Question 9.9.5: Determine whether the vector field F(x, y) = (x² - 2y³)i + (......
Determine whether the vector field \mathrm{F}(x, y)=\left(x^{2}-2 y^{3}\right) \mathrm{i}+(x+5 y) \mathrm{j} is conservative.
Step-by-Step
With P=x^{2}-2 y^{3} and Q=x+5 y, we find
\frac{\partial P}{\partial y}=-6 y^{2} \text { and } \frac{\partial Q}{\partial x}=1 .
Because \partial P / \partial y \neq \partial Q / \partial x for all points in the plane, it follows from Theorem 9.9.4 that \mathrm{F} is not conservative.
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