Prove that $\triangledown\times\left[\frac 1r(A\times R)\right]=\frac 1rA+\frac{A\cdot R}{r^3}R$ for any constant vector $A$.

Question

Question : Prove that $$\triangledown\times\left[\frac 1r(A\times R)\right]=\frac 1rA+\frac{A\cdot R}{r^3}R$$ for any constant vector $A$.

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Proof

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Doubt : I don't know how we split the term into two terms in the first line. I understand the first term, but not the second.

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Answer 1

Actually, we can use an simpler example to illustrate this, consider a vector field $U:\mathbb R^3\to\mathbb R^3$ and a scalar field $g:\mathbb R^3\to\mathbb R$.

Then we compute its curl. We can use the index formula for curl $$ \nabla\times(gU)=\sum_{ijk}\epsilon_{ijk} e_i\partial_j (g U_k)\\ =\sum_{ijk}\epsilon_{ijk} e_i[(\partial_j g) U_k+g(\partial_j U_k)]\\ =(\nabla g)\times U+g\nabla\times U $$ Then use the Leibniz rule to separate it into two terms, and represent it back in terms of outer product.

In your first line, substitute $g=\frac 1r$, $U=A\times R$, then you get your 2 terms