Discussion
If \(\overline{A} = x\hat{e}_{x} + y\hat{e}_{y} + z\hat{e}_{j}\), then \(\nabla^{2}\overline{A}\) will be :
Answer: C
The Laplacian of the field is 0.
The field is the position vector, \(\overline{A} = x\hat{e}_{x} + y\hat{e}_{y} + z\hat{e}_{z}\) (the paper prints the last unit vector as \(\hat{e}_{j}\)). In Cartesian coordinates the vector Laplacian acts on each component separately, and each component is linear in the coordinates:
\[ \nabla^{2}x = \nabla^{2}y = \nabla^{2}z = 0 \]
so \(\nabla^{2}\overline{A} = 0\). The value 3 is the divergence \(\nabla \cdot \overline{A}\), a common confusion; 1 and -3 do not arise from either operation.
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