This came up in an assignment problem.

I am tasked to use

1) ▽ /: ▽[f_] := {D[f, x], D[f, y], D[f, z]}

2) ▽ /: ▽ · {fx_, fy_, fz_} := D[fx, x] + D[fy, y] + D[fz, z]


3) ▽ /: Cross[▽, {fx_, fy_, fz_}] := {D[fz, y] - D[fy, z], D[fx, z] - D[fz, x], D[fy, x] - D[fx, y]}

to show that

▽[a.b] == Cross[a, Cross[▽, b]] + Cross[b, Cross[▽, a]] + a.▽[b] + b.▽[a]

is true.

However, I am unable to figure out how the given operators work. I understand that the given operators are defined functions. However, treating the given operators as though they were a defined function -in fact they are!- doesn't work.

For instance, referring to 1) and trying to evaluate a scalar number "1":

▽ /: ▽[1]

Mathematica returns nothing.


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