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- Apr 14, 2013

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Determine if the vector field $\overrightarrow{F}=y\hat{i}+(x+z)\hat{j}-y\hat{k}$ is conservative or not.

The vector field $\overrightarrow{F}=M\hat{i}+N\hat{j}+P\hat{k}$ is conservative if $$\frac{\partial{M}}{\partial{y}}=\frac{\partial{N}}{\partial{x}}, \frac{\partial{M}}{\partial{z}}=\frac{\partial{P}}{\partial{x}}, \frac{\partial{N}}{\partial{z}}=\frac{\partial{P}}{\partial{y}}$$

In this case:

$$\frac{\partial{M}}{\partial{y}}=1=\frac{\partial{N}}{\partial{x}}, \frac{\partial{M}}{\partial{z}}=0=\frac{\partial{P}}{\partial{x}}, \frac{\partial{N}}{\partial{z}}=1 \neq \frac{\partial{P}}{\partial{y}}=-1$$

Does this mean that the vector field is not conservative? Is it a ~if and only if~ condition?