Line integrals of scalar fields are a key concept in multivariable calculus. They allow us to integrate a function along a curve in space, combining ideas from single-variable calculus and vector calculus.

This topic builds on our understanding of scalar fields and parametric curves. We'll learn how to calculate line integrals, explore their properties, and see how they're used to solve real-world problems like finding the work done by a force.

Scalar Fields and Parametric Curves

Scalar Fields

• Scalar field $f(x, y, z)$ assigns a scalar value to each point $(x, y, z)$ in a subset of $\mathbb{R}^3$
• Examples of scalar fields include temperature distribution in a room or electric potential in a region of space
• Scalar fields can be visualized using level curves (2D) or level surfaces (3D), which are sets of points where the scalar field has a constant value

Parametric Curves and Arc Length

• Parametric curve $\mathbf{r}(t) = (x(t), y(t), z(t))$ represents a curve in space as a function of a parameter $t$, typically defined on an interval $[a, b]$
• Examples of parametric curves include a helix $(a\cos t, a\sin t, bt)$ or a circle $(a\cos t, a\sin t, 0)$
• Arc length of a parametric curve $\mathbf{r}(t)$ on the interval $[a, b]$ is given by the integral: $L = \int_a^b |\mathbf{r}'(t)| dt = \int_a^b \sqrt{(x'(t))^2 + (y'(t))^2 + (z'(t))^2} dt$
• Arc length measures the distance traveled along the curve from $\mathbf{r}(a)$ to $\mathbf{r}(b)$

Line Integrals and Their Properties

Line Integrals

• Line integral $\int_C f(x, y, z) ds$ integrates a scalar field $f(x, y, z)$ along a curve $C$ parameterized by $\mathbf{r}(t)$, $t \in [a, b]$
• Evaluation of a line integral involves substituting the parametric equations into the scalar field and integrating with respect to the parameter: $\int_C f(x, y, z) ds = \int_a^b f(x(t), y(t), z(t)) |\mathbf{r}'(t)| dt$
• Line integrals can be used to calculate the mass of a wire with variable density or the work done by a force along a path

Properties of Line Integrals

• Orientation of the curve affects the sign of the line integral; reversing the direction of integration changes the sign of the result
• Path independence: If $f(x, y, z)$ is a conservative field (i.e., it has a potential function), then the line integral of $f$ along any curve connecting two points depends only on the endpoints and not on the path taken
• Closed curve: If $C$ is a closed curve (i.e., it starts and ends at the same point), then the line integral of a conservative field over $C$ is zero

Applications and Theorems

Work and Line Integrals

• Work done by a force $\mathbf{F}(x, y, z) = (P(x, y, z), Q(x, y, z), R(x, y, z))$ along a curve $C$ is given by the line integral: $W = \int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) dt$
• This line integral represents the sum of the dot products of the force and the tangent vector to the curve at each point along the path
• Examples of work 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 $f(x, y, z)$ is a conservative field with potential function $\phi(x, y, z)$ such that $\nabla \phi = f$, then the line integral of $f$ along any curve $C$ from $\mathbf{r}(a)$ to $\mathbf{r}(b)$ is equal to the change in the potential function: $\int_C f(x, y, z) ds = \phi(\mathbf{r}(b)) - \phi(\mathbf{r}(a))$
• This theorem establishes a connection between line integrals and the gradient of a potential function
• The fundamental theorem of line integrals is a powerful tool for simplifying the evaluation of line integrals of conservative fields