Sorting terms in a huge equation by grouping specific products

Question

I have a boundary condition for a problem, which contains around 100 terms! I have no trouble getting Mathematica to compute it, but the terms are sorted in a seemingly random order and I would like to organise them to make it a bit easier to deal with without having to manual.

Consider the following expression:

\begin{align*} & \dfrac{\partial A_{1}}{\partial x}\dfrac{\partial A_{2}}{\partial y}+\dfrac{\partial A_{2}}{\partial x}\dfrac{\partial A_{3}}{\partial z}+\dfrac{\partial A_{3}}{\partial y}\dfrac{\partial A_{1}}{\partial z}\\ & +\dfrac{\partial A_{1}}{\partial x}\dfrac{\partial B_{2}}{\partial y}+\dfrac{\partial A_{2}}{\partial x}\dfrac{\partial B_{3}}{\partial z}+\dfrac{\partial A_{3}}{\partial y}\dfrac{\partial B_{1}}{\partial z}\\ & +\dfrac{\partial B_{1}}{\partial x}\dfrac{\partial C_{2}}{\partial y}+\dfrac{\partial B_{2}}{\partial x}\dfrac{\partial C_{3}}{\partial z}+\dfrac{\partial B_{3}}{\partial y}\dfrac{\partial C_{1}}{\partial z}\\ & +\dfrac{\partial B_{1}}{\partial x}\dfrac{\partial C_{2}}{\partial y}+\dfrac{\partial C_{1}}{\partial x}\dfrac{\partial C_{1}}{\partial z}+\dfrac{\partial C_{1}}{\partial y}\dfrac{\partial C_{1}}{\partial z}\\ & +\dfrac{\partial A_{1}}{\partial x}C_{1}+\dfrac{\partial A_{2}}{\partial y}C_{2}+\dfrac{\partial A_{3}}{\partial z}C_{3} \end{align*}

I am very new to patterns/matching, but could somebody please explain how I can use these to extract all terms from the above expression which contain:

• $\dfrac{\partial A_i}{\partial x_k}\dfrac{\partial A_j}{\partial x_\ell }$
• $\left(\dfrac{\partial A_i}{\partial x_k}\right)^2$
• $C_i\dfrac{\partial A_j}{\partial x_k}$

where $i,j,k,\ell\in\{1,2,3\}$ and $x_1=x,x_2=y,x_3=z$.

Obviously the way I have specified the terms above lends itself to the use of a Table command, but I would like to be able to do this in such a way which is independent of the choice of subscripts, and can be picked up from whatever the expression is. How can I elegantly answer the above questions?

Answer 1

Note, do not use capitalized variable names, those are used by the system. E.g. "C" is used for integration constants.

To be able to choose the functions by index, we use indexed function names. I assume that you are dealing with code that can be evaluated. Unfortunately, in MMA we need to write functions with arguments, what makes it a bit clumsy.

Further, to minimize my effort in writing a complicated test expression, I use a helper function that creates the complicated derivatives:

f[a1_, a2_, x1_] := D[a1[x, y, z], x1] a2[x, y, z];
f[a1_, a2_, x1_, x2_] := D[a1[x, y, z], x1] D[a2[x, y, z], x2];

With this we create some teste expression:

ex = f[a[1], a[2], x, y] + f[a[1], a[2], y, z] + f[a[1], b[3], x, y] +
   f[a[1], c[3], x]

We can now pick out e.g. all terms that are derivatives from a[1] and a[2]:

Cases[ex, Derivative[_, _, _][a[1]][x, y, z]  Derivative[_, _, _][a[2]][x, y, z]]

Or cases with a derivative of a[1] and c[3]:

Cases[ex, Derivative[_, _, _][a[1]][x, y, z]  c[3][x, y, z]]

Or all terms that contain a derivative relative to x:

Cases[ex, Derivative[1, _, _][_][x, y, z]  _ ]

To make life easier we may define a function that helps us pick terms. The first 2 arguments are the function names, the next 2 arguments are 1,2,3, indicating the derivative relative to x,y,z. 0 means no derivative:

pick[a1_, a2_, x1_ : 0, x2_ : 0] := (der = {_, _, _}; 
  der1 = If[x1 > 0, der1 = der; 
    der1[[x1]] = 1; (Derivative @@ der1)[a1], a1]; 
  der2 = If[x2 > 0, der2 = der; 
    der2[[x2]] = 1; (Derivative @@ der2)[a2], a2]; 
  Cases[ex, der1[x, y, z] der2[x, y, z]])

Now we can say, e.g.:

pick[a[1], b[3], 1, 2]

Or all terms with a x derivative of a[1]