Exercise 1210Calculus › Multivariable calculus › Vector calculusL1Added 3 Nov 2026

Conservative vector field

#conservative#gradient#path independence

Statement

Show that $\vec F(x,y)=(2x,2y)$ is conservative and compute $\int_C\vec F\cdot d\vec r$ along any path from $(0,0)$ to $(1,2)$ .

Hint

Find a potential $\phi$ with $\nabla\phi=\vec F$ .

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Potential. $\phi(x,y)=x^2+y^2$ satisfies $\nabla\phi=(2x,2y)=\vec F$ .

Path independence. Integral $=\phi(1,2)-\phi(0,0)=1+4=5$ .

Conclusion. The field is conservative; integral $=5$ regardless of path.