Curl of the product of a scalar and a vector using Levi-Civita | Quantum Science Philippines
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Curl of the product of a scalar and a vector using Levi-Civita

\vec{\nabla} \times f \vec{A}=f \vec{\nabla} \times \vec{A} \ + \ \vec{\nabla} f \times \vec{A}

By Eliezer Estrecho

To prove this formula, we use the following:

\vec{\nabla} \times \vec{A}=\epsilon _{ijk} \hat{e}_{i} \nabla_{j} A_{k}

Where: \vec{\nabla}=\nabla_{j} \hat{e}_{j} and \vec{A}=A_{k} \hat{e}_{k}

Using the equation above:

  \vec{\nabla} \times f \vec{A} =\epsilon _{ijk} \hat{e}_{i} \nabla_{j} f A_{k} \\  \vec{\nabla} \times f \vec{A} =\epsilon _{ijk} \hat{e}_{i} \ (f \nabla_{j} A_{k} + A_{k}\nabla_{j}f) \ \ \ \ \ \ \ product \ rule\\  \vec{\nabla} \times f \vec{A} =\epsilon_{ijk}\hat{e}_{i}f \nabla_{j} A_{k} + \epsilon_{ijk}\hat{e}_{i} A_{k} \nabla_{j} f

We can factor out  f in the first term to give:

f \epsilon_{ijk}\hat{e}_{i} \nabla_{j} A_{k}=f \vec{\nabla} \times \vec{A}

Note that for the second term, the permutation of indices are odd, rearranging them to ijk will give the negative:

\epsilon_{ijk}\hat{e}_{i} A_{k} \nabla_{j} f = -\epsilon_{ijk}\hat{e}_{i} A_{j} \nabla_{k} f = -\vec{A} \times \vec{\nabla}f = \vec{\nabla}f \times \vec{A}

Thus,

\vec{\nabla} \times f \vec{A}=f \vec{\nabla} \times \vec{A} \ + \ \vec{\nabla} f \times \vec{A}

About the author: Eliezer Estrecho is currently a MS Physics student of MSU-IIT.

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