18.3 Properties and evaluation of line integrals

Properties of Line Integrals

Line integrals let you compute quantities accumulated along a curve, whether that's work done by a force, mass of a wire, or flux across a boundary. This section covers the core properties that make line integrals manageable and the techniques you'll use to actually evaluate them.

One distinction to keep in mind throughout: properties of scalar line integrals ($\int_C f\, ds$) and vector line integrals ($\int_C \mathbf{F} \cdot d\mathbf{r}$) sometimes behave differently, especially when it comes to reversing orientation.

Fundamental Properties

Linearity is the most frequently used property. For any constants $a$ and $b$ and integrable functions $f$ and $g$ on a curve $C$:

$\int_C (af + bg)\, ds = a\int_C f\, ds + b\int_C g\, ds$

This combines two ideas: additivity (you can split a sum of integrands) and homogeneity (you can pull out scalar constants). The same linearity holds for vector line integrals with $\mathbf{F} \cdot d\mathbf{r}$.

Path additivity lets you break a curve into pieces. If $C$ is composed of consecutive smooth segments $C_1$ and $C_2$ (written $C = C_1 + C_2$), then:

$\int_C f\, ds = \int_{C_1} f\, ds + \int_{C_2} f\, ds$

This is essential for piecewise-smooth curves, where you parameterize and integrate each smooth segment separately, then add the results.

Orientation reversal is where scalar and vector line integrals diverge. Let $-C$ denote the curve $C$ traversed in the opposite direction.

• For scalar line integrals: $\int_{-C} f\, ds = \int_C f\, ds$ (the sign does not change, because $ds > 0$ regardless of direction)
• For vector line integrals: $\int_{-C} \mathbf{F} \cdot d\mathbf{r} = -\int_C \mathbf{F} \cdot d\mathbf{r}$ (the sign does flip, because $d\mathbf{r}$ reverses)

This distinction trips people up on exams. The scalar element $ds$ measures arc length (always positive), while $d\mathbf{r}$ carries directional information.

Applications and Implications

These properties give you a toolkit for simplifying problems before you compute anything:

• Use linearity to split a complicated integrand into parts you can handle separately.
• Use path additivity to deal with curves that have corners or switch formulas (e.g., a path that follows a line segment then an arc).
• Use orientation reversal strategically: if a vector line integral is easier to compute in the reverse direction, compute it that way and negate the result.

Evaluation Techniques

Parameterization

The standard method for evaluating a line integral is to parameterize the curve and reduce everything to a single-variable integral.

For a scalar line integral $\int_C f\, ds$:

1. Parameterize the curve: $\mathbf{r}(t) = (x(t), y(t), z(t))$ for $a \leq t \leq b$.
2. Compute the speed: $\lVert \mathbf{r}'(t) \rVert = \sqrt{(x'(t))^2 + (y'(t))^2 + (z'(t))^2}$.
3. Substitute and integrate:

$\int_C f\, ds = \int_a^b f(\mathbf{r}(t))\, \lVert \mathbf{r}'(t) \rVert\, dt$

For a vector line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$:

1. Parameterize the curve the same way.
2. Compute $\mathbf{r}'(t)$ (you need the vector, not just its magnitude).
3. Substitute and integrate:

$\int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)\, dt$

Notice the difference: scalar line integrals use the magnitude $\lVert \mathbf{r}'(t) \rVert$, while vector line integrals use the vector $\mathbf{r}'(t)$ in a dot product.

Quick example: Evaluate $\int_C y\, ds$ where $C$ is the upper half of the unit circle.

1. Parameterize: $\mathbf{r}(t) = (\cos t, \sin t)$, $0 \leq t \leq \pi$.
2. Speed: $\lVert \mathbf{r}'(t) \rVert = \sqrt{\sin^2 t + \cos^2 t} = 1$.
3. Integrate: $\int_0^{\pi} \sin t \cdot 1\, dt = [-\cos t]_0^{\pi} = 2$.

Numerical Methods

When an antiderivative doesn't exist in closed form, numerical approximation methods like the trapezoidal rule or Simpson's rule work on line integrals too. You divide the parameter interval $[a, b]$ into subintervals, approximate the integrand on each, and sum. This is the same idea as in single-variable calculus, applied after parameterization converts the line integral into a definite integral.

Fundamental Theorem for Line Integrals

This is the vector-calculus analogue of the Fundamental Theorem of Calculus, and it's a major time-saver. If $\mathbf{F} = \nabla f$ (meaning $\mathbf{F}$ is a conservative vector field with potential function $f$), and $C$ is any smooth curve from point $A$ to point $B$, then:

$\int_C \mathbf{F} \cdot d\mathbf{r} = f(B) - f(A)$

The integral depends only on the endpoints, not on the path taken. This means:

• You never need to parameterize the curve. Just evaluate the potential function at the two endpoints and subtract.
• The integral around any closed curve is zero: $\oint_C \mathbf{F} \cdot d\mathbf{r} = 0$ when $\mathbf{F}$ is conservative.

To use this theorem, you first need to verify that $\mathbf{F}$ is conservative (check that $\frac{\partial F_1}{\partial y} = \frac{\partial F_2}{\partial x}$ in 2D, or that $\nabla \times \mathbf{F} = \mathbf{0}$ in 3D on a simply connected domain), then find the potential function $f$.

Change of Variables / Coordinate Systems

Sometimes a curve is more naturally described in polar, cylindrical, or spherical coordinates. The idea is straightforward: parameterize the curve in whatever coordinate system is most convenient, compute $\mathbf{r}'(t)$, and proceed as usual. For instance, a spiral is much easier to parameterize in polar coordinates than in Cartesian. The evaluation formula doesn't change; you're just choosing a smarter parameterization.

Advanced Topics

Stokes' Theorem and Its Significance

Stokes' theorem connects line integrals to surface integrals. If $\mathbf{F}$ is a smooth vector field defined on an oriented surface $S$ with boundary curve $C$ (oriented by the right-hand rule), then:

$\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$

This generalizes the Fundamental Theorem of Calculus to higher dimensions. It also generalizes Green's theorem, which is the special case where $S$ is a flat region in the $xy$-plane.

Stokes' theorem matters because it lets you convert between two types of integrals, and you can choose whichever is easier to compute. In physics, it shows up constantly:

• In electromagnetism, it connects the circulation of the electric field to the rate of change of magnetic flux (Faraday's law).
• In fluid dynamics, it relates the circulation of a velocity field to the vorticity flux through a surface.

Applying Stokes' Theorem

To apply Stokes' theorem in practice:

1. Identify the pieces: the vector field $\mathbf{F}$, the surface $S$, and its boundary curve $C$.
2. Check orientation: the surface normal and the boundary curve must be consistently oriented via the right-hand rule. If you curl the fingers of your right hand along $C$, your thumb should point in the direction of the surface normal.
3. Decide which side to compute. If the line integral looks hard, compute the surface integral of $\nabla \times \mathbf{F}$ instead (or vice versa).
4. Compute. Calculate $\nabla \times \mathbf{F}$, parameterize the surface, and evaluate the surface integral. Or parameterize $C$ and evaluate the line integral directly.

One powerful application: if $\nabla \times \mathbf{F} = \mathbf{0}$ everywhere on $S$, then $\oint_C \mathbf{F} \cdot d\mathbf{r} = 0$. This gives another way to identify conservative fields on simply connected domains.

Stokes' theorem also connects to the divergence theorem (Gauss' theorem), which relates a surface integral over a closed surface to a volume integral of divergence. Together, these three results (Green's, Stokes', divergence) form the backbone of multivariable integral theorems.