Question

Which of the following holds for any non-zero vector \(\vec a\)?

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

Hint:

Divergence of curl is always zero: \(\nabla\cdot(\nabla\times \vec A)=0\).

Answer 1

The Correct Option is C

Concept:
A very important vector identity is: \[ \nabla\cdot(\nabla\times \vec A)=0 \] for any sufficiently differentiable vector field \(\vec A\). This means the divergence of the curl of any vector field is always zero.

Step 1: Understand the identity. The curl of a vector field is \[ \nabla\times \vec a \] Taking divergence of this curl gives \[ \nabla\cdot(\nabla\times \vec a) \] This expression is always zero.

Step 2: Check option (A). \[ \nabla\cdot \vec a=0 \] This is not always true for every vector field. It is true only for solenoidal vector fields.

Step 3: Check option (B). \[ \nabla\times \vec a=0 \] This is not always true for every vector field. It is true only for irrotational vector fields.

Step 4: Select the universal identity. The identity that always holds is \[ \nabla\cdot(\nabla\times \vec a)=0 \]

Step 5: Final answer. \[ \boxed{\nabla\cdot(\nabla\times \vec a)=0} \]