Relative scalar
In mathematics, a relative scalar (of weight w) is a scalar-valued function whose transform under a coordinate transform, x ¯ j = x ¯ j ( x i ) {\displaystyle {\bar {x}}^{j}={\bar {x}}^{j}(x^{i})} on an n-dimensional manifold obeys the following equation f ¯ ( x ¯ j ) = J w f ( x i ) {\displaystyle {\bar {f}}({\bar {x}}^{j})=J^{w}f(x^{i})} where J = | ∂ ( x 1 , … , x n ) ∂ ( x ¯ 1 , … , x ¯ n ) | , {\displaystyle J=\left|{\dfrac {\partial (x_{1},\ldots ,x_{n})}{\partial ({\bar {x}}^{1},\ldots ,{\bar {x}}^{n})}}\right|,} that is, the determinant of the Jacobian of the transformation. A scalar density refers to the w = 1 {\displaystyle w=1} case.