# Confusing arken/weber problem (del in triple vector product)

the following problem is in the arfken/weber textbook and was also on a practice exam for my mathematical methods course:

Verify that
$$\mathbf{A} \times (\nabla \times \mathbf{A}) = \frac{1}{2} \nabla(A^2) - (\mathbf{A} \cdot \nabla)\mathbf{A}.$$

i used the BAC-CAB rule, but i don't get the factor of 1/2.

the solutions booklet that came with the textbook very tersely explains, "the factor of 1/2 occurs because the del only operates on one of the A's."

i would very much appreciate an explanation that is perhaps more informative than this one sentence blurb! :tongue:

thank you.

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robphy
Homework Helper
Gold Member
I think the "product rule" was used.
Have you tried an index-based calculation...
$$\epsilon_{ijk}A^j\epsilon^{klm}\nabla_l A_m =\left( \delta_i^l\delta_j^m- \delta_i^m\delta_j^l)A^j\nabla_l A_m$$

lurflurf
Homework Helper
You can't just go around baccabing things. In particular as the one sentence you dislike states the del only operates on one of the A's. That other half is busy over here
$$(\mathbf{A} \times \nabla) \times \mathbf{A}= \frac{1}{2} \nabla(A^2) - \mathbf{A}(\nabla\cdot\mathbf{A})$$
since
$$(a\times\nabla)\times b+a\nabla\cdot b=a\times(\nabla\times b)+(a\cdot\nabla)b$$

lurflurf
Homework Helper
what is
$$\nabla(A^2)$$
If one A is constant?
say A(x) then find
$$\nabla(A(x).A(0))|_{x=0}$$
Compare the 1-var analog
D[y(x)*y(x)]|{x=0}=2y(0)*y'(0)
D[y(0)*y(x)]|{x=0}=y(0)*y'(0)

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lurflurf said:
You can't just go around baccabing things. In particular as the one sentence you dislike states the del only operates on one of the A's. That other half is busy over here
$$(\mathbf{A} \times \nabla) \times \mathbf{A}= \frac{1}{2} \nabla(A^2) - \mathbf{A}(\nabla\cdot\mathbf{A})$$
since
$$(a\times\nabla)\times b+a\nabla\cdot b=a\times(\nabla\times b)+(a\cdot\nabla)b$$

i still don't understand. :grumpy:

lurflurf
Homework Helper
i still don't understand. :grumpy:
It is like asking where the 1/2 comes from in
$$y \ \frac{d}{dx}y=\frac{1}{2} \ \frac{d}{dx}y^2$$
You should avoid trying to use vector identities on del that do not apply.
$$\mathbf{A_1} \times (\nabla_2 \times \mathbf{A_2}) =\nabla_2(\mathbf{A_1}\cdot\mathbf{A_2}) - (\mathbf{A_1} \cdot \nabla_2)\mathbf{A_2}$$
is true and makes what is happening clear. A1 and A2 are equal, but nabla2 acts only on A2. The grad(A^2) part is the place where the distinction matters. In order to switch back to more standard notation we note that
$$\nabla_1(\mathbf{A_1}\cdot\mathbf{A_2})=\nabla_2(\mathbf{A_1}\cdot\mathbf{A_2})$$
and
$$\nabla(\mathbf{A}\cdot\mathbf{A})=\nabla_1(\mathbf{A_1}\cdot\mathbf{A_2})+\nabla_2(\mathbf{A_1}\cdot\mathbf{A_2})$$

You infact did not use B(A.C)-C(A.B), you used B(A.C)-(A.B)C
so you saw that not all vector idenetities hold if a vector is replaced by del, but you still used an identity that was not true.

for comparison
$$(\mathbf{A_1} \times \nabla_2) \times \mathbf{A_2} =\nabla_2(\mathbf{A_1}\cdot\mathbf{A_2}) - \mathbf{A_1}(\nabla_2\cdot\mathbf{A_2})$$
thus
$$(\mathbf{A} \times \nabla) \times \mathbf{A}= \frac{1}{2} \nabla(A^2) - \mathbf{A}(\nabla\cdot\mathbf{A})$$

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lurflurf said:
It is like asking where the 1/2 comes from in
$$y \ \frac{d}{dx}y=\frac{1}{2} \ \frac{d}{dx}y^2$$
You should avoid trying to use vector identities on del that do not apply.
$$\mathbf{A_1} \times (\nabla_2 \times \mathbf{A_2}) =\nabla_2(\mathbf{A_1}\cdot\mathbf{A_2}) - (\mathbf{A_1} \cdot \nabla_2)\mathbf{A_2}$$
is true and makes what is happening clear. A1 and A2 are equal, but nabla2 acts only on A2. The grad(A^2) part is the place where the distinction matters. In order to switch back to more standard notation we note that
$$\nabla_1(\mathbf{A_1}\cdot\mathbf{A_2})=\nabla_2(\mathbf{A_1}\cdot\mathbf{A_2})$$
and
$$\nabla(\mathbf{A}\cdot\mathbf{A})=\nabla_1(\mathbf{A_1}\cdot\mathbf{A_2})+\nabla_2(\mathbf{A_1}\cdot\mathbf{A_2})$$

You infact did not use B(A.C)-C(A.B), you used B(A.C)-(A.B)C
so you saw that not all vector idenetities hold if a vector is replaced by del, but you still used an identity that was not true.

for comparison
$$(\mathbf{A_1} \times \nabla_2) \times \mathbf{A_2} =\nabla_2(\mathbf{A_1}\cdot\mathbf{A_2}) - \mathbf{A_1}(\nabla_2\cdot\mathbf{A_2})$$
thus
$$(\mathbf{A} \times \nabla) \times \mathbf{A}= \frac{1}{2} \nabla(A^2) - \mathbf{A}(\nabla\cdot\mathbf{A})$$

ah!

thank you very much. that is exactly what i needed.

thanks a lot.

found another way to do the problem (without invoking the scary kronecker deltas and epsilon tensors!). this is the proof suggested by david griffiths' intro to electrodynamics:

$$\nabla (\mathbf{A} \cdot \mathbf{B}) = \mathbf{A} \times (\nabla \times \mathbf{B}) + \mathbf{B} \times (\nabla \times \mathbf{A}) + (\mathbf{A} \cdot \nabla)\mathbf{B} + (\mathbf{B} \cdot \nabla)\mathbf{A}.$$

when $$\mathbf{A} = \mathbf{B},$$

$$\nabla (\mathbf{A} \cdot \mathbf{A}) = 2\mathbf{A} \times (\nabla \times \mathbf{A}) + 2(\mathbf{A} \cdot \nabla)\mathbf{A}.$$

note: $$\mathbf{A} \cdot \mathbf{A} = A^2.$$

so

$$\mathbf{A} \times (\nabla \times \mathbf{A}) = \frac{1}{2} \nabla(A^2) - (\mathbf{A} \cdot \nabla)\mathbf{A}.$$

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robphy