19.2 The fundamental theorem for line integrals

Fundamental Theorem and Conservative Fields

The Fundamental Theorem for Line Integrals

The fundamental theorem for line integrals is the multivariable generalization of the Fundamental Theorem of Calculus. Just as the single-variable FTC lets you evaluate a definite integral by finding an antiderivative at two endpoints, this theorem lets you evaluate a line integral through a vector field by evaluating a potential function at two endpoints.

Here's the precise statement: if $\mathbf{F}$ is a conservative vector field with potential function $f$ (meaning $\mathbf{F} = \nabla f$), and $C$ is a smooth curve parameterized by $\mathbf{r}(t)$ for $a \leq t \leq b$, then

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

The line integral depends only on the values of $f$ at the endpoints of $C$. You don't need to parameterize the curve or set up a complicated integral. Find the potential function, plug in the endpoints, subtract.

Conservative Vector Fields and Path Independence

A vector field $\mathbf{F}$ is conservative if there exists a scalar function $f$ such that $\nabla f = \mathbf{F}$. Conservative fields have a powerful property: path independence. This means the line integral of $\mathbf{F}$ between two points gives the same value regardless of which path you take between them.

A direct consequence of path independence is that the line integral over any closed curve is zero:

$\oint_C \mathbf{F} \cdot d\mathbf{r} = 0$

This makes sense if you think about the fundamental theorem: on a closed curve, the start and end points are the same, so $f(\text{end}) - f(\text{start}) = 0$.

Gravitational and electrostatic force fields are classic physical examples of conservative fields. In both cases, the work done moving an object between two points depends only on where you start and where you end, not on the route you take.

Scalar Fields and Their Relationship to Vector Fields

A scalar field assigns a single number to each point in space. Temperature distribution across a room is a good example: every point has a temperature value, but that value isn't a vector.

When you take the gradient of a scalar field $f$, you get a vector field $\nabla f$. If a given vector field $\mathbf{F}$ happens to equal $\nabla f$ for some scalar function $f$, then $f$ is called a potential function for $\mathbf{F}$, and $\mathbf{F}$ is conservative. This relationship is what makes the fundamental theorem work: the potential function is the bridge that lets you convert a line integral into a simple endpoint calculation.

Gradients and Potential Functions

Gradients and Their Properties

The gradient of a scalar function $f$ is a vector field written $\nabla f$. In Cartesian coordinates:

$\nabla f = \left(\frac{\partial f}{\partial x},\; \frac{\partial f}{\partial y},\; \frac{\partial f}{\partial z}\right)$

Two key geometric facts about the gradient:

• It points in the direction of greatest rate of increase of $f$ at each point.
• Its magnitude $\|\nabla f\|$ equals the rate of change of $f$ in that direction.

Think of a topographic map. At any point on the surface, the gradient tells you which direction is the steepest uphill climb and how steep that climb is. Where $\nabla f = \mathbf{0}$, you're at a critical point (a peak, valley, or saddle).

Potential Functions and Their Relationship to Vector Fields

A potential function (or scalar potential) for a vector field $\mathbf{F}$ is a scalar function $f$ satisfying $\nabla f = \mathbf{F}$. A potential function exists if and only if $\mathbf{F}$ is conservative.

One thing to watch: the potential function is never unique. If $f$ is a potential function for $\mathbf{F}$, then $f + C$ for any constant $C$ is also a valid potential function, since the gradient of a constant is zero. This parallels the "+C" ambiguity in single-variable antiderivatives.

Physical examples include gravitational potential energy (whose gradient gives the gravitational force field) and electric potential (whose gradient gives the electric field).

Finding a Potential Function

Finding a potential function is the multivariable analog of finding an antiderivative. If $\mathbf{F} = (P, Q, R)$ is conservative, you need a function $f$ satisfying:

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

Here's the standard process:

1. Integrate one component. Start by integrating $P$ with respect to $x$: $f(x, y, z) = \int P\, dx + g(y, z)$. The "constant" of integration here is a function $g(y, z)$ since it could depend on the other variables.

2. Determine the unknown function. Differentiate your result with respect to $y$ and set it equal to $Q$: $\frac{\partial f}{\partial y} = Q$. This lets you solve for $\frac{\partial g}{\partial y}$, which you then integrate with respect to $y$ to get $g(y, z) = (\text{something}) + h(z)$.

3. Pin down the last piece. Differentiate with respect to $z$ and set it equal to $R$ to find $h(z)$.

4. Assemble $f$. Combine everything into a single expression for $f(x, y, z)$.

Example: Suppose $\mathbf{F} = (2xy,\; x^2 + z,\; y)$. Integrate $P = 2xy$ with respect to $x$ to get $f = x^2 y + g(y,z)$. Differentiate with respect to $y$: $x^2 + g_y = x^2 + z$, so $g_y = z$, giving $g = yz + h(z)$. Differentiate with respect to $z$: $y + h'(z) = y$, so $h'(z) = 0$ and $h(z) = C$. The potential function is $f(x,y,z) = x^2 y + yz + C$.

Once you have $f$, any line integral of $\mathbf{F}$ reduces to evaluating $f$ at the endpoints and subtracting.