Calculating Metric Matrix with Basis (1,0)^T and (1,1)^T | Homework Question • Physics Forums

narmer · 2007-04-15T20:49:44+00:00

Homework Statement If I have a basis: [(1,0)^T, (1,1)^T], how do I compute the metric matrix, which is [(2,1)^T, (1,1)^T]? Homework Equations The Attempt at a Solution

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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Calculating Metric Matrix with Basis (1,0)^T and (1,1)^T | Homework Question

Homework Statement

Homework Equations

The Attempt at a Solution

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

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