Question

If \( \vec{F} \) is a vector point function and \( \phi \) is a scalar point function, then match List-I with List-II and choose the correct option:

LIST-I	LIST-II
(A) \( \text{div (grad } \phi) \)	(IV) \( \nabla \cdot \nabla \phi \)
(B) \( \text{curl (grad } \phi) \)	(III) \( \vec{0} \)
(C) \( \vec{F} \times \text{curl } \vec{F} \)	(I) \( \frac{1}{2} \nabla F^2 - (\vec{F} \cdot \nabla) \vec{F} \)
(D) \( \text{curl (curl } \vec{F}) \)	(II) \( \text{grad(div } \vec{F}) - \nabla^2 \vec{F} \)

Choose the correct answer from the options given below:

• A. A-I, B-II, C-III, D-IV
• B. A-IV, B-III, C-II, D-I
• C. A-I, B-II, C-IV, D-III
• D. A-IV, B-III, C-I, D-II

Hint:

It is highly recommended to memorize the two "zero" identities: \( \text{curl(grad } \phi) = \vec{0} \) and \( \text{div(curl } \vec{F}) = 0 \). Also, the "curl of curl" identity \( \nabla \times (\nabla \times \vec{F}) = \nabla(\nabla \cdot \vec{F}) - \nabla^2 \vec{F} \) is extremely common in physics and engineering and is worth memorizing.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This question tests knowledge of standard vector calculus identities involving the operators gradient (\(\nabla\)), divergence (\(\nabla \cdot\)), and curl (\(\nabla \times\)).

Step 2: Detailed Explanation:
Let's analyze each identity in List-I.
A. div(grad \(\phi\)): This translates to \( \nabla \cdot (\nabla \phi) \), which is the definition of the Laplacian operator applied to \(\phi\), written as \( \nabla^2 \phi \). Option IV, \( \nabla \cdot \nabla \phi \), is the direct expression for this. Match: A - IV
B. curl(grad \(\phi\)): This translates to \( \nabla \times (\nabla \phi) \). This is a fundamental vector identity which is always equal to the zero vector, \( \vec{0} \). Match: B - III
C. \( \vec{F} \times \text{curl } \vec{F} \): This translates to \( \vec{F} \times (\nabla \times \vec{F}) \). This is a standard vector identity which expands to \( \frac{1}{2}\nabla(\vec{F} \cdot \vec{F}) - (\vec{F} \cdot \nabla)\vec{F} \). Note that \( \vec{F} \cdot \vec{F} = |\vec{F}|^2 = F^2 \). This matches the expression in option I. Match: C - I
D. curl(curl \( \vec{F} \)): This translates to \( \nabla \times (\nabla \times \vec{F}) \). This is the "vector triple product" identity for nabla, which expands to \( \nabla(\nabla \cdot \vec{F}) - (\nabla \cdot \nabla)\vec{F} \). In words, this is grad(div \( \vec{F} \)) - (Laplacian of \( \vec{F} \)). This matches the expression in option II. Match: D - II
Step 3: Final Answer:
Combining the matches gives A-IV, B-III, C-I, D-II. This corresponds to option (D).