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A vector field p and a scalar field r are given by p = (2x² - 3xy + z²) î + (2y² - 3yz + x²) ĵ + (2z² - 3xz + x²) k̂ r = 6x² + 4y² - z² - 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 p is zero. Which one of the following options is CORRECT?

1. Both P and Q are FALSE
2. P is TRUE and Q is FALSE
3. P is FALSE and Q is TRUE
4. Both P and Q are TRUE

Correct answer: Both P and Q are TRUE

Solution

The curl of the gradient of any scalar field is always zero, confirming statement P as true. Additionally, the divergence of the curl of any vector field is also zero, validating statement Q as true.

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