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
open all close allBasic Examples (1)
Convert a scalar field from polar to Cartesian coordinates:
TransformedField["Polar" -> "Cartesian", r ^ 2 Cos[θ], {r, θ} -> {x, y}]Change a vector field from Cartesian to polar coordinates:
TransformedField[ "Cartesian" -> "Polar", {x, y}, {x, y} -> {r, θ}]//SimplifyScope (4)
TransformedField["Cartesian" -> "Spherical", x ^ 2 + y ^ 2, {x, y, z} -> {r, θ, φ}]//SimplifyConvert a spherical unit vector to Cartesian coordinates:
TransformedField["Spherical" -> "Cartesian", {0, 0, 1}, {r, θ, φ} -> {x, y, z}]Convert the vertical unit vector to prolate spheroidal coordinates, specifying both metric and coordinate system:
TransformedField[{"Cartesian" -> "ProlateSpheroidal", "Euclidean"}, {0, 0, 1}, {x, y, z} -> {ξ, η, φ}]//SimplifyConvert a rank-2 tensor from polar to Cartesian coordinates:
TransformedField["Polar" -> "Cartesian", {{1, 0}, {0, r ^ 2 Sin[θ] ^ 2}}, {r, θ} -> {x, y}]//SimplifyApplications (2)
Re-express spherical harmonics in Cartesian coordinates:
SphericalHarmonicY[3, 1, θ, ϕ]FullSimplify[TransformedField["Spherical" -> "Cartesian", SphericalHarmonicY[3, 1, θ, ϕ], {r, θ, ϕ} -> {x, y, z}]]An electric dipole of dipole moment 
located at the origin and aligned with the 
axis has the following electric potential in spherical coordinates:
Vs = (p Cos[θ]) / r ^ 2Compute the corresponding expression in Cartesian coordinates:
Vc = TransformedField[ "Spherical" -> "Cartesian", Vs, {r, θ, φ} -> {x, y, z}]Derive the dipole electric field in spherical coordinates:
Es = -Grad[Vs, {r, θ, φ}, "Spherical"]Transform this expression to Cartesian coordinates:
Ec = TransformedField["Spherical" -> "Cartesian", Es, {r, θ, φ} -> {x, y, z}]The same expression is obtained by differentiating the Cartesian potential function:
Ec == -Grad[Vc, {x, y, z}]//SimplifyPlot the lines of force in the 
plane:
StreamPlot[Rest[Ec /. {p -> 1, x -> 0}]//Evaluate, {y, -1, 1}, {z, -1, 1}]Properties & Relations (2)
Use Map to transform a list as a list of scalars rather than as a vector:
TransformedField[{"Cartesian" -> {"Bipolar", a}}, #, {x, y} -> {μ, ν}]& /@ {x, y, x ^ 2 + y ^ 2}The same principle applies to lists of vectors and higher-rank tensors:
TransformedField[{"Cartesian" -> {"Bipolar", a}}, #, {x, y} -> {μ, ν}]& /@ {{0, 1}, {1, 0}, {x, y}}//FullSimplifyTransformedField changes the coordinate expressions for fields:
TransformedField["Cartesian" -> "Polar", {x, y}, {x, y} -> {r, θ}]//SimplifyCoordinateTransform changes the coordinate values of points:
CoordinateTransform["Cartesian" -> "Polar", {x, y}]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