Cadabra manual: Evaluate

evaluate

Evaluate components of a tensor expression.

Given an abstract tensor expression and a set of rules for the components of tensors occurring in this expression, evaluate the components of the full expression. The minimal information needed for this to work is a declaration of the indices used, and a declaration of the values that those indices use:

{r,t}::Coordinate. {m,n,p,s}::Indices(values={t,r}). ex:= A_{n m} B_{m n p} ( C_{p s} + D_{s p} );

\(\displaystyle{}A_{n m} B_{m n p} \left(C_{p s} +D_{s p}\right)\)

A_{n m} B_{m n p} (C_{p s} + D_{s p})

The list of component values should be given just like the list of rules for the substitute algorithm, that is, as equalities

rl:= [ A_{r t} = 3, B_{t r t} = 2, B_{t r r} = 5, C_{t r} = 1, D_{r t} = r**2*t, D_{t r}=t**2 ];

\(\displaystyle{}\left[A_{r t} = 3, B_{t r t} = 2, B_{t r r} = 5, C_{t r} = 1, D_{r t} = {r}^{2} t, D_{t r} = {t}^{2}\right]\)

{A_{r t} = 3, B_{t r t} = 2, B_{t r r} = 5, C_{t r} = 1, D_{r t} = (r)**2 t, D_{t r} = (t)**2}

The evaluate algorithm then works out the values of the components of the ex expression, which will be denoted with a 'box' in its output,

evaluate(ex, rl);

\(\displaystyle{}\square{}_{s}\left\{\begin{aligned}\square{}_{r}& = 6{r}^{2} t +6\\[-.5ex] \square{}_{t}& = 15{t}^{2}\\[-.5ex] \end{aligned}\right. \)

\components_{s}({{r} = 6(r)**2 t + 6, {t} = 15(t)**2})

Note how the composite tensor of the original expression is now replaced with an 'anonymous tensor' represented by a box (which carries the free indices of the original expression), and all the components of the tensor are listed.

If your indices take numerical values, you can also declare them as Integer. The evaluate algorithm will then figure out the values from there:

{i,j}::Indices(space). {i,j}::Integer(1..3). ex:= A_{i} B^{i}. evaluate(ex);

\(\displaystyle{}A_{1} B^{1} +A_{2} B^{2} +A_{3} B^{3}\)

A_{1} B^{1} + A_{2} B^{2} + A_{3} B^{3}

If you have more than one list of rules, first join them together using the join function before passing them into evaluate:

ex:= A_{i} B^{i}; rl1:= { A_{1} = q1, A_{2} = q2 }; rl2:= { B^{1} = q3, B^{2} = q4 }; evaluate(ex, join(rl1, rl2));

\(\displaystyle{}A_{i} B^{i}\)

A_{i} B^{i}

\(\displaystyle{}\left[A_{1} = {q_{1}}, A_{2} = {q_{2}}\right]\)

{A_{1} = q1, A_{2} = q2}

\(\displaystyle{}\left[B^{1} = {q_{3}}, B^{2} = {q_{4}}\right]\)

{B^{1} = q3, B^{2} = q4}

\(\displaystyle{}{q_{1}} {q_{3}} +{q_{2}} {q_{4}}\)

q1 q3 + q2 q4

It often happens that you want to evaluate the components of a tensor only on the right-hand side of an equality, for instance when you are trying to build substitution or evaluation rules for subsequent steps. To do this, use the rhsonly=True parameter,

ex2:= T_{m n} = A_{m p} B_{p n}; evaluate(ex2, rl, rhsonly=True);

\(\displaystyle{}T_{m n} = A_{m p} B_{p n}\)

T_{m n} = A_{m p} B_{p n}

\(\displaystyle{}T_{m n} = \square{}_{m}{}_{n}\left\{\begin{aligned}\square{}_{r}{}_{t}& = 3B_{t t}\\[-.5ex] \square{}_{r}{}_{r}& = 3B_{t r}\\[-.5ex] \end{aligned}\right. \)

T_{m n} = \components_{m n}({{r, t} = 3B_{t t}, {r, r} = 3B_{t r}})