Chapter 35 — Key Takeaways
The Big Idea
A line integral is a Riemann sum that follows a curve. Chop the curve into tiny pieces, evaluate something on each piece, multiply by the size of the piece, and sum — the same logic as every integral since Chapter 13, now with the pieces lying along a curve. Two genuinely different flavors result, and this chapter culminates in two theorems — the Fundamental Theorem for Line Integrals and Green's theorem — that are the Fundamental Theorem of Calculus (Chapter 14) generalized to two dimensions.
Two Types of Line Integral
| Scalar line integral | Vector line integral | |
|---|---|---|
| Integrand | scalar field $f$ | vector field $\mathbf{F}$ |
| Element | $ds = \|\mathbf{r}'(t)\|\,dt$ | $d\mathbf{r} = \mathbf{r}'(t)\,dt$ |
| Formula | $\int_C f\,ds = \int_a^b f(\mathbf{r}(t))\,\|\mathbf{r}'(t)\|\,dt$ | $\int_C\mathbf{F}\cdot d\mathbf{r} = \int_a^b\mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt$ |
| Archetype | mass of a wire | work done by a force |
| Orientation | blind to direction | flips sign when reversed |
Scalar (§35.2): weights $f$ by arc length. Applications: mass $M=\int_C\rho\,ds$, center of mass, average value, curve length ($\int_C 1\,ds$). The most common error is dropping the $\|\mathbf{r}'\|$ factor; the check is that $\int_C 1\,ds$ must equal the length of $C$.
Vector (§35.3): extracts the tangential component $\mathbf{F}\cdot\mathbf{T}$ and accumulates it. For a closed curve it is called circulation, written $\oint_C\mathbf{F}\cdot d\mathbf{r}$. In components, $\int_C\mathbf{F}\cdot d\mathbf{r}=\int_C P\,dx+Q\,dy$.
Procedure for a Vector Line Integral
- Check for a potential first (curl test, §35.6). If $\mathbf{F}=\nabla f$, the answer is $f(B)-f(A)$ — skip to the end.
- Parametrize $C$: $\mathbf{r}(t)$, $a\le t\le b$, in the correct direction.
- Differentiate: compute $\mathbf{r}'(t)$.
- Substitute into the field: form $\mathbf{F}(\mathbf{r}(t))$.
- Dot and integrate: $\int_a^b\mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt$.
- For a closed loop, consider Green's theorem instead — a double integral of the curl is often far easier.
(For a scalar integral, step 5 uses $f(\mathbf{r}(t))\,\|\mathbf{r}'(t)\|$ — the magnitude of velocity, not a dot product.)
Fundamental Theorem for Line Integrals (§35.5)
For a conservative field $\mathbf{F}=\nabla f$ and any curve $C$ from $A$ to $B$: $$\int_C\nabla f\cdot d\mathbf{r} = f(B) - f(A).$$
- It is FTC again. Integrate a gradient (the multivariable derivative), get the net change of the potential across the endpoints — exactly $\int_a^b F'\,dx = F(b)-F(a)$, one dimension up. The proof is the multivariable chain rule (Chapter 30) followed by the single-variable FTC (Chapter 14).
- Path-independence: the value depends only on the endpoints, never on the route.
- Closed-loop corollary: $\oint_C\nabla f\cdot d\mathbf{r}=0$. This is the calculus face of conservation of energy: a conservative force does zero net work around any closed loop.
Conservative ⇔ Path-Independent ⇔ Curl-Free (§35.6)
On a simply-connected domain (no holes), these three conditions are equivalent:
- $\mathbf{F}=\nabla f$ for some potential $f$ (conservative);
- $\nabla\times\mathbf{F}=\mathbf{0}$, i.e. in 2D, $\dfrac{\partial Q}{\partial x}=\dfrac{\partial P}{\partial y}$ (curl-free);
- $\int_C\mathbf{F}\cdot d\mathbf{r}$ is path-independent, equivalently $\oint_C\mathbf{F}\cdot d\mathbf{r}=0$ on every closed loop.
Warning: curl-free implies conservative only on a simply-connected domain. The vortex $\mathbf{F}=\langle -y,x\rangle/(x^2+y^2)$ is curl-free but has $\oint=2\pi$ around the origin, because its domain (the punctured plane) has a hole.
Green's Theorem (§35.7)
For a positively oriented (counterclockwise), piecewise-smooth, simple closed curve $C$ bounding region $D$: $$\oint_C P\,dx + Q\,dy = \iint_D\left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dA, \qquad\text{i.e.}\qquad \oint_C\mathbf{F}\cdot d\mathbf{r} = \iint_D(\nabla\times\mathbf{F})_z\,dA.$$
Total circulation around the boundary equals total microscopic curl over the interior. Interior edges of a fine paving cancel in pairs; only the outer boundary survives — the telescoping idea behind every theorem in the family.
Orientation matters: a clockwise loop gives the negative; flip the sign before applying the theorem.
