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Function Repository Resource:

 TensorPureFunction

Source Notebook

Get a pure function whose argument is a matrix for a given tensor

Contributed by: E. Chan-López & Víctor Castellanos

ResourceFunction["TensorPureFunction"][t,argvars]

gives the corresponding pure function representation of a tensor t for a matrix of variables argvars.

ResourceFunction["TensorPureFunction"][t]

represents an operator form that can be applied to arguments.

Details

The t must be a tensor of any rank.
The argvars must be a tensor of rank 2 (matrix) containing the variables of interest.
ResourceFunction["TensorPureFunction"][t,argvars ] can be used to obtain different functions with array arguments in the basic vector calculus and with multilinear functions to study normal forms in applied bifurcation theory.

Examples

Basic Examples (3) 

Define a function that takes a matrix of variables as input and returns a vector:

In[1]:=
func = ResourceFunction["TensorPureFunction"][{x^2 + y}, {{x, y}}]
Out[1]=

Apply the function:

In[2]:=
func[{a, b}]
Out[2]=

Use TensorPureFunction with a tensor of rank 3:

In[3]:=
t3 = ResourceFunction["TensorPureFunction"][ D[{x^3 y - 2 x y^4 - y^6, x^2 y^2 - 5 x y^3}, {{x, y}, 2}], {{x, y}}]
Out[3]=

Apply the function:

In[4]:=
t3[{1, 1}] // MatrixForm
Out[4]=

Use TensorPureFunction with a tensor of rank 4:

In[5]:=
t4 = ResourceFunction["TensorPureFunction"][ D[{x^3 y - 2 x y^4 - y^6, x^2 y^2 - 5 x y^3}, {{x, y}, 3}], {{x, y}}]
Out[5]=

Apply the function:

In[6]:=
t4[{1, 1}] // MatrixForm
Out[6]=

Scope (3) 

Use TensorPureFunction in combination with Grad:

In[7]:=
ResourceFunction["TensorPureFunction"][ Grad[x y Sin[z], {x, y, z}], {{x, y, z}}]
Out[7]=

Use TensorPureFunction in combination with Curl:

In[8]:=
ResourceFunction["TensorPureFunction"][ Curl[{Exp[y] Sin[w], Exp[x] Cos[w], x y}, {x, y, w}], {{x, y, w}}]
Out[8]=

Use TensorPureFunction to compute a Jacobian matrix:

In[9]:=
jacobian = ResourceFunction["TensorPureFunction"][ Simplify@D[{x^3 - 2 x y - y^6, x y^2 - 5 x y}, {{x, y}}], {{x, y}}]
Out[9]=

Apply the Jacobian function:

In[10]:=
jacobian[{1, 1}] // MatrixForm
Out[10]=

Use TensorPureFunction to compute a Hessian matrix:

In[11]:=
hessian = ResourceFunction["TensorPureFunction"][ D[x^3 - 2 x y - y^6, {{x, y}, 2}], {{x, y}}]
Out[11]=

Apply the Hessian function:

In[12]:=
hessian[{1, 1}] // MatrixForm
Out[12]=

Applications (4) 

Multilinear Functions (2) 

Use TensorPureFunction with the following trilinear function:

In[13]:=
d3 = ResourceFunction[ "TensorPureFunction"][{-2 x3 y1 y2 - 2 x2 y1 y3 - 2 x1 y2 y3, 4 x2 x3 y1 + 4 x1 x3 y2 + 4 x1 x2 y3}, {{x1, y1}, {x2, y2}, {x3, y3}}]
Out[13]=

Apply the trilinear function:

In[14]:=
d3[{{1, 1}, {1, 2}, {1, -1}}] // MatrixForm
Out[14]=

Deformation Gradient and Vorticity Tensor (2) 

Define a simple function to compute the deformation gradient given a velocity field at the point :

In[15]:=
DeformationGradient[velocityField_List, coordinates_List] := 1/2 (Grad[velocityField, coordinates] + Transpose[Grad[velocityField, coordinates]])

