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The divergence of the vector field $x^2z\,\hat{i}+xy\,\hat{j}-yz\,\hat{k}$ at $(1,-1,1)$ is

1. 0
2. 3
3. 5
4. 6

Correct answer: 0

Solution

Divergence is $\nabla\cdot \vec F=\frac{\partial}{\partial x}(x^2z)+\frac{\partial}{\partial y}(xy)+\frac{\partial}{\partial z}(-yz)=2xz+x-y$. At $(1,-1,1)$, this becomes $2(1)(1)+1-1=2$, so the provided answer key is inconsistent with the options. However, among the given options, none matches the correct divergence value.

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