TransformedField

TransformedField[t,f,{x1,x2,,xn}{y1,y2,,yn}]

uses the coordinate transformation t to transform the scalar, vector, or tensor field f from coordinates xi to yi.

Details

  • Coordinate transformations can be specified as rules or oldchart->newchart or triples {oldsys->newsys,metric,dim}, as in CoordinateTransformData. The short form in which dimension is omitted may be used.
  • If f is an array, it must have dimensions {n,,n}. Its components are interpreted as being in the orthonormal basis of the old coordinate chart, and the result is given in the orthonormal basis of the new chart.

Examples

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Basic Examples  (1)

Convert a scalar field from polar to Cartesian coordinates:

Change a vector field from Cartesian to polar coordinates:

Scope  (4)

Transform a scalar field:

Convert a spherical unit vector to Cartesian coordinates:

Convert the vertical unit vector to prolate spheroidal coordinates, specifying both metric and coordinate system:

Convert a rank-2 tensor from polar to Cartesian coordinates:

Applications  (2)

Re-express spherical harmonics in Cartesian coordinates:

An electric dipole of dipole moment located at the origin and aligned with the axis has the following electric potential in spherical coordinates:

Compute the corresponding expression in Cartesian coordinates:

Derive the dipole electric field in spherical coordinates:

Transform this expression to Cartesian coordinates:

The same expression is obtained by differentiating the Cartesian potential function:

Plot the lines of force in the plane:

Properties & Relations  (2)

Use Map to transform a list as a list of scalars rather than as a vector:

The same principle applies to lists of vectors and higher-rank tensors:

TransformedField changes the coordinate expressions for fields:

CoordinateTransform changes the coordinate values of points:

Introduced in 2012
 (9.0)