Is there a handy informal argument to derive change of variable volume element for spherical coordinate $r^2 \sin\phi$. Possibly some sort of geometric interpretation? The purpose is to not memorize this formula (say sit for an exam 1 month later). The standard derivation by computing determinant is too lengthy to do it quickly.
2026-03-26 04:30:23.1774499423
How to derive volume element for spherical coordinate fast (maybe informally)?
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Think of an infinitesimal volume cube in spherical coordinates. The three side lengths of the cube are $dr$, $rd\phi$ and $r\sin\phi d\theta$, respectively, and its volume is
$r^2\sin\phi dr d\phi d\theta$
Hence, the variable volume element is just $r^2\sin\phi$.
This brief geometric argument often comes pretty handy.