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. Relative scalars are an important special case of the more general concept of a relative tensor.
Ordinary scalar An ordinary scalar or absolute scalar refers to the w = 0 {\displaystyle w=0} case. If x i {\displaystyle x^{i}} and x ¯ j {\displaystyle {\bar {x}}^{j}} refer to the same point P {\displaystyle P} on the manifold, then we desire f ¯ ( x ¯ j ) = f ( x i ) {\displaystyle {\bar {f}}({\bar {x}}^{j})=f(x^{i})} . This equation can be interpreted two ways when x ¯ j {\displaystyle {\bar {x}}^{j}} are viewed as the "new coordinates" and x i {\displaystyle x^{i}} are viewed as the "original coordinates". The first is as f ¯ ( x ¯ j ) = f ( x i ( x ¯ j ) ) {\displaystyle {\bar {f}}({\bar {x}}^{j})=f(x^{i}({\bar {x}}^{j}))} , which "converts the function to the new coordinates". The second is as f ( x i ) = f ¯ ( x ¯ j ( x i ) ) {\displaystyle f(x^{i})={\bar {f}}({\bar {x}}^{j}(x^{i}))} , which "converts back to the original coordinates. Of course, "new" or "original" is a relative concept. There are many physical quantities that are represented by ordinary scalars, such as temperature and pressure.
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