Differential Geometry/notation help

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Homework Statement



I have this theorem which I am having trouble understanding due to notation.

Theorem: Define S,S'\subseteqℝ3 to be surfaces. Let f:S→S' be a smooth map. f is a local isometry if for all p in S, and all w1,w2 in TpS,
<w1,w2>=<dfp(w1),dfp(w2)>.


The thing I don't get, is what does dfp(w1) mean?
What action is going on between dfp and w1?


Thank you for your time.
 
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df_p is the derived mapping from the tangent space at p in S to the tangent space at f(p) in S'.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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