- #1
jack476
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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?