Question

Which of the following are true:
A. $\text{div }(\vec{F} \times \vec{G}) = \vec{G} \cdot \text{curl } \vec{F} - \vec{F} \cdot \text{curl } \vec{G}$ B. $\text{div }(\text{curl } \vec{F}) = 0$ C. $\text{curl }(\vec{\nabla} f) = 0$ D. $\text{div }(\vec{\nabla} f \times \vec{\nabla} g) \neq 0$ E. $\vec{\nabla}(fg) = f \vec{\nabla} g + g \vec{\nabla} f$ Choose the correct answer from the options given below:

• A. A, B, C, D, E
• B. A, B, C Only
• C. A, B, C, E Only
• D. A, B, E Only

Hint:

Two famous null identities in vector calculus: 1. $\text{div}(\text{curl } \vec{A}) = 0$ 2. $\text{curl}(\text{grad } f) = \vec{0}$

Answer 1

The Correct Option is C

Step 1 : Concept:
This question tests standard vector calculus identities involving gradient ($\vec{\nabla}$), divergence ($\text{div}$), and curl ($\text{curl}$).

Step 2 : Key Formulas and Approach:
1. $\nabla \cdot (\vec{A} \times \vec{B}) = \vec{B} \cdot (\nabla \times \vec{A}) - \vec{A} \cdot (\nabla \times \vec{B})$
2. $\nabla \cdot (\nabla \times \vec{A}) = 0$
3. $\nabla \times (\nabla f) = 0$
4. Product rule for gradient: $\nabla (fg) = f \nabla g + g \nabla f$

Step 3 : Step-by-step Explanation:

• Statement A:
Using vector differential operations: \[ \text{div}(\vec{F} \times \vec{G}) = \nabla \cdot (\vec{F} \times \vec{G}) = \vec{G} \cdot (\nabla \times \vec{F}) - \vec{F} \cdot (\nabla \times \vec{G}) = \vec{G} \cdot \text{curl } \vec{F} - \vec{F} \cdot \text{curl } \vec{G} \] Statement A is correct.

• Statement B:
Divergence of curl of any vector field $\vec{F}$ is identically zero ($\nabla \cdot (\nabla \times \vec{F}) = 0$). Statement B is correct.

• Statement C:
Curl of gradient of any scalar field $f$ is identically zero ($\nabla \times (\nabla f) = 0$). Statement C is correct.

• Statement D:
Applying identity from Statement A with $\vec{F} = \vec{\nabla} f$ and $\vec{G} = \vec{\nabla} g$: \[ \text{div}(\vec{\nabla} f \times \vec{\nabla} g) = \vec{\nabla} g \cdot \text{curl}(\vec{\nabla} f) - \vec{\nabla} f \cdot \text{curl}(\vec{\nabla} g) \] Since $\text{curl}(\vec{\nabla} f) = 0$ and $\text{curl}(\vec{\nabla} g) = 0$, we get: \[ \text{div}(\vec{\nabla} f \times \vec{\nabla} g) = \vec{\nabla} g \cdot \vec{0} - \vec{\nabla} f \cdot \vec{0} = 0 \] Statement D claims this divergence is non-zero ($\neq 0$), which is false. Statement D is incorrect.

• Statement E:
The product rule for the gradient operator is $\vec{\nabla}(fg) = f \vec{\nabla} g + g \vec{\nabla} f$. Statement E is correct.

Step 4 : Final Answer:
Statements A, B, C, and E are correct. Therefore, option (C) is the correct answer.