# What is the significance of this curl product rule?

## Homework Statement

Verify the identity:

## \nabla \times ( A \times B) = (B\bullet \nabla)A - (A\bullet\nabla)B + A(\nabla \bullet B)-B(\nabla\bullet A)##

My issue here is I don't understand the significance of why a term has B or A on the left of the dot product, and another has B or A on the right? (Eg ## B(\nabla)

Also, what is the significance of the ## \nabla ## being on the left, or right, of what it is multiplying? (Eg ## \langle \frac {d} {dx}, \frac {d} {dy}, \frac {d} {dz} \rangle \bullet \langle a_1, a_2, a_3 \rangle = \langle a_1, a_2, a_3 \rangle \bullet \langle \frac {d} {dx}, \frac {d} {dy}, \frac {d} {dz} \rangle##, or if the ## \nabla ## is on the right, does that leave the operator open to work on whichever comes AFTER the dot product?

## The Attempt at a Solution

I have my solution attached as a thumbnail, because it's pretty hefty.

Why does my solution look different than the given solution at the top?

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andrewkirk
Homework Helper
Gold Member
does that leave the operator open to work on whichever comes AFTER the dot product?
Yes.
A somewhat less confusing - but perhaps less memorable - way to write the identity is:

$$\nabla\times (A\times B)\equiv B\cdot[\nabla A]-A\cdot[\nabla B]+(\nabla\cdot B)A-(\nabla\cdot A)B$$
the items in square brackets are vector fields and those in round parentheses are scalar fields. So the latter can be applied to vector fields by scalar multiplication.

Yes.
A somewhat less confusing - but perhaps less memorable - way to write the identity is:

$$\nabla\times (A\times B)\equiv B\cdot[\nabla A]-A\cdot[\nabla B]+(\nabla\cdot B)A-(\nabla\cdot A)B$$
the items in square brackets are vector fields and those in round parentheses are scalar fields. So the latter can be applied to vector fields by scalar multiplication.
Okay, I think I get it, but how come in my solution it seems like the A always comes first? I'm not sure what I did wrong, and why am I interpreting the ## \nabla ## as always being on the left?

andrewkirk