18.2 Line integrals of vector fields

Vector Fields and Line Integrals

Defining and Visualizing Vector Fields

A vector field assigns a vector to each point in a subset of space. In 2D, you might write $\mathbf{F}(x, y) = (P(x,y),\, Q(x,y))$; in 3D, $\mathbf{F}(x,y,z) = (P, Q, R)$. Vector fields model physical phenomena like fluid flow, electromagnetic fields, and gravitational fields.

Two common ways to visualize them:

• Arrow plots show the direction and magnitude of vectors at sampled points. Longer arrows mean larger magnitude.
• Streamlines are curves that are tangent to the vector field at every point, tracing out the flow pattern. A particle released into the field would follow a streamline.

The Line Integral of a Vector Field

The central object here is the line integral of a vector field $\mathbf{F}$ along a curve $C$, written:

$\int_C \mathbf{F} \cdot d\mathbf{r}$

This measures the accumulated component of $\mathbf{F}$ along the direction of travel. Physically, if $\mathbf{F}$ is a force, this integral gives the work done by the force on an object moving along $C$.

To compute it, parametrize the curve as $\mathbf{r}(t)$ for $a \le t \le b$, then:

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

The dot product at each point picks out how much of $\mathbf{F}$ points along the curve's tangent direction. Where $\mathbf{F}$ is perpendicular to the curve, the contribution is zero.

Circulation and Flux

Circulation measures the tendency of a vector field to push along a closed curve $C$:

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

• Positive circulation indicates net counterclockwise rotation; negative indicates clockwise.
• If circulation is zero around every closed curve, the field has no rotational tendency.

Flux across a curve (in 2D) or through a surface (in 3D) measures how much of the field passes through:

• In 2D, flux across a closed curve $C$ is $\oint_C \mathbf{F} \cdot \mathbf{n}\, ds$, where $\mathbf{n}$ is the outward unit normal.
• Positive flux means net outward flow; negative means net inward flow.

These two quantities capture complementary information: circulation is about flow along a curve, while flux is about flow across it.

Path Independence and Conservative Fields

A vector field is path independent if the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ between any two points depends only on the endpoints, not on which curve connects them. This is a strong condition with a direct physical meaning: the work done by $\mathbf{F}$ moving an object from point $A$ to point $B$ is the same no matter what route you take.

A vector field with this property is called conservative. Conservative fields have a potential function $\phi$ such that $\mathbf{F} = \nabla \phi$. The potential function acts like a stored-energy landscape: the field always points "downhill" on that landscape, and the work between two points is just the difference in potential values.

Conservative Vector Fields

Properties and Conditions

Conservative vector fields satisfy two equivalent properties:

• Path independence: $\int_C \mathbf{F} \cdot d\mathbf{r}$ depends only on the endpoints of $C$.
• Zero circulation on closed loops: $\oint_C \mathbf{F} \cdot d\mathbf{r} = 0$ for every closed curve $C$ in the domain.

These are genuinely equivalent: if every closed-loop integral is zero, you can show any two paths between the same endpoints give the same integral (subtract them to form a closed loop).

For a vector field $\mathbf{F}(x, y, z) = (P, Q, R)$ with continuous first partial derivatives, the component test checks whether $\mathbf{F}$ could be conservative:

• $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$
• $\frac{\partial P}{\partial z} = \frac{\partial R}{\partial x}$
• $\frac{\partial Q}{\partial z} = \frac{\partial R}{\partial y}$

This is equivalent to requiring $\nabla \times \mathbf{F} = \mathbf{0}$ (the curl is zero). But there's a catch: the domain must be simply connected (no holes). On a domain with holes, a field can pass the component test yet still fail to be conservative. The classic example is the angle field around the origin in 2D.

Potential Functions and Gradients

If $\mathbf{F} = (P, Q, R)$ is conservative, there exists a scalar function $\phi(x,y,z)$ with:

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

The Fundamental Theorem for Line Integrals then gives you a powerful shortcut:

$\int_C \mathbf{F} \cdot d\mathbf{r} = \phi(\text{endpoint}) - \phi(\text{start point})$

No parametrization needed. Just evaluate $\phi$ at two points and subtract.

Finding the potential function step by step:

1. Integrate $P$ with respect to $x$: $\phi(x,y,z) = \int P\, dx + g(y,z)$, where $g(y,z)$ is an unknown function (it plays the role of the "constant" of integration, but it can depend on the other variables).
2. Differentiate your result with respect to $y$ and set it equal to $Q$. This lets you solve for $\frac{\partial g}{\partial y}$.
3. Integrate to update $g(y,z) = h(y) + k(z)$, where $k(z)$ is still unknown.
4. Differentiate with respect to $z$ and set equal to $R$ to find $k(z)$.
5. Combine everything. The potential is determined up to an additive constant.

Applications and Examples

Gravitational and electrostatic fields are the classic conservative fields. The gravitational field $\mathbf{F} = -\frac{GMm}{r^3}\mathbf{r}$ has potential function $\phi = -\frac{GMm}{r}$, and the work done moving a mass between two points depends only on the change in gravitational potential energy.

Non-conservative fields, like friction or magnetic forces on moving charges, do not have potential functions. For these fields, the work done genuinely depends on the path, and closed-loop integrals can be nonzero.

Green's Theorem

Statement and Formulation

Green's theorem connects a line integral around a closed curve to a double integral over the region it encloses. It's a 2D result that converts between two types of computation, letting you pick whichever is easier.

For a vector field $\mathbf{F}(x,y) = (P(x,y),\, Q(x,y))$ where $P$ and $Q$ have continuous partial derivatives on an open region containing $D$ and its boundary $C$:

$\oint_C (P\, dx + Q\, dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA$

The curve $C$ must be positively oriented (counterclockwise) and simple (no self-intersections). The left side is the circulation of $\mathbf{F}$ around $C$. The right side integrates the scalar curl $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$ over the enclosed region.

Notice that if $\mathbf{F}$ is conservative, the scalar curl is zero everywhere, so the double integral is zero, confirming that circulation around any closed curve vanishes.

Applications and Examples

Green's theorem is useful in several ways:

• Simplifying line integrals: When the boundary curve $C$ is complicated but the region $D$ has a simple shape, convert to a double integral.
• Computing areas: Choosing $P$ and $Q$ so that $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 1$ turns the double integral into the area of $D$. A common choice is $P = -y/2$, $Q = x/2$, giving $\text{Area} = \frac{1}{2}\oint_C (x\, dy - y\, dx)$.

Example: Find the circulation of $\mathbf{F}(x,y) = (xy,\, x^2)$ around the unit circle.

1. Compute the scalar curl: $\frac{\partial}{\partial x}(x^2) - \frac{\partial}{\partial y}(xy) = 2x - x = x$.

2. Apply Green's theorem: $\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_D x\, dA$.

3. Over the unit disk, $x$ is an odd function in $x$ and the region is symmetric about the $y$-axis, so the integral is $0$.

Green's theorem is a special case of the more general Stokes' theorem, which relates the surface integral of $\nabla \times \mathbf{F}$ over a surface to the line integral of $\mathbf{F}$ around the surface's boundary. In 2D with a flat region, Stokes' theorem reduces exactly to Green's theorem.