GATE CE 2024 Set 1 — Question 37
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Engineering Mathematics → Vector Calculus → Gradient, Divergence & Curl
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Question
A vector field and a scalar field are given byConsider the statements P and Q.
P: Curl of the gradient of the scalar field is a null vector.
Q: Divergence of curl of the vector field is zero.
Which one of the following options is CORRECT?
P: Curl of the gradient of the scalar field is a null vector.
Q: Divergence of curl of the vector field is zero.
Which one of the following options is CORRECT?
Correct answer
(D) Both P and Q are TRUE
Solution
Statement P is true because of the vector identity (curl of gradient is always zero) for any twice-differentiable scalar field . Statement Q is true because of the vector identity (divergence of curl is always zero) for any twice-differentiable vector field . Since both and are polynomial functions, they are infinitely differentiable, and the identities hold. Thus, both P and Q are TRUE.
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