19.1 Path independence and conservative vector fields

Conservative Vector Fields and Path Independence

Properties of Conservative Vector Fields

A conservative vector field is one where the line integral between two points doesn't depend on which path you take. Only the starting and ending points matter. This single property has several powerful consequences.

• The line integral $\int_C \vec{F} \cdot d\vec{r}$ yields the same value for every path $C$ connecting the same two endpoints.
• The work done by a conservative field on an object traversing any closed path is exactly zero: no net energy is gained or lost over a round trip.
• Conservative vector fields have zero curl everywhere in their domain: $\nabla \times \vec{F} = \vec{0}$.

These three properties are tightly linked. Zero curl, path independence, and vanishing closed-loop integrals are all equivalent characterizations of a conservative field (provided the domain is simply connected).

Path Independence and Closed Paths

Path independence means that for a conservative field $\vec{F}$, the integral $\int_C \vec{F} \cdot d\vec{r}$ from point $A$ to point $B$ produces the same result regardless of which curve $C$ you choose.

A closed path is any curve that starts and ends at the same point. If you split a closed path into two separate paths from $A$ to $B$ (one going "forward," one going "back"), path independence forces the two integrals to cancel. That's why the closed-loop integral vanishes:

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

The converse also holds: if the integral around every closed loop is zero, then the field is path-independent. These two conditions are logically equivalent.

Simply Connected Regions

The equivalence between zero curl and conservativeness requires a topological condition on the domain: it must be simply connected.

A simply connected region is one where every closed loop can be continuously shrunk to a single point without leaving the region. Intuitively, the region has no holes that a loop could get "stuck" around.

• Simply connected examples: a disk, a rectangular box, all of $\mathbb{R}^3$.
• Not simply connected: an annulus (ring with a hole), the punctured plane $\mathbb{R}^2 \setminus \{(0,0)\}$, a torus.

Why does this matter? On a domain with holes, a field can have zero curl everywhere yet still fail to be conservative. The classic example is $\vec{F} = \left(\frac{-y}{x^2+y^2},\, \frac{x}{x^2+y^2}\right)$, which is irrotational on $\mathbb{R}^2 \setminus \{(0,0)\}$ but has a nonzero integral around any loop encircling the origin. The hole in the domain is what breaks the implication.

On a simply connected domain, zero curl guarantees conservativeness with no exceptions.

Potential Functions and Gradient Fields

Potential Functions

A potential function is a scalar function $\phi(x, y, z)$ satisfying $\vec{F} = \nabla \phi$. When such a $\phi$ exists, $\vec{F}$ is conservative, and computing line integrals becomes dramatically simpler.

Instead of parametrizing a curve and integrating, you evaluate $\phi$ at the endpoints:

$\int_A^B \vec{F} \cdot d\vec{r} = \phi(B) - \phi(A)$

This is the Fundamental Theorem for Line Integrals, and it's the vector-calculus analogue of the single-variable Fundamental Theorem of Calculus.

The potential function is unique only up to an additive constant: if $\phi$ is a potential function, so is $\phi + C$ for any constant $C$, since the constant vanishes under differentiation.

Finding a potential function (step-by-step):

Given $\vec{F} = (P, Q, R)$:

1. Integrate $P$ with respect to $x$: $\phi = \int P\, dx + g(y, z)$, where $g(y,z)$ is an unknown function playing the role of the "constant" of integration.
2. Differentiate your result with respect to $y$ and set it equal to $Q$: $\frac{\partial \phi}{\partial y} = Q$. Solve for $g(y,z)$ up to an unknown function $h(z)$.
3. Differentiate with respect to $z$ and set it equal to $R$: $\frac{\partial \phi}{\partial z} = R$. Solve for $h(z)$.
4. Combine everything to write $\phi(x,y,z)$.

Gradient Fields

A gradient field is any vector field that can be written as $\nabla \phi$ for some scalar function $\phi$:

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

On a simply connected domain, "gradient field" and "conservative field" mean exactly the same thing. On domains with holes, a field can be conservative along certain restricted families of paths yet fail to be a global gradient field. For this course, the key takeaway is: on simply connected domains, the terms are interchangeable.

Geometrically, the gradient vector at any point is perpendicular to the level surface $\phi = \text{const}$ passing through that point and points in the direction of steepest increase of $\phi$. Its magnitude equals the rate of that increase.

Line Integrals and the Curl

Line Integrals

The line integral $\int_C \vec{F} \cdot d\vec{r}$ measures the accumulated effect of a vector field $\vec{F}$ along a curve $C$. Physically, it represents the work done by the force $\vec{F}$ on a particle traveling along $C$.

Computing a line integral (step-by-step):

1. Parametrize the path: write $\vec{r}(t) = (x(t),\, y(t),\, z(t))$ for $t \in [a, b]$.
2. Compute the derivative $\vec{r}\,'(t) = (x'(t),\, y'(t),\, z'(t))$.
3. Substitute the parametrization into $\vec{F}$ to get $\vec{F}(\vec{r}(t))$.
4. Take the dot product $\vec{F}(\vec{r}(t)) \cdot \vec{r}\,'(t)$.
5. Integrate with respect to $t$:

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

If $\vec{F}$ is conservative, you can skip all of this and just use $\phi(B) - \phi(A)$. That's the whole payoff of identifying a field as conservative.

The Curl

The curl of a vector field $\vec{F} = (P, Q, R)$ measures the local rotational tendency of the field:

$\nabla \times \vec{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z},\;\; \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x},\;\; \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)$

A field with $\nabla \times \vec{F} = \vec{0}$ everywhere is called irrotational. On a simply connected domain, irrotational is equivalent to conservative.

The curl connects to line integrals through Stokes' theorem:

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

where $S$ is any oriented surface bounded by the closed curve $C$. If $\nabla \times \vec{F} = \vec{0}$ throughout a simply connected domain, the right side is zero for every surface, confirming that every closed-loop integral vanishes and the field is conservative.

The curl is also a practical screening tool. If you compute $\nabla \times \vec{F}$ and get something nonzero, the field is definitely not conservative, and you can stop looking for a potential function.

Quick check for 2D fields: For $\vec{F} = (P, Q)$ in the plane, the only nontrivial component of the curl is $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$. If this expression equals zero throughout a simply connected region, $\vec{F}$ is conservative.