Under a smooth locally invertible coordinate map into Euclidean space, the positive volume element in a multiple integral is multiplied by the absolute Jacobian determinant. In three-dimensional spherical coordinates, ; in cylindrical coordinates, . The absolute value distinguishes positive integration measure from a signed orientation convention.
A volume element is a differential quantity used in mathematics and physics, typically in the context of calculus and geometric analysis. It represents an infinitesimally small portion of space, allowing for the integration and measurement of quantities over three-dimensional regions.
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