Once given the right coordinates (polar, sphere, cylindric) I am able to determine the value of a given integral. But how do I know, if the coordinates are not explicitly given, which coordinates to chose?
2026-03-26 20:36:23.1774557383
Intuition behind chosing coordinates
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The whole point of switching coordinates is to exploit symmetries. The typical example is radial symmetry. Assume you want to integrate a function $$ f=f(x_1\ldots x_n)$$ that is radially symmetric, that is $f(\boldsymbol x)=g(|\boldsymbol x|)$ for a function $g=g(r)$ of a single variable, over a ball (=a radially symmetric domain). Then choosing polar coordinates the multiple integral reduces to a single-variable integral, with a significant simplification.
This is a paradigm in the choice of coordinate system. It is generally a good idea to choose a system that is adapted to the symmetries of the problem.