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Have you looked into the package suite called xAct? It was basically designed for precisely this purpose. It can be found here:

http://xact.es/index.html

An example of how to do your specific problem in xAct is the following:

In[2]:= << xAct`xTensor` In[4]:= DefConstantSymbol[dim] In[5]:= DefManifold[M, dim, {μ, ν, ρ1, ρ2, ρ3}] In[7]:= DefTensor[A[-μ], M] In[8]:= DefTensor[F[-μ, -ν], M] In[10]:= A[ν] A[ρ2] F[-ρ2, -ρ3] F[-ν, ρ3] \ - A[ρ1] A[ρ3] F[-ρ3, -μ] F[-ρ1, μ] ToCanonical[%]  Out[10]= A[ν]  A[ρ2]  F[-ν, ρ3]   F[-ρ2, -ρ3] - A[ρ1]  A[ρ3]  F[-ρ1, μ]  F[-ρ3, -μ] Out[11]= 0 

Have you looked into the package suite called xAct? It was basically designed for precisely this purpose. It can be found here:

http://xact.es/index.html

An example of how to do your specific problem in xAct is the following:

In[2]:= << xAct`xTensor` In[4]:= DefConstantSymbol[dim] In[5]:= DefManifold[M, dim, {μ, ν, ρ1, ρ2, ρ3}] In[7]:= DefTensor[A[-μ], M] In[8]:= DefTensor[F[-μ, -ν], M] In[10]:= A[ν] A[ρ2] F[-ρ2, -ρ3] F[-ν, ρ3] \ - A[ρ1] A[ρ3] F[-ρ3, -μ] F[-ρ1, μ] Out[10]= A[ν] A[ρ2] F[-ν, ρ3] F[-ρ2, -ρ3] - A[ρ1] A[ρ3] F[-ρ1, μ] F[-ρ3, -μ] In[11]:= ToCanonical[%] Out[11]= 0 

Have you looked into the package suite called xAct? It was basically designed for precisely this purpose. It can be found here:

http://xact.es/index.html

An example of how to do your specific problem in xAct is the following:

In[2]:= << xAct`xTensor` In[4]:= DefConstantSymbol[dim] In[5]:= DefManifold[M, dim, {\[Mu], \[Nu], \[Rho]1, \[Rho]2, \[Rho]3}] In[7]:= DefTensor[A[-\[Mu]], M] In[8]:= DefTensor[F[-\[Mu], -\[Nu]], M] In[10]:= A[\[Nu]] A[\[Rho]2] F[-\[Rho]2, -\[Rho]3] F[-\[Nu], \[Rho]3] \ - A[\[Rho]1] A[\[Rho]3] F[-\[Rho]3, -\[Mu]] F[-\[Rho]1, \[Mu]] ToCanonical[%] Out[10]= A[\[Nu]] A[\[Rho]2] F[-\[Nu], \[Rho]3] F[-\[Rho]2, -\[Rho]3] - A[\[Rho]1] A[\[Rho]3] F[-\[Rho]1, \[Mu]] F[-\[Rho]3, -\[Mu]] Out[11]= 0 

Have you looked into the package suite called xAct? It was basically designed for precisely this purpose. It can be found here:

http://xact.es/index.html

An example of how to do your specific problem in xAct is the following:

In[2]:= << xAct`xTensor` In[4]:= DefConstantSymbol[dim] In[5]:= DefManifold[M, dim, {μ, ν, ρ1, ρ2, ρ3}] In[7]:= DefTensor[A[-μ], M] In[8]:= DefTensor[F[-μ, -ν], M] In[10]:= A[ν] A[ρ2] F[-ρ2, -ρ3] F[-ν, ρ3] \ - A[ρ1] A[ρ3] F[-ρ3, -μ] F[-ρ1, μ] ToCanonical[%] Out[10]= A[ν] A[ρ2] F[-ν, ρ3] F[-ρ2, -ρ3] - A[ρ1] A[ρ3] F[-ρ1, μ] F[-ρ3, -μ] Out[11]= 0 

Have you looked into the package suite called xAct? It was basically designed for precisely this purpose. It can be found here:

http://xact.es/index.html

Have you looked into the package suite called xAct? It was basically designed for precisely this purpose. It can be found here:

http://xact.es/index.html

An example of how to do your specific problem in xAct is the following:

In[2]:= << xAct`xTensor` In[4]:= DefConstantSymbol[dim] In[5]:= DefManifold[M, dim, {\[Mu], \[Nu], \[Rho]1, \[Rho]2, \[Rho]3}] In[7]:= DefTensor[A[-\[Mu]], M] In[8]:= DefTensor[F[-\[Mu], -\[Nu]], M] In[10]:= A[\[Nu]] A[\[Rho]2] F[-\[Rho]2, -\[Rho]3] F[-\[Nu], \[Rho]3] \ - A[\[Rho]1] A[\[Rho]3] F[-\[Rho]3, -\[Mu]] F[-\[Rho]1, \[Mu]] ToCanonical[%] Out[10]= A[\[Nu]] A[\[Rho]2] F[-\[Nu], \[Rho]3] F[-\[Rho]2, -\[Rho]3] - A[\[Rho]1] A[\[Rho]3] F[-\[Rho]1, \[Mu]] F[-\[Rho]3, -\[Mu]] Out[11]= 0