Exercise 1215Calculus › Multivariable calculus › Vector calculusL1Added 3 Nov 2026

Testing whether a field is conservative

#conservative#curl#test

Statement

Is $\vec F(x,y)=(x^2y,\,xy^2)$ conservative?

Hint

For a field $(P,Q)$ in $\mathbb{R}^2$ : check $P_y=Q_x$ .

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Compute. $P=x^2y$ , $Q=xy^2$ . Then $P_y=x^2$ , $Q_x=y^2$ .

Test. $P_y\neq Q_x$ in general, so $\vec F$ is not conservative.

Conclusion. $\vec F$ is not conservative (curl nonzero).