Find the correct identity, if $f$ is a scalar field, $\mathbf { F }$ and $\mathbf { G }$ are vector fields.
A) $\operatorname { div } ( f \mathbf { F } ) = f \operatorname { div } ( \mathbf { F } ) + \mathbf { F } \cdot \nabla f$
B) $\quad \operatorname { curl } ( f \mathbf { F } ) = f \operatorname { div } ( \mathbf { F } ) + \mathbf { F } \cdot \nabla f$
C) $\operatorname { div } ( f \mathbf { F } ) = f \operatorname { curl } ( \mathbf { F } ) + ( \nabla f ) \times \mathbf { F }$
D) None of these

Question

Quizplus · Accepted Answer

This is the correct form of the product rule for divergence. Option B is incorrect because curl and divergence are two different operations in vector calculus, and the product rule does not involve the curl. Option C is incorrect because the product of a scalar field and a vector field cannot produce a vector in the form of the cross product. Option D is obviously not the answer because one of the options must be correct.

Find the Correct Identity, If fff Is a Scalar Field F\mathbf { F }F

Find the Correct Identity, If $f$ Is a Scalar Field $\mathbf { F }$