MHB Liouville's extension of Dirichlet's theorem

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What is liouville's extension of dirichlet's theorem ?
and where can I use such a theorem ?
Can I apply Integration like this ?
$$\int_0^{\frac{\pi}{2}} \cos^2(x)\sin^2(x)\,dx$$
 
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Re: liouville's extension of dirichlet's theorem

I can help you evaluate that definite integral using elementary techniques if you would like.

Even so, I have moved this topic to Analysis instead. :D
 
Re: liouville's extension of dirichlet's theorem

MarkFL said:
I can help you evaluate that definite integral using elementary techniques if you would like.

Even so, I have moved this topic to Analysis instead. :D

Thank you.
But i want to know how we apply above theorem for integration like this
 
ksananthu said:
What is liouville's extension of dirichlet's theorem ?
and where can I use such a theorem ?
Can I apply Integration like this ?
$$\int_0^{\frac{\pi}{2}} \cos^2(x)\sin^2(x)\,dx$$
You can find a statement of those theorems here. They provide a way to calculate triple integrals of certain functions over the region of three-dimensional space given by $x\geqslant0,\: y\geqslant0,\: z\geqslant0,\: x+y+z\leqslant1.$ I cannot see any way in which these results could have anything to do with an integral such as $$\int_0^{\pi/2} \!\!\!\!\!\!\cos^2(x)\sin^2(x)\,dx$$, which as MarkFL points out can be evaluated using far more elementary techniques.
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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