Exercise 1211Calculus › Multivariable calculus › Vector calculusL1Added 3 Nov 2026

Fundamental theorem for line integrals

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Statement

State the fundamental theorem for line integrals: if $\vec F=\nabla\phi$ , then $\int_C\vec F\cdot d\vec r=\phi(B)-\phi(A)$ .

Hint

Apply the univariate chain rule along a parametrization.

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Solution

Proof. Let $\gamma(t)$ , $t\in[a,b]$ , parametrize $C$ from $A$ to $B$ . Then
$\int_C\vec F\cdot d\vec r=\int_a^b\nabla\phi(\gamma(t))\cdot\gamma'(t)\,dt=\int_a^b\frac{d}{dt}[\phi(\gamma(t))]\,dt=\phi(B)-\phi(A).$

Conclusion. For conservative fields, the integral only depends on endpoints.

Fundamental theorem for line integrals

Statement

Solution

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