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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 $0$.

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

2 Answers 2

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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 $$x_\theta(\pi/4,\pi/4)=(-1/2,1/2,0) \\ x_\phi(\pi/4,\pi/4)=(1/2,1/2,-1/\sqrt2)$$

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but for which theta doesnot work? –  moji Feb 19 at 21:49
    
And how can I construct a vector field on $\mathbb{S}^2$ with just one zero –  moji Feb 19 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. $$ p^\perp=\{x\in\mathbb R^3 : (x,p)=0\}. $$

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