Question

If $f$ and $F$ are scalar and vector functions respectively, then which of the following is Not Correct?

• A. $\nabla \cdot \nabla f = \nabla^{2}f$
• B. $\nabla \cdot \nabla \times F = 0$
• C. $\nabla \times \nabla f = 0$
• D. $\nabla \times (\nabla \times F) = \nabla \cdot (\nabla F)$

Hint:

Remember: "Curl of Gradient" and "Divergence of Curl" are always zero.

Answer 1

The Correct Option is D

Step 1: Concept
This question tests standard vector identities involving gradient, divergence, and curl.

Step 2: Meaning
Options (A), (B), and (C) are fundamental identities: $\text{div}(\text{grad } f) = \text{Laplacian } f$; $\text{div}(\text{curl } F) = 0$; and $\text{curl}(\text{grad } f) = 0$.

Step 3: Analysis
The correct identity for the double curl is $\nabla \times (\nabla \times F) = \nabla(\nabla \cdot F) - \nabla^{2}F$.

Step 4: Conclusion
Option (D) incorrectly represents this identity and is therefore the "not correct" choice. Final Answer: (D)