The calculation of the deformation gradient for the -velocity is as follows:

In[16]:=
u[x_, y_, z_] := {x*z, y*z, x*y} DeformationGradient[u[x, y, z], {x, y, z}]
Out[17]=

Now, use TensorPureFunction to obtain a pure function of the above array:

In[18]:=
dg = ResourceFunction["TensorPureFunction"][%, {{x, y, z}}]
Out[18]=
In[19]:=
dg[{1, 1, 1}] // MatrixForm
Out[19]=

Define a simple function to compute the vorticity tensor given a velocity field at the point :

In[20]:=
VorticityTensor[velocityField_, coordinates_List] := 1/2 (Grad[velocityField, coordinates] - Transpose[Grad[velocityField, coordinates]])

The calculation of the vorticity tensor for the -velocity is as follows:

In[21]:=
u[x_, y_, z_] := {-y*x, x*y, y*z^2} VorticityTensor[u[x, y, z], {x, y, z}]
Out[22]=

Now, use TensorPureFunction to obtain a pure function of the above array:

In[23]:=
vt = ResourceFunction["TensorPureFunction"][%, {{x, y, z}}]
Out[23]=
In[24]:=
vt[{1, 1, 1}] // MatrixForm
Out[24]=

Properties and Relations (3) 

Use TensorPureFunction with the resource function JacobianMatrix:

In[25]:=
ResourceFunction["TensorPureFunction"][ Simplify@ ResourceFunction["JacobianMatrix"][{x^3 - 2 x y - y^6, x y^2 - 5 x y}, {x, y}], {{x, y}}]
Out[25]=

This is equivalent the computing the Jacobian directly:

In[26]:=
jacobian = ResourceFunction["TensorPureFunction"][ Simplify@D[{x^3 - 2 x y - y^6, x y^2 - 5 x y}, {{x, y}}], {{x, y}}]
Out[26]=
In[27]:=
% === jacobian
Out[27]=

Use TensorPureFunction with the resource function HessianMatrix:

In[28]:=
ResourceFunction["TensorPureFunction"][ ResourceFunction["HessianMatrix"][ x^3 - 2 x y - y^6, {x, y}], {{x, y}}]
Out[28]=

This is equivalent the computing the Hessian directly:

In[29]:=
hessian = ResourceFunction["TensorPureFunction"][ D[x^3 - 2 x y - y^6, {{x, y}, 2}], {{x, y}}]
Out[29]=
In[30]:=
% === hessian
Out[30]=

Use TensorPureFunction with the resource function DVectorField:

In[31]:=
d21 = ResourceFunction["TensorPureFunction"][ DVectorField[{x - y^2, 2 y/x}, {x, y}, {1, 2}, 2, "MultilinearFunction"], {{x1, y1}, {x2, y2}}];
In[32]:=
d22 = DVectorField[{x - y^2, 2 y/x}, {x, y}, {1, 2}, 2, "PureMultilinearFunction"];

The previous tensors are identical, as seen below:

In[33]:=
SameQ[d21, d22]
Out[33]=

Publisher

Ramón Eduardo Chan López

Version History

  • 2.0.0 – 02 October 2023
  • 1.0.0 – 08 November 2022

Source Metadata

  • Citation:
    • Marsden, J. E., & Tromba, A. (2003). Vector calculus. Macmillan.
    • Kuznetsov, Y. A. (2013). Elements of applied bifurcation theory (Vol. 112). Springer Science & Business Media.
    • Kuznetsov YA (1999) Numerical normalization techniques for all codim 2 bifurcations of equilibria in ODE's. SIAM J Numer Anal 36: 1104–1124.

Related Resources

Author Notes

The current implementation draws inspiration from the in-depth discussion and contributions of various users on the Mathematica Stack Exchange Community in the post titled Can a pure function be constructed whose argument list is a matrix?.

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