# Product Rule

Hello everyone. I'm trying to get my head around this product rule:

$$\nabla \times (A\times B) = (B\cdot \nabla )A - (A\cdot \nabla )B + A(\nabla \cdot B) - B(\nabla \cdot A)$$

Ok, we have this

$$\nabla = (\partial /\partial x,\partial/\partial y,\partial /\partial z)$$

and for dot products

$$a\cdot b = b\cdot a$$

Therefore in the product rule given above, is it not the case

$$(B\cdot \nabla )A = A(\nabla \cdot B)$$

and similarly, the other two terms on the RHS are equal?
Thank-you for your help.

## Answers and Replies

G01
Homework Helper
Gold Member
Therefore in the product rule given above, is it not the case

$$(B\cdot \nabla )A = A(\nabla \cdot B)$$

Be careful. $$\nabla$$ is a vector operator not a vector. It will not commute the way you expect it to.

HallsofIvy
Homework Helper
In fact, I would much prefer the notation $B\cdot(\nabla A)$ to $(B\cdot\nabla)A$.

In fact, I would much prefer the notation $B\cdot(\nabla A)$ to $(B\cdot\nabla)A$.

Aha! I think I get it now! The brackets were confusing, because usually we have to evaluate the stuff in the brackets first right?

Does this mean $$\nabla$$ always acts on the vector directly to its right?

In fact, I would much prefer the notation $B\cdot(\nabla A)$ to $(B\cdot\nabla)A$.

Hold on, this can't be right can it? Then we would have
$$\nabla \times (A\times B) = 0$$

wouldn't we? Can someone please tell me what $$(B\cdot\nabla)A[/itex] is? dx Homework Helper Gold Member Can someone please tell me what [tex](B\cdot\nabla)A[/itex] is? First you evaluate [tex] \nabla A$$. You get a vector field, i.e. a vector at every point of space. Then $$B \cdot(\nabla A)$$ would be vector field whose value at any point is the dot of B with vector $$\nabla A$$ at that point.

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I'm sorry for being extremely thick, but then doesn't that mean that

$$\nabla \times (A\times B) = 0$$?

This makes no sense, because it means the cross product of any two vectors has zero curl? Surely $$(B\cdot \nabla)\cdot A$$ is not the same thing as $$B\cdot (\nabla \cdot A)$$?

dx
Homework Helper
Gold Member
I'm sorry for being extremely thick, but then doesn't that mean that

$$\nabla \times (A\times B) = 0$$?

why does it mean that?

The second operation not a dot product. del.A is a scalar. So (del.A)B is the scalar (d/dxAx + d/dyAy + d/dzAz) times the vector B = d/dxAx*B + d/dyAy*B + d/dzAz*B

Also, remember the del is an operator so del.A is not the same as A.del. A.del is still a scalar though being applied to B to it gets pretty messy looking.

http://mathworld.wolfram.com/ConvectiveOperator.html

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dx
Homework Helper
Gold Member
Surely $$(B\cdot \nabla)\cdot A$$ is not the same thing as $$B\cdot (\nabla \cdot A)$$?

No, what I said was $$B \cdot (\nabla A)$$ is the same thing as $$(B \cdot \nabla)A$$

Er, ok, what's the difference? A and B are vectors.

The second operation not a dot product. del.A is a scalar. So (del.A)B is the scalar (d/dxAx + d/dyAy + d/dzAz) times the vector B = d/dxAx*B + d/dyAy*B + d/dzAz*B

Also, remember the del is an operator so del.A is not the same as A.del. A.del is still a scalar though being applied to B to it gets pretty messy looking.

http://mathworld.wolfram.com/ConvectiveOperator.html

Oh, ok, this site explains it to me. Thanks for all your help everyone, taking time to answer my stupid questions

lurflurf
Homework Helper
In fact, I would much prefer the notation $B\cdot(\nabla A)$ to $(B\cdot\nabla)A$.

Some people do not like to take the gradient of a vector since it is a dyad, and it makes them feel ickky inside.

lurflurf
Homework Helper
Aha! I think I get it now! The brackets were confusing, because usually we have to evaluate the stuff in the brackets first right?

Does this mean $$\nabla$$ always acts on the vector directly to its right?

This is a confusing aspect of vector calculus.
$$\nabla$$ acts to the right only.
One may define a bidirectional del/nabla
consider this bidirectional derivative
it is a good and correct habit when working with vectors to switch to bidirectional form
-hold all function left of operators constant
-change operators to birectional
-perfom manipulations ending with a form easy to conver to unidirectional form
-convert

recall this identity when working with products

$$\mathbf{(a\times\nabla)\times b+a\nabla\cdot b=a\times(\nabla\times b)+(a\cdot\nabla)b}$$

Dick