A derivative identity (Zangwill)

[jack476]

Homework Statement

Without using vector identities, show that $\nabla \cdot [\vec{A}(r) \times \vec{r}] = 0$.

Homework Equations

The definitions and elementary properties of the dot and cross products in terms of Levi-Civita symbols. The "standard" calculus III identities for the divergence and curl are not allowed.

The Attempt at a Solution

$$
\begin{align*}
\nabla \cdot(\vec{A}\times \vec{r}) &= \epsilon_{ijk}\partial_i(A_jr_k)\\
&= \epsilon_{ijk}r_k\partial_iA_j + \epsilon_{ijk}A_j\partial_ir_k\\
&= \epsilon_{ijk}r_k\partial_iA_j + \epsilon_{ijk}A_j\delta_{ik}
\end{align*}$$

The $\epsilon_{ijk}A_j\delta_{ik}$ term disappears because $\delta_{ik}\epsilon_{ijk}=0$ and the $\epsilon_{ijk}r_k\partial_iA_j$ term disappears because $\epsilon_{ijk}r_k\partial_iA_j = \epsilon_{ijk}\delta_{ik}r_i\partial_iA_j = 0$ for the same reason. Therefore $\nabla \cdot [\vec{A}(r) \times \vec{r}] = 0$.

Is the use of the Kronecker delta in the final step valid?

 

[Ray Vickson]

Homework Statement

Without using vector identities, show that $\nabla \cdot [\vec{A}(r) \times \vec{r}] = 0$.

Homework Equations

The definitions and elementary properties of the dot and cross products in terms of Levi-Civita symbols. The "standard" calculus III identities for the divergence and curl are not allowed.

The Attempt at a Solution

$$
\begin{align*}
\nabla \cdot(\vec{A}\times \vec{r}) &= \epsilon_{ijk}\partial_i(A_jr_k)\\
&= \epsilon_{ijk}r_k\partial_iA_j + \epsilon_{ijk}A_j\partial_ir_k\\
&= \epsilon_{ijk}r_k\partial_iA_j + \epsilon_{ijk}A_j\delta_{ik}
\end{align*}$$

The $\epsilon_{ijk}A_j\delta_{ik}$ term disappears because $\delta_{ik}\epsilon_{ijk}=0$ and the $\epsilon_{ijk}r_k\partial_iA_j$ term disappears because $\epsilon_{ijk}r_k\partial_iA_j = \epsilon_{ijk}\delta_{ik}r_i\partial_iA_j = 0$ for the same reason. Therefore $\nabla \cdot [\vec{A}(r) \times \vec{r}] = 0$.

Is the use of the Kronecker delta in the final step valid?

Are you sure it does not depend on the form of $\vec{A}(r)?$ If that is the case, can you prove it?

 

[Delta2]

Lets cheat a bit and use the vector calculus identity $\nabla \cdot (A\times B)=(\nabla \times A)\cdot B - A\cdot(\nabla\times B)$ then it seems we can prove it but we need that $\nabla\times A=\vec{0}$. Is this fact about the curl of A given as additional assumption which you omitted to write here?

 

[MathematicalPhysicist]

You know there's a solution manual in the internet, right?