Nikhil Kumar Singh

∫ [Li₂(x²-1)]/(1-x²) dx from 0 to 1 -∫ [Li₂(x²-1)]/(x²-1) dx from 0 to 1 Introduce summation and compose well. We later arrive at Li₃(-¹/₄)

This integral diverges, Yes you're right in saying that the indefinite integration is equal to $\tan x$ but when you apply your limits, it has discontinuity which means it probably does not converge.

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a map that continuous in each variable separately. Assume that for any compact subset $K\subset\mathbb{R}^2$, $f(K)$ is compact. Prove that $f$ is continuous.

I have known that RP^nis smooth manifold.What is the difference of?the projective map between CPn and RPn

If you are interested to find the surface area of the strip $(h)$: You can use integration: However, $V = {\pi}(R^2h-\frac{h^3}{3})$ is your formula It's for hemisphere: The height$(h)$ is ⊥ to the base of the hemisphere. Let: h = R $V(R) = 2{\pi}R^3/3$ Which is of hemisphere.

Okay i know nothing about math, i didn't read all comments, it was Abit above my head, but i think what OP meant in their question is something along the lines.. i don't even know is what I'm trying to explain here is correct in any way, but myself coming here for wondering pretty much the same t...