18.1 Line integrals of scalar fields

Scalar Fields and Parametric Curves

Scalar Fields

A scalar field $f(x, y, z)$ assigns a single real number to each point $(x, y, z)$ in some subset of $\mathbb{R}^3$. Think of temperature at every point in a room, or electric potential throughout a region of space.

You can visualize scalar fields using level curves (in 2D) or level surfaces (in 3D), which are the sets of points where the field takes a constant value.

Parametric Curves and Arc Length

A parametric curve $\mathbf{r}(t) = (x(t), y(t), z(t))$ traces out a path in space as the parameter $t$ varies over an interval $[a, b]$. Common examples:

• A helix: $(a\cos t,\; a\sin t,\; bt)$
• A circle in the $xy$-plane: $(a\cos t,\; a\sin t,\; 0)$

The arc length of $\mathbf{r}(t)$ on $[a, b]$ measures the total distance traveled along the curve:

$L = \int_a^b \|\mathbf{r}'(t)\|\, dt = \int_a^b \sqrt{(x'(t))^2 + (y'(t))^2 + (z'(t))^2}\, dt$

The quantity $\|\mathbf{r}'(t)\|\, dt$ is the infinitesimal arc length element $ds$. This is exactly what shows up inside line integrals.

Line Integrals and Their Properties

Definition and Evaluation

The line integral of a scalar field $f$ along a curve $C$ is written $\int_C f\, ds$. It accumulates the values of $f$ along the curve, weighted by arc length. If $C$ is parameterized by $\mathbf{r}(t)$ for $t \in [a, b]$, the evaluation formula is:

$\int_C f(x, y, z)\, ds = \int_a^b f\bigl(x(t),\, y(t),\, z(t)\bigr)\, \|\mathbf{r}'(t)\|\, dt$

Here's the step-by-step process:

1. Parameterize the curve. Write $\mathbf{r}(t) = (x(t), y(t), z(t))$ on $[a, b]$.
2. Compute $\mathbf{r}'(t)$ and then its magnitude $\|\mathbf{r}'(t)\|$.
3. Substitute $x(t), y(t), z(t)$ into $f$ to get $f(\mathbf{r}(t))$.
4. Multiply $f(\mathbf{r}(t))$ by $\|\mathbf{r}'(t)\|$ and integrate from $a$ to $b$.

A classic application: if a wire has shape $C$ and density function $\rho(x,y,z)$, its total mass is $\int_C \rho\, ds$.

Properties of Scalar Line Integrals

A crucial property of scalar line integrals is independence of orientation. Unlike line integrals of vector fields, reversing the direction you traverse $C$ does not change the value of $\int_C f\, ds$. The scalar line integral depends on the geometric curve itself, not on which direction you walk along it.

The integral is also independent of parameterization: any smooth reparameterization of the same curve gives the same result, as long as the curve is traversed exactly once.

Applications and Theorems

Work and Vector Line Integrals

The section above covers scalar line integrals (integrating $f\, ds$). There's a closely related but distinct object: the line integral of a vector field, which computes work. The work done by a force $\mathbf{F}(x,y,z) = (P, Q, R)$ along $C$ is:

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

This integral does depend on orientation: reversing the curve flips the sign of $W$. That makes physical sense, since walking against a force versus with it should give opposite work values.

Examples include the work done by gravity on an object moving along a curved path, or the work done by an electric field on a charged particle.

Fundamental Theorem of Line Integrals

If $\mathbf{F}$ is a conservative vector field with potential function $\phi$ (meaning $\mathbf{F} = \nabla \phi$), then:

$\int_C \mathbf{F} \cdot d\mathbf{r} = \phi(\mathbf{r}(b)) - \phi(\mathbf{r}(a))$

This is the multivariable analogue of the Fundamental Theorem of Calculus. Two important consequences:

• The integral depends only on the endpoints, not on the particular path $C$ connecting them.
• Over any closed curve (where $\mathbf{r}(a) = \mathbf{r}(b)$), the integral is zero.

Note that this theorem applies to vector line integrals ($\int_C \mathbf{F} \cdot d\mathbf{r}$), not directly to scalar line integrals ($\int_C f\, ds$). The original guide conflated these two types. Keep them straight: scalar line integrals use $ds$ and measure accumulated value along a curve; vector line integrals use $d\mathbf{r}$ and measure work-like quantities.