Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Basic Tensor Computation
Plus two tensors together:
In[10]:=
T=
["T",{0,2},{x,y},Array["T",{2,2}]];S=
["S",{0,2},{x,y},Array["S",{2,2}]];sum=S["ab"]+T["ab"]
 | CreateTensor |
| CreateTensor |
Out[12]=
STensor
[,ab]
 |  |
|
In[13]:=
sum["Components"]//MatrixForm
Out[13]//MatrixForm=
S[1,1]+T[1,1] | S[1,2]+T[1,2] |
S[2,1]+T[2,1] | S[2,2]+T[2,2] |
Two tensor with different rank or coordinate system can not be added together. The result will be unevaluated.
In[14]:=
T=
["T",{0,2},{x,y},Array["T",{2,2}]];S=
["S",{1,1},{x,y},Array["S",{2,2}]];P=
["P",{0,2},{r,θ},Array["S",{2,2}]];
| CreateTensor |
| CreateTensor |
| CreateTensor |
In[18]:=
{T+S,T+P}
Out[18]=
STensor
+STensor
,STensor
+STensor
|  |
|
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Scalar multiplication can be done directly as any other expressions:
In[23]:=
T=
["T",{0,2},{x,y},Array["T",{2,2}]];kT
| CreateTensor |
Out[24]=
STensor
 | |
|
Tensor product can be done with indices:
In[5]:=
T=
["T",{0,2},{x,y},Array["T",{2,2}]];S=
["S",{1,1},{x,y},Array["S",{2,2}]];
| CreateTensor |
| CreateTensor |
In[7]:=
T["ab"]S["c","d"]
Out[7]=
STensor
[c,abd]
|  |
|
Tensor contract can be done when there are common indices both in sup and sub:
In[8]:=
{T["ab"]S["a","c"],S["a","a"]}
Out[8]=
STensor
[,bc],S[1,1]+S[2,2]
| |
|
When calculating product of two tensors with different coordinate system, a warning will be given:
In[9]:=
T=
["T",{0,2},{x,y},Array["T",{2,2}]];S=
["S",{1,1},{r,θ},Array["S",{2,2}]];
| CreateTensor |
| CreateTensor |
In[12]:=
T["ab"]S["c","d"]
Out[12]=
STensor
[c,abd]
| |
|
Compute wedge product of two differential forms:
In[13]:=
T=
"T",{0,2},{x,y,z},
[Array["T",{3,3}],Antisymmetric[All]];S=
"T",{0,2},{x,y,z},
[Array["T",{3,3}],Antisymmetric[All]];T["ab"]⋀S["cd"]
| CreateTensor |
 | Symmetrize |
| CreateTensor |
| Symmetrize |
Out[15]=
STensor
[abcd]
|  |
|
The tensors involved will be antisymmetrized automatically:
In[16]:=
T=
["T",{0,2},{x,y,z},Array["T",{3,3}]];S=
["T",{0,2},{x,y,z},Array[&,{3,3}]];T["ab"]⋀S["cd"]
| CreateTensor |
| CreateTensor |
"s"
##
Out[18]=
STensor
[abcd]
| |
|
Derivative Operator will hold the same if the metric is not set:
In[19]:=
T=
["T",{0,2},{r,θ,ϕ},Array["T",{3,3}]];Grad[T["ab"],"c"]
| CreateTensor |
Out[20]=
It will operate on tensors when the metric is given:
In[21]:=
g=
"g",{r,θ,ϕ},DiagonalMatrix[{1,r^2,r^2Sin[θ]^2}],
True;T=
["T",{0,2},{r,θ,ϕ},Array["T"[r],{3,3}],g];dT=Grad[T["ab"],"c"]
| CreateTensor |
| IsMetric |
| CreateTensor |
Out[23]=
STensor
[,abc]
|  |
|
Similar to Grad, D represents the ordinary derivative operator replying on coordinate system:
In[24]:=
g=
"g",{r,θ,ϕ},DiagonalMatrix[{1,r^2,r^2Sin[θ]^2}],
True;T=
["T",{0,2},{r,θ,ϕ},Array["T"[r],{3,3}],g];DT=D[T["ab"],"c"]
| CreateTensor |
| IsMetric |
| CreateTensor |
Out[26]=
STensor
[,abc]
| |
|
Check all non-zero components
In[27]:=
DT["Non-zero Components"]//TableForm
Out[27]//TableForm=
∂(T) 1,1,1 ′ T |
∂(T) 1,2,1 ′ T |
∂(T) 1,3,1 ′ T |
∂(T) 2,1,1 ′ T |
∂(T) 2,2,1 ′ T |
∂(T) 2,3,1 ′ T |
∂(T) 3,1,1 ′ T |
∂(T) 3,2,1 ′ T |
∂(T) 3,3,1 ′ T |
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