# 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:

Wolfram Research (2012), TransformedField, Wolfram Language function, https://reference.wolfram.com/language/ref/TransformedField.html.

#### Text

Wolfram Research (2012), TransformedField, Wolfram Language function, https://reference.wolfram.com/language/ref/TransformedField.html.

#### CMS

Wolfram Language. 2012. "TransformedField." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/TransformedField.html.

#### APA

Wolfram Language. (2012). TransformedField. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TransformedField.html

#### BibTeX

@misc{reference.wolfram_2022_transformedfield, author="Wolfram Research", title="{TransformedField}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/TransformedField.html}", note=[Accessed: 05-June-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_transformedfield, organization={Wolfram Research}, title={TransformedField}, year={2012}, url={https://reference.wolfram.com/language/ref/TransformedField.html}, note=[Accessed: 05-June-2023 ]}