| | 2025.01.28

6: Line Integrals

Integrating functions along curves.

Line Integrals of Scalar Functions

For a curve $C$ parametrized by $\mathbf{r}(t)$ , $a \leq t \leq b$ :

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

where $ds = |\mathbf{r}'(t)| \, dt$ is the arc length element.

For a vector field $\mathbf{F} = \langle P, Q, R \rangle$ along curve $C$ :

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

\int_C \mathbf{F} \cdot d\mathbf{r} = \int_C P \, dx + Q \, dy + R \, dz = \int_C \mathbf{F} \cdot \mathbf{T} \, ds

If $\mathbf{F}$ is a force field, $\int_C \mathbf{F} \cdot d\mathbf{r}$ is the work done moving along $C$ .

Depends on direction: $\int_{-C} \mathbf{F} \cdot d\mathbf{r} = -\int_C \mathbf{F} \cdot d\mathbf{r}$
Additive over paths: $\int_{C_1 + C_2} = \int_{C_1} + \int_{C_2}$

If $\mathbf{F} = \nabla f$ (conservative field) and $C$ goes from $A$ to $B$ :

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

Key insight: For conservative fields, the line integral depends only on endpoints, not the path!

A vector field $\mathbf{F}$ is conservative if $\mathbf{F} = \nabla f$ for some scalar function $f$ (called the potential function).

The following are equivalent for $\mathbf{F}$ on a simply connected domain:

$\mathbf{F} = \langle P, Q \rangle$ is conservative iff:

\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}

$\mathbf{F} = \langle P, Q, R \rangle$ is conservative iff:

\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}, \quad \frac{\partial P}{\partial z} = \frac{\partial R}{\partial x}, \quad \frac{\partial Q}{\partial z} = \frac{\partial R}{\partial y}

If $\mathbf{F} = \langle P, Q \rangle$ is conservative:

W = \int_C \mathbf{F} \cdot d\mathbf{r}

For a closed curve $C$ :

\text{Circulation} = \oint_C \mathbf{F} \cdot d\mathbf{r}

Measures the tendency of the field to circulate around $C$ .

\text{Flux} = \int_C \mathbf{F} \cdot \mathbf{n} \, ds

where $\mathbf{n}$ is the outward normal.

Integral	Formula	Physical Meaning
Scalar line integral	$\int_C f \, ds$	Mass of wire, arc length
Vector line integral	$\int_C \mathbf{F} \cdot d\mathbf{r}$	Work done by force
Conservative field	$\int_C \nabla f \cdot d\mathbf{r} = f(B) - f(A)$	Path-independent
Circulation	$\oint_C \mathbf{F} \cdot d\mathbf{r}$	Rotational tendency