Exercise 592Calculus › Multivariable calculus › Vector calculusL1Added 3 Nov 2026

Closed paths in conservative fields

#closed path#conservative#zero integral

Statement

Show that if $\vec F$ is conservative and $C$ is closed, then $\oint_C\vec F\cdot d\vec r=0$ .

Hint

Use the fundamental theorem: $\phi(B)-\phi(A)$ with $A=B$ .

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Apply. $\oint_C\vec F\cdot d\vec r=\phi(A)-\phi(A)=0$ .

Conclusion. Conservative fields have zero circulation on closed loops.