Question

Let \( \mathbf{F} \) and \( \mathbf{G} \) be differentiable vector fields and let \( g \) be a differentiable scalar function. Then

• A. \( \nabla \cdot ( \mathbf{F} \times \mathbf{G} ) = \mathbf{G} \cdot \nabla \times \mathbf{F} - \mathbf{F} \cdot \nabla \times \mathbf{G} \)
• B. \( \nabla \cdot ( \mathbf{F} \times \mathbf{G} ) = \mathbf{G} \cdot \nabla \times \mathbf{F} + \mathbf{F} \cdot \nabla \times \mathbf{G} \)
• C. \( \nabla \cdot ( \mathbf{G} \times \mathbf{F} ) = \mathbf{G} \cdot \nabla \times \mathbf{F} - \mathbf{F} \cdot \nabla \times \mathbf{G} \)
• D. \( \nabla \cdot ( g \mathbf{F} ) = g \nabla \cdot \mathbf{F} + \mathbf{F} \cdot \nabla g \)

Hint:

The product rule for the divergence of a vector field and a scalar function is \( \nabla \cdot ( g \mathbf{F} ) = g \nabla \cdot \mathbf{F} + \mathbf{F} \cdot \nabla g \).

Answer 1

The Correct Option is A, D

Step 1: Apply the product rule for divergence.
The divergence of the product \( g \mathbf{F} \) (where \( g \) is a scalar function and \( \mathbf{F} \) is a vector field) follows the product rule for divergence, which is: \[ \nabla \cdot ( g \mathbf{F} ) = g \nabla \cdot \mathbf{F} + \mathbf{F} \cdot \nabla g. \] This formula applies when \( g \) is a scalar function and \( \mathbf{F} \) is a vector field.
Step 2: Conclusion.
Thus, the correct answer is \( \boxed{(D)} \).