Question

Which of the following is NOT TRUE?

• A. \(\nabla\cdot\nabla f=0\)
• B. \(\nabla\times\nabla f=0\)
• C. \(\nabla\cdot(\nabla\times \vec f)=0\)
• D. \(\nabla\times(\nabla\times \vec f)=0\)

Hint:

Remember: curl of gradient is zero, divergence of curl is zero, but divergence of gradient is the Laplacian and need not be zero.

Answer 1

The Correct Option is A

Concept:
We use standard vector identities: \[ \nabla\times(\nabla f)=0 \] and \[ \nabla\cdot(\nabla\times \vec F)=0 \] But \[ \nabla\cdot(\nabla f)=\nabla^2 f \] which is the Laplacian of \(f\). It is not always zero.

Step 1: Analyze option (A). \[ \nabla\cdot\nabla f=\nabla^2 f \] The Laplacian \(\nabla^2 f\) is not always zero. For example, if \[ f=x^2+y^2+z^2 \] then \[ \nabla^2 f=2+2+2=6 \] So, \[ \nabla\cdot\nabla f=0 \] is not always true.

Step 2: Analyze option (B). \[ \nabla\times\nabla f=0 \] This is a standard identity. The curl of a gradient is always zero.

Step 3: Analyze option (C). \[ \nabla\cdot(\nabla\times \vec f)=0 \] This is also a standard identity. The divergence of a curl is always zero.

Step 4: Final answer. The statement that is not true in general is \[ \boxed{\nabla\cdot\nabla f=0} \]