Discussion
If a vector field \(\overline{F} = x\hat{i} + 2y\hat{j} + 3z\hat{k}\), then \(\overline{\nabla} \times (\overline{\nabla} \times \overline{F})\) is :
Answer: A
The double curl is Zero.
Each component of \(\overline{F} = x\hat{i} + 2y\hat{j} + 3z\hat{k}\) depends only on its own coordinate, so every cross derivative vanishes; the x-component of the curl, for example, is
\[ \begin{aligned} (\nabla \times \overline{F})_{x} &= \frac{\partial (3z)}{\partial y} - \frac{\partial (2y)}{\partial z} \\ &= 0 - 0 = 0 \end{aligned} \]
The curl of the zero vector is again zero, so \(\nabla \times (\nabla \times \overline{F}) = 0\). The options \(\hat{j}\), \(2\hat{j}\) and \(3\hat{k}\) would need a non-zero curl to begin with; the field is conservative (it is the gradient of \(x^{2}/2 + y^{2} + 3z^{2}/2\)).
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