JacobianDeterminant
VectorAnalysis`
VectorAnalysis`
JacobianDeterminant
As of Version 9.0, vector analysis functionality is built into the Wolfram Language »
JacobianDeterminant[pt]
gives the determinant of the Jacobian matrix of the transformation from the default coordinate system to the Cartesian coordinate system at the point pt.
JacobianDeterminant[pt,coordsys]
gives the determinant of the Jacobian matrix of the transformation from the coordinate system coordsys to the Cartesian coordinate system at the point pt.
Details and Options
- To use JacobianDeterminant, you first need to load the Vector Analysis Package using Needs["VectorAnalysis`"].
- JacobianDeterminant[pt,coordsys] is equivalent to Det[JacobianMatrix[pt,coordsys]].
- The coordinates of pt should be given in the coordinate system coordsys, if specified, or the default coordinate system otherwise.
- If pt is not given, the default coordinate variables for coordsys are used.
Examples
Basic Examples (1)
Needs["VectorAnalysis`"]Jacobian determinant for transformation from cylindrical to Cartesian coordinates:
jdet = JacobianDeterminant[Cylindrical]Use the Jacobian determinant to change variables in a multiple integral:
Integrate[E ^ (-x ^ 2 - y ^ 2 - z), {x, 0, Infinity}, {y, 0, Infinity},
{z, 0, Infinity}]Integrate[E ^ (-Rr ^ 2 - z) * jdet, {Rr, 0, Infinity}, {Ttheta, 0, Pi / 2}, {z, 0, Infinity}]