Relative scalar

A relative scalar (of weight $w$) is a scalar-valued function whose transform under a coordinate transform,

${\bar {x}}^{j}={\bar {x}}^{j}(x^{i})$

on an $n$-dimensional manifold obeys the following equation

${\bar {f}}({\bar {x}}^{j})=J^{w}f(x^{i})$

where $J={\frac {\partial ({\bar {x}}^{1},\ldots ,{\bar {x}}^{n})}{\partial (x_{1},\ldots ,x_{n})}}$, that is, the determinate of the transformation.

Scalar density

A scalar density refers to the $w=1$ case.

Generalization

Scalar densities are special cases of tensor densities.

See also

• Jacobian matrix and determinant