Exercise 1919Calculus › Multivariable calculus › Vector calculusL1Added 3 Nov 2026

Finding a potential function

#potential#conservative#integration

Statement

Find a potential for $\vec F(x,y)=(e^x\cos y,\,-e^x\sin y)$ .

Hint

Integrate $P=e^x\cos y$ in $x$ , then use the condition on $Q$ .

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Integrate $P$ . $\phi=\int e^x\cos y\,dx=e^x\cos y+g(y)$ .

Use $Q$ . $\phi_y=-e^x\sin y+g'(y)=-e^x\sin y$ , so $g'(y)=0$ .

Potential. $\phi(x,y)=e^x\cos y$ .

Conclusion. $\vec F=\nabla\phi$ with $\phi=e^x\cos y$ .