20.2 Applications of Green's theorem

Green's Theorem connects line integrals around closed curves to double integrals over the enclosed regions. It's a powerful tool for calculating area, flux, and circulation in vector fields, often turning a difficult integral into a straightforward one.

This section covers practical applications of Green's Theorem: finding areas of irregular shapes, computing fluid flow through curves, and analyzing conservative vector fields with their potential functions.

Area and Flux

Calculating Area and Flux

Green's Theorem gives you a slick way to calculate the area enclosed by a closed curve $C$ in the plane. Instead of setting up a double integral over the region, you can work directly with the boundary curve.

If you parametrize $C$ as $x = x(t)$, $y = y(t)$ for $a \leq t \leq b$, the area formula is:

$A = \frac{1}{2} \oint_C (x\,dy - y\,dx) = \frac{1}{2} \int_a^b \bigl(x(t)\,y'(t) - y(t)\,x'(t)\bigr)\,dt$

This works because if you set $P = -y$ and $Q = x$ in Green's Theorem, the integrand $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$ simplifies to $1 + 1 = 2$, so the double integral becomes $2A$. Dividing by 2 gives the area formula above.

Flux measures how much of a vector field flows outward through a curve. For a vector field $\mathbf{F}(x,y) = P\,\mathbf{i} + Q\,\mathbf{j}$ and a closed curve $C$ enclosing region $D$:

$\text{Flux} = \oint_C \mathbf{F} \cdot \mathbf{n}\,ds = \iint_D \nabla \cdot \mathbf{F}\,dA = \iint_D \left(\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}\right) dA$

Here $\mathbf{n}$ is the outward unit normal to $C$. This is the two-dimensional divergence theorem, which is itself a form of Green's Theorem.

Circulation measures the tendency of a vector field to rotate around a closed curve:

$\text{Circulation} = \oint_C \mathbf{F} \cdot d\mathbf{r} = \oint_C P\,dx + Q\,dy$

Positive circulation indicates counterclockwise rotation; negative indicates clockwise.

Relationship between Area, Flux, and Circulation

Green's Theorem in its standard (circulation) form states:

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

• The right side is the double integral of the scalar curl of $\mathbf{F}$ over the region $D$. It measures the total rotational tendency inside the region.
• The left side is the circulation of $\mathbf{F}$ along the boundary curve $C$.

Be careful with terminology here: the left side is the circulation (work done along the boundary), not the flux. The flux form of Green's Theorem is the separate identity involving divergence shown above. These are two distinct forms of the same theorem, and mixing them up is a common exam mistake.

This relationship lets you convert between line integrals and double integrals, choosing whichever is easier to evaluate for a given problem.

Vector Fields

Conservative Vector Fields

A vector field $\mathbf{F}(x,y) = P\,\mathbf{i} + Q\,\mathbf{j}$ is conservative if there exists a scalar function $f(x,y)$ such that $\nabla f = \mathbf{F}$. That means:

$P(x,y) = \frac{\partial f}{\partial x} \quad \text{and} \quad Q(x,y) = \frac{\partial f}{\partial y}$

Conservative fields have two key properties:

• Path independence: The line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ depends only on the endpoints, not on the path between them. Specifically, $\int_C \mathbf{F} \cdot d\mathbf{r} = f(\text{endpoint}) - f(\text{start point})$.
• Zero curl: The scalar curl is always zero: $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 0$ everywhere in the domain.

Green's Theorem makes the connection between these properties concrete. If $\mathbf{F}$ is conservative, then the scalar curl is zero throughout $D$, so:

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

This confirms that the circulation around any closed curve is zero for a conservative field. Conversely, if you compute a line integral around a closed curve and get a nonzero answer, the field cannot be conservative.

Potential Functions

For a conservative field $\mathbf{F}$, the scalar function $f$ with $\nabla f = \mathbf{F}$ is called a potential function. Here's how to find one:

1. Integrate $P(x,y)$ with respect to $x$: $f(x,y) = \int P(x,y)\,dx + g(y)$, where $g(y)$ is an unknown function of $y$ alone (it plays the role of the "constant" of integration).
2. Differentiate your result with respect to $y$: compute $\frac{\partial f}{\partial y}$.
3. Set that equal to $Q(x,y)$ and solve for $g'(y)$.
4. Integrate $g'(y)$ to find $g(y)$, then substitute back into $f$.

The potential function is unique up to an additive constant (just like antiderivatives in single-variable calculus).

Equipotential curves are the level curves $f(x,y) = c$. The vector field $\mathbf{F}$ is always perpendicular to these curves, since the gradient of a function points normal to its level curves.

Applications

Work Done by a Vector Field

The work done by a force field $\mathbf{F}$ along a curve $C$ is:

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

For a conservative field, this simplifies to:

$W = f(\text{endpoint}) - f(\text{start point})$

The path doesn't matter, only where you start and end. For a closed curve in particular, the start and end points coincide, so $W = 0$.

For a non-conservative field, the work depends on the specific path taken. This is where Green's Theorem becomes especially useful: rather than parametrizing a complicated closed curve, you can convert to a double integral:

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

If the double integral is easier to evaluate (and it often is), this saves significant effort.

Fluid Flow and Velocity Fields

A velocity field $\mathbf{v}(x,y)$ assigns a velocity vector to each point in the plane, describing how a fluid moves. The field is tangent to streamlines, which trace the paths of individual fluid particles.

Flux through a curve represents the volume of fluid crossing the curve per unit time:

$\text{Flux} = \oint_C \mathbf{v} \cdot \mathbf{n}\,ds$

By the divergence form of Green's Theorem, this equals $\iint_D \nabla \cdot \mathbf{v}\,dA$. For an incompressible fluid, the divergence is zero everywhere ($\nabla \cdot \mathbf{v} = 0$), so the net flux through any closed curve is zero. Physically, this means fluid neither accumulates nor depletes inside any region.

Circulation of a velocity field along a closed curve measures net rotation:

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

Nonzero circulation indicates the presence of vortices or rotational flow within the enclosed region. By Green's Theorem, this equals the double integral of the scalar curl over the region, so you can detect rotation by integrating $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$ over $D$ instead.