= Area formula
{disambiguate=geometric measure theory}
{wiki}
= Area formula
{synonym}
For a <locally Lipschitz function> $T$ between Euclidean spaces of the same dimension, the area formula integrates its absolute <Jacobian determinant> against the multiplicity of its fibers. In particular, for an injective $C^1$ map $T:X\to\mathbb R^d$ and nonnegative measurable $a$,
$$
\int_X a(T(x))|\det DT(x)|\,dx=\int_{T(X)}a(y)\,dy.
$$
This injective version extends the <change of variables formula> to maps with a possibly singular derivative.
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