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Let $\pi: S^2-\{N\}\to \mathbb R^2$ be the stereographic projection map.

Let $\sigma:\mathbb R^2\to S^2-\{N\}$ be its inverse.

Let $p\in S^2-\{N\}$ and $x_1,x_2\in$ the tangent space of $S^2$

Would someone be nice enough to explain to me then what the following mean intuitively? And hopefully also a way to visualize them/gain some sort of physical intuition on them?

1) $(d\sigma)_{\pi(p)}$

2) $d\pi_p$

3) Why $x_1\cdot x_2=(d\pi_p(x_1),d\pi_p(x_2))_{\pi(p)}$

4) $d\pi_p\circ (d\sigma)_{\pi(p)}=$ identity

I have read the differential forms article on Wikipedia in hope to learn more, but I still don't quite get the idea. I know for example that (1) is the differential of $\sigma$ at the point $\pi(p)$ but I don't understand what that means. I hope that someone could give me a geometric picture of some kind. And if there should be such a saint out there, I would like to thank you very much (in advance).

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