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$$ \iint_S f(\nabla \times \mathbf{A}) \cdot d\mathbf{S} = \oint_{\partial S} f \mathbf{A} \cdot d\mathbf{r} - \iint_S \mathbf{A} \times (\nabla f) \cdot d\mathbf{S} $$

Is anything wrong with this. My textbook says it should be

$$ \iint_S f(\nabla \times \mathbf{A}) \cdot d\mathbf{S} = \oint_{\partial S} f \mathbf{A} \cdot d\mathbf{r} + \iint_S \mathbf{A} \times (\nabla f) \cdot d\mathbf{S} $$

Which of the two is correct? I used integration by parts and Stoke's theorem.

sarango24
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1 Answers1

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Your textbook is correct. It is straighfoward to prove the identity

$$ \vec{\nabla}\times(f\vec{A})=(\vec{\nabla}f)\times\vec{A}+f(\vec{\nabla}\times\vec{A}) $$

Taking a surface integral of both sides, applying Stokes theroem to the left-hand side, and using the antisymmetry of the cross product gets the answer in your textbook.

See also this handy link of vector calculus identities on Wiki.

CW279
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