperspace to the vector $p$, i.e. $$p^\perp=\{x\in\mathbb R^3 : (x,p)=0\}.$$