Question

If \(\vec{V} = a\hat{i} + b\hat{j} + c\hat{k}\), identify the INVALID operation on \(\vec{V}\).

• A. \(\nabla \cdot \nabla \times (\vec{V})\)
• B. \(\nabla \times \nabla \cdot (\vec{V})\)
• C. \(\nabla(\nabla \cdot (\vec{V}))\)
• D. \(\nabla \times \vec{V}\)

Hint:

Curl requires a vector input, while divergence requires a vector input and produces a scalar—always check the type of input and output.

Answer 1

The Correct Option is B

Step 1: Understand each vector calculus operator.
- \(\nabla \cdot \vec{V}\) gives a scalar.
- \(\nabla \times \vec{V}\) gives a vector.
- \(\nabla (\text{scalar})\) gives a vector.
- \(\nabla \cdot (\text{vector})\) gives a scalar.
- \(\nabla \times (\text{vector})\) gives a vector.
Step 2: Check validity of each option.
(A) \(\nabla \cdot (\nabla \times \vec{V})\) is valid and always equals 0.
(B) \(\nabla \times (\nabla \cdot \vec{V})\) is the INVALID operation because the curl of a scalar (\(\nabla \cdot \vec{V}\)) does not make sense. The curl requires a vector.
(C) \(\nabla (\nabla \cdot \vec{V})\) is valid — gradient of divergence.
(D) \(\nabla \times \vec{V}\) is valid — curl of a vector.
Step 3: Conclusion.
The invalid operation is (B), as curl of a scalar is not a valid operation. Thus, the correct answer is (B).