Area corollary: choosing $Q_x-P_y=1$ (e.g. $P=-y/2,\ Q=x/2$) turns the double integral into the area: $$\text{Area}(D)=\frac12\oint_C(x\,dy - y\,dx).$$ This is the principle of the planimeter and the continuous form of the shoelace formula of computational geometry (Case Study 2).
Flux form (the 2D Divergence Theorem): integrating the outward-normal component instead of the tangential one gives $\oint_C\mathbf{F}\cdot\mathbf{n}\,ds=\iint_D(\nabla\cdot\mathbf{F})\,dA$. Circulation form measures swirling; flux form measures spreading.
Where Green's Theorem Sits (§35.8)
| Theorem | Statement | Chapter |
|---|---|---|
| Fundamental Theorem of Calculus | $\int_a^b F'\,dx = F(b)-F(a)$ | 14 |
| Fund. Thm. for Line Integrals | $\int_C\nabla f\cdot d\mathbf{r} = f(B)-f(A)$ | 35 |
| Green's Theorem | $\oint_{\partial D}\mathbf{F}\cdot d\mathbf{r}=\iint_D(\nabla\times\mathbf{F})_z\,dA$ | 35 |
| Stokes' Theorem | $\oint_{\partial S}\mathbf{F}\cdot d\mathbf{r}=\iint_S(\nabla\times\mathbf{F})\cdot d\mathbf{S}$ | 37 |
| Divergence Theorem | $\oiint_{\partial V}\mathbf{F}\cdot d\mathbf{S}=\iiint_V(\nabla\cdot\mathbf{F})\,dV$ | 37 |
| Generalized Stokes | $\int_{\partial M}\omega=\int_M d\omega$ | 38 |
Green's theorem is Stokes' theorem flattened into the plane. Every row says the same thing: the integral of a derivative over a region equals boundary data.
Applications in One Glance
| Reading | Field | Where |
|---|---|---|
| Work against gravity = $mg\,\Delta h$ (path-independent) | physics | Case Study 1 |
| Friction work = $\int_C ds$ (path-dependent, non-conservative) | physics | Case Study 1 |
| Magnetic force does zero work ($\mathbf{F}\perp\mathbf{v}$) | physics | §35.9 |
| Circulation $\Gamma=\oint\mathbf{u}\cdot d\mathbf{r}$ and lift (Kutta–Joukowski) | fluid dynamics | §35.9 |
| Area from boundary; planimeter; shoelace formula | surveying / GIS | Case Study 2 |
| Ampère's & Faraday's laws as loop integrals | electromagnetism | §35.9 |
Common Errors
- Dropping $\|\mathbf{r}'\|$ in a scalar integral. $\int_C 1\,ds$ must return the curve's length.
- Confusing the flavors. Scalar uses $\|\mathbf{r}'\|\,dt$; vector uses a dot product $\mathbf{F}\cdot\mathbf{r}'\,dt$.
- Wrong orientation. Vector integrals flip sign with direction; a clockwise loop needs a sign flip before Green's theorem.
- Green's on a non-closed curve. It requires a closed boundary; for an open arc, parametrize directly or close it and subtract.
- Hunting for a potential without the curl test. Non-conservative fields produce contradictory "potentials" (the $\langle -y,x\rangle$ trap, §35.6).
- Ignoring connectivity. Curl-free ⇒ conservative fails on domains with holes.
Connections
- Chapter 25: parametric curves — every line integral relies on a parametrization $\mathbf{r}(t)$.
- Chapter 28: vector-valued functions — source of $\mathbf{r}'(t)$ and arc length.
- Chapter 30: the multivariable chain rule — the engine of the Fundamental Theorem for Line Integrals.
- Chapter 32: double integrals — the right-hand side of Green's theorem.
- Chapter 34: vector fields, conservative fields, and the curl — the integrand and the screening test.
- Chapter 36: surface integrals — line integrals one dimension up; flux through a surface.
- Chapter 37: Stokes' and the Divergence Theorem — the full 3D generalizations of Green's theorem.
- Chapter 38: the universal FTC $\int_{\partial M}\omega=\int_M d\omega$ that contains all of the above.
What's Next
Chapter 36 lifts integration to surfaces: scalar surface integrals (area and mass of a shell) and vector surface integrals — flux, $\iint_S\mathbf{F}\cdot d\mathbf{S}$, how much of a field passes through a surface. Chapter 37 then delivers the climax: Stokes' theorem and the Divergence theorem, with Green's theorem revealed as their flat, two-dimensional shadow.
Reflection
A line integral is a Riemann sum that has learned to follow a curve. Scalar line integrals give masses and lengths; vector line integrals give work and circulation, the energy interpretation that unifies physics. The Fundamental Theorem for Line Integrals shows that conservative fields forget their paths and remember only their endpoints — conservation of energy, written as calculus. And Green's theorem delivers the first true statement that a region's boundary determines what happens throughout its interior, the pattern that organizes all of vector calculus. You have taken the Fundamental Theorem of Calculus and bent it around a curve.