The Fundamental Theorem of Line Integrals

The fundamental theorem of line integrals extends the fundamental theorem of calculus to line integrals of gradient fields, establishing path independence for conservative vector fields.

The Theorem

Theorem10.4Fundamental Theorem of Line Integrals

Let $f : D \to \mathbb{R}$ be a $C^1$ function on an open set $D \subseteq \mathbb{R}^n$ , and let $C$ be a piecewise smooth curve in $D$ from point $\mathbf{a}$ to point $\mathbf{b}$ . Then $\int_C \nabla f \cdot d\mathbf{r} = f(\mathbf{b}) - f(\mathbf{a})$ In particular, the integral depends only on the endpoints and is independent of the path $C$ .

Proof: If $\mathbf{r}(t)$ , $a \leq t \leq b$ , parametrizes $C$ , then by the chain rule: $\int_C \nabla f \cdot d\mathbf{r} = \int_a^b \nabla f(\mathbf{r}(t)) \cdot \mathbf{r}'(t)\,dt = \int_a^b \frac{d}{dt}[f(\mathbf{r}(t))]\,dt = f(\mathbf{r}(b)) - f(\mathbf{r}(a))$

Applications

ExampleWork in a gravitational field

The gravitational potential near Earth's surface is $f(x,y,z) = -mgz$ . The work done by gravity $\mathbf{F} = -mg\,\hat{\mathbf{k}} = \nabla f$ along any path from height $z_1$ to height $z_2$ is $W = f(z_2) - f(z_1) = -mg(z_2 - z_1) = mg(z_1 - z_2)$ This is independent of the horizontal motion — only the height difference matters.

ExampleElectrostatic potential energy

The electrostatic potential of a point charge $q$ at the origin is $V(r) = \frac{q}{4\pi\epsilon_0 r}$ . The electric field $\mathbf{E} = -\nabla V$ . The work done by $\mathbf{E}$ moving a test charge from distance $r_1$ to $r_2$ is $W = V(r_1) - V(r_2) = \frac{q}{4\pi\epsilon_0}\left(\frac{1}{r_1} - \frac{1}{r_2}\right)$

Exact Differentials

Theorem10.5Criterion for Exactness

The differential expression $P\,dx + Q\,dy$ is exact on a simply connected domain $D$ (i.e., equals $df$ for some $f$ ) if and only if $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$ throughout $D$ .

RemarkConnection to differential forms

In the language of differential forms, the fundamental theorem of line integrals states $\int_C df = f(\mathbf{b}) - f(\mathbf{a})$ , which is the generalized Stokes' theorem applied to $0$ -forms. The exactness condition $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$ says the $1$ -form $\omega = P\,dx + Q\,dy$ is closed ( $d\omega = 0$ ), and on simply connected domains, closed forms are exact.