Function Repository Resource:

CoordinateMappingData

Source Notebook

Calculate characteristic properties for a generalized mapping between two coordinate systems

Contributed by: Lars Ulke-Winter

ResourceFunction["CoordinateMappingData"][mapping,property]

gives the value of the specified property for the coordinate transformation mapping.

ResourceFunction["CoordinateMappingData"][mapping,property,{x₁,x₂,…,x_n}]

gives the value of the property evaluated at the point {x₁,x₂,…,x_n}.

Details and Options

Metrics and Jacobians are fundamental objects in differential geometry and tensor analysis. ResourceFunction["CoordinateMappingData"] generalizes CoordinateChartData and CoordinateTransformData for arbitrary mappings between systems.

The value for mapping should be a Function or List of relations between coordinates.

Available properties include:

"CovariantBaseVectors"

covariant base vectors b_i (tangential to the coordinates lines)

"ContravariantBaseVectors"

contravariant base vectors b^j

"Metric"

components of the 2-rank covariant metric tensor b_ij=b_i·b_j

"InverseMetric"

components of the 2-rank contravariant metric tensor b^kl=b^k·b^l, with

"VolumeFactor"

coefficient of the differential in volume (surface, line) integrals,

"LeviCivitaCovariant"

, where e_i…n is LeviCivitaTensor[n]

"LeviCivitaContravariant"

"MappingJacobian"

Jacobian matrix of the mapping

"MappingJacobianDeterminant"

determinant of the mapping Jacobian matrix

"InverseMappingJacobian"

inverse of the Jacobian matrix of the mapping

Contravariant base vectors b^j are also known as dual base vectors because the satisfy the relation

For mappings to a lower-dimensional manifold, the "MappingJacobian" property returns the mixed shift tensor.

ResourceFunction["CoordinateMappingData"] takes the following option:

"UnitVectors"

False

whether the components of the basis should be normalized

Examples

Basic Examples (5)

Define local covariant base vectors in cylindrical coordinates:

In[1]:=

Out[1]=

Get covariant base vectors at point {r,ϕ,z}:

In[2]:=

$ResourceFunction[ "CoordinateMappingData"][{Cos[#1[[2]]] #1[[1]], Sin[#1[[2]]] #1[[1]], #1[[3]]} &, "CovariantBaseVectors"][{r, \[Phi], z}]$

Out[2]=

Define a mapping and coordinate system:

In[3]:=

$mapping = {r Cos[\[Phi]], r Sin[\[Phi]], z}; coords = {r, \[Phi], z};$

Calculate the associated covariant and contravariant base vectors:

In[4]:=

Out[4]=

In[5]:=

Out[5]=

Get normalized covariant and contravariant base vectors:

In[6]:=

$mapping = {r Cos[\[Phi]], r Sin[\[Phi]], z}; coords = {r, \[Phi], z};$

In[7]:=

$covBasisNormalized = ResourceFunction["CoordinateMappingData"][mapping, "CovariantBaseVectors", coords, "UnitVectors" -> True] // Simplify[#, {r > 0, \[Phi] > 0}] &$

Out[7]=

In[8]:=

$conBasisNormalized = ResourceFunction["CoordinateMappingData"][mapping, "ContravariantBaseVectors", coords, "UnitVectors" -> True] // Simplify[#, {r > 0, \[Phi] > 0}] &$

Out[8]=

Verify their inverse relationships:

In[9]:=

Out[9]=

Get the covariant metric of mapping to a cylindrical system:

In[10]:=

$mapping = {r Cos[\[Phi]], r Sin[\[Phi]], z}; coords = {r, \[Phi], z};$

In[11]:=

Out[11]=

Compare the result with the cataloged named system of CoordinateChartData:

In[12]:=

Out[12]=

Do a similar computation to get the inverse metric:

In[13]:=

Out[13]=

In[14]:=

Out[14]=

Verify their inverse relationship:

In[15]:=

Out[15]=

Identify the volume factor of a mapping:

In[16]:=

$mapping = {r Cos[\[Phi]], r Sin[\[Phi]], z}; coords = {r, \[Phi], z};$

In[17]:=

Out[17]=

Compare with CoordinateChartData:

In[18]:=

Out[18]=

Calculate the covariant components of the Levi-Civita tensor :

In[19]:=

Out[19]=

Get the contravariant components of the Levi-Civita tensor :

In[20]:=

Out[20]=

Find out which properties are available:

In[21]:=

Out[21]=

Scope (9)

Use the contravariant Levi-Civita symbol to evaluate the cross product :

In[22]:=

$mapping = {r Cos[\[Phi]], r Sin[\[Phi]], z}; coords = {r, \[Phi], z};$

In[23]:=

$wCon = ResourceFunction["CoordinateMappingData"][mapping, "LeviCivitaContravariant", coords] . {0, 1, 0} . {1, 0, 0} // Simplify[#, r > 0] &; covBasis = ResourceFunction["CoordinateMappingData"][mapping, "CovariantBaseVectors", coords]; \!$ \*UnderoverscriptBox[\(\[Sum]$, $i = 1$, $3$]$wCon[[ i]] covBasis[[i]]$\)$

Out[19]=

Check the above result by calculating the cross product of contravariant base vectors b³=b¹⨯b²:

In[24]:=

$conBasis = ResourceFunction["CoordinateMappingData"][mapping, "ContravariantBaseVectors", coords] // Simplify; (conBasis[[1]]\[Cross]conBasis[[2]]) // Simplify$

Out[15]=

Affine coordinate transformation (1)

Find out the properties of an affine coordinate transformation:

In[25]:=

In[26]:=

{conBasis, covBasis, invJacobian, conMetric, conLeviCivita, covLeviCivita, jacobian, jacobianDet, covMetric, volumefactor} = ResourceFunction["CoordinateMappingData"][mapping, #, coords] & /@ (ResourceFunction["CoordinateMappingData"][
"Properties"]);
MatrixForm /@ %

Out[18]=

Polar coordinate transformation (1)

Compute properties for polar coordinates:

In[27]:=

$mapping = {r Cos[\[Phi]], r Sin[\[Phi]]}; coords = {r, \[Phi]}; ResourceFunction["CoordinateMappingData"][mapping, #, coords] & /@ ResourceFunction["CoordinateMappingData"]["Properties"] // Simplify[#, r > 0] &$

Out[26]=

Surfaces embedded in 3D Euclidean space (4)

Identify some metrics on the surface of a sphere:

In[28]:=

Get the tangent space (represented by the covariant vectors):

In[29]:=