Software Tools that allow you to do symbolic Ricci Calculus / Tensor Notation?

Question

Here are some examples of problems that I want to be able to solve for symbolically:

Let $R = \sqrt{x_k x_k}$. Calculate $\displaystyle{\partial R \over \partial x_i}$ and $\displaystyle{\partial^2 R \over \partial x_i \partial x_j}$

Given that tensor R has component of $R_{ij} = \cos(\theta) \delta_{ij} + n_i n_j(1 - \cos (\theta)) - \sin(\theta) \varepsilon_{ijk} n_k$, where $n_k$ are the unit vector components, calculate $R_{ik} R'_{jk}$

Let a tensor A be given by: $$ \mathbf{A} = \alpha(\mathbf{I} - \mathbf{e_1} \otimes \mathbf{e_2}) + \beta (\mathbf{e_1} \otimes \mathbf{e_2} + \mathbf{e_2} \otimes \mathbf{e_1})$$ Calculate the eigenvalues and derive the associated normalized eigenvectors.

In other words, I want to be able to use some package that has symbolic manipulations that can work with these sorts of problems. The emphasis is on symbolic - I am dealing with theoretical problems, not once that I have numerical answers in. I need the inputs to be symbolic and the outputs to be symbolic, with both in the form of the examples that I have given. These sorts of theoretical problems are also not given to me in Voigt notation, which I want to avoid using if possible. If I have any software that I would be biased towards wanting the package to be in, then I would prefer Mathematica. However, any software package that meets my need here would be highly appreciated. Half a year ago I attempted to find a package that did this, and ended up wasting four hours because the package could only spit out numerical answers.

Answer 1

There are several software packages available, Xact and Cadabra come to mind. However, the examples you provide are easily handled using the the built in Mathematica commands TensorProduct and TensorContract.

I will illustrate using your first example. Let $x_k=(x_1,x_2)$ be a coordinate vector. We can start by making a new tensor $T_{jk}=x_jx_k$ using TensorProduct:

TensorProduct[xk, xk]

Next we contract the two indices using TensorContract:

TensorContract[TensorProduct[xk, xk], {1, 2}]

Finally we take the square root and inspect the result:

Sqrt[TensorContract[TensorProduct[xk, xk], {1, 2}]]

$R=\sqrt{x_1^2+x_2^2}$

Now if we want to calculate the gradient, we just apply the D operator on R, $T_i=\frac{\partial R}{\partial x_i}$:

Ti = D[Sqrt[TensorContract[TensorProduct[xk, xk], {1, 2}]], {xk,1}]

Or even two times for the Hessian $T_{ij}=\frac{\partial^2 R}{\partial x_i\partial x_j}$

Tij = D[Sqrt[TensorContract[TensorProduct[xk, xk], {1, 2}]],{xk,2}];

$$\left( \begin{array}{cc} \frac{x_2^2}{\left(x_1^2+x_2^2\right){}^{3/2}} & -\frac{x_1 x_2}{\left(x_1^2+x_2^2\right){}^{3/2}} \\ -\frac{x_1 x_2}{\left(x_1^2+x_2^2\right){}^{3/2}} & \frac{x_1^2}{\left(x_1^2+x_2^2\right){}^{3/2}} \\ \end{array} \right)$$ Finally,just as a neat trick, we contract $T_{ij}$ (which is the same thing as the matrix trace in this case)

FullSimplify[TensorContract[Tij, {1, 2}]]

$$T_{kk}=\frac{1}{\sqrt{x_1^2+x_2^2}}$$ Your next examples are also trivial, use the built in command LeviCivitaTensor for the Levi-Civita "tensor".

Full Mathematica code for this illustration:

xk = {Subscript[x, 1], Subscript[x, 2]};
Ti = D[Sqrt[TensorContract[TensorProduct[xk, xk], {1, 2}]], {xk, 1}];
Tij = D[Sqrt[TensorContract[TensorProduct[xk, xk], {1, 2}]], {xk, 2}]
FullSimplify[TensorContract[Tij, {1, 2}]]