Question

A vector field \( \vec{p} \) and a scalar field \( r \) are given by \[ \vec{p} = \left( 2x^2 - 3xy + z^2 \right) \hat{i} + \left( 2y^2 - 3yz + x^2 \right) \hat{j} + \left( 2z^2 - 3xz + x^2 \right) \hat{k} \] \[ r = 6x^2 + 4y^2 - z^2 - 9xyz - 2xy + 3xz - yz \] Consider the statements P and Q: P: Curl of the gradient of the scalar field \( r \) is a null vector. Q: Divergence of curl of the vector field \( \vec{p} \) is zero. Which one of the following options is CORRECT?

• A. Both P and Q are FALSE
• B. P is TRUE and Q is FALSE
• C. P is FALSE and Q is TRUE
• D. Both P and Q are TRUE

Hint:

Remember the fundamental vector calculus identities: \[ \nabla \times (\nabla r) = 0, \quad \nabla \cdot (\nabla \times \vec{p}) = 0 \]

Answer 1

The Correct Option is D

Statement P: The curl of the gradient of any scalar field \( r \) is always zero, meaning: \[ \nabla \times (\nabla r) = \vec{0} \] Thus, statement P is TRUE. Statement Q: For any vector field \( \vec{p} \), the divergence of its curl is always zero by vector calculus identity: \[ \nabla \cdot (\nabla \times \vec{p}) = 0 \] Thus, statement Q is TRUE. Conclusion: Both statements P and Q are correct, so the correct answer is option (D).