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, simplifying complex calculations.

This section explores practical applications of Green's Theorem. We'll use it to find areas of irregular shapes, compute fluid flow through curves, and analyze conservative vector fields and their potential functions.

Area and Flux

Calculating Area and Flux

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• Green's Theorem provides a way to calculate the area enclosed by a closed curve $C$ in the plane
  • Parametrize the curve $C$ as $x = x(t)$, $y = y(t)$, $a \leq t \leq b$
  • Area is given by $A = \frac{1}{2} \oint_C (x\,dy - y\,dx) = \frac{1}{2} \int_a^b (x(t)y'(t) - y(t)x'(t))\,dt$
• Flux measures the amount of a vector field flowing through a surface
  • For a vector field $\mathbf{F}(x, y) = P(x, y)\,\mathbf{i} + Q(x, y)\,\mathbf{j}$ and a curve $C$, the flux is $\iint_D \nabla \cdot \mathbf{F}\,dA = \oint_C \mathbf{F} \cdot \mathbf{n}\,ds$
  • $\mathbf{n}$ is the outward unit normal vector to the curve $C$
  • $D$ is the region enclosed by $C$
• Circulation measures the tendency of a vector field to rotate around a point or axis
  • For a vector field $\mathbf{F}(x, y) = P(x, y)\,\mathbf{i} + Q(x, y)\,\mathbf{j}$ and a curve $C$, the circulation is $\oint_C \mathbf{F} \cdot d\mathbf{r} = \oint_C P\,dx + Q\,dy$
  • Positive circulation indicates counterclockwise rotation, negative indicates clockwise rotation

Relationship between Area, Flux, and Circulation

• Green's Theorem relates the double integral of the curl of a vector field over a region $D$ to the line integral of the field over the boundary curve $C$ of $D$
  • $\iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\,dA = \oint_C P\,dx + Q\,dy$
• The left side of Green's Theorem represents the curl (circulation) of the vector field
• The right side represents the work done by the field along the boundary curve (flux)
• This relationship allows for the calculation of area, flux, and circulation using line integrals instead of double integrals

Vector Fields

Conservative Vector Fields

• A vector field $\mathbf{F}(x, y) = P(x, y)\,\mathbf{i} + Q(x, y)\,\mathbf{j}$ is conservative if there exists a scalar function $f(x, y)$ such that $\nabla f = \mathbf{F}$
  • $P(x, y) = \frac{\partial f}{\partial x}$ and $Q(x, y) = \frac{\partial f}{\partial y}$
• For a conservative vector field, the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ is independent of the path and only depends on the endpoints
  • $\int_C \mathbf{F} \cdot d\mathbf{r} = f(\text{end point}) - f(\text{start point})$
• The curl of a conservative vector field is always zero: $\nabla \times \mathbf{F} = 0$

Potential Functions

• For a conservative vector field $\mathbf{F}$, a scalar function $f$ such that $\nabla f = \mathbf{F}$ is called a potential function for $\mathbf{F}$
• To find a potential function, integrate the components of $\mathbf{F}$:
  • $f(x, y) = \int P(x, y)\,dx + C(y)$ where $C(y)$ is a function of $y$
  • Differentiate $f$ with respect to $y$ and equate it to $Q(x, y)$ to find $C(y)$
• The potential function is unique up to a constant
• Equipotential curves are level curves of the potential function $f(x, y) = c$ where $c$ is a constant
  • The vector field $\mathbf{F}$ is always perpendicular to its equipotential curves

Applications

Work Done by a Vector Field

• The work done by a force field $\mathbf{F}$ along a curve $C$ is given by the line integral $W = \int_C \mathbf{F} \cdot d\mathbf{r}$
  • If $\mathbf{F}$ is conservative, the work done is independent of the path and equals the change in the potential function
  • $W = \int_C \mathbf{F} \cdot d\mathbf{r} = f(\text{end point}) - f(\text{start point})$
• For a closed curve, the work done by a conservative field is always zero
  • $\oint_C \mathbf{F} \cdot d\mathbf{r} = 0$ for conservative $\mathbf{F}$
• The work done by a non-conservative field depends on the path taken

Fluid Flow and Velocity Fields

• A velocity field $\mathbf{v}(x, y)$ describes the velocity of a fluid at each point $(x, y)$
  • The velocity field is tangent to the streamlines (path of fluid particles)
• The flux of a velocity field through a curve represents the volume of fluid flowing through the curve per unit time
  • Flux = $\int_C \mathbf{v} \cdot \mathbf{n}\,ds$ where $\mathbf{n}$ is the outward unit normal vector
• The circulation of a velocity field along a closed curve measures the net rotation of the fluid
  • Circulation = $\oint_C \mathbf{v} \cdot d\mathbf{r}$
  • Non-zero circulation indicates the presence of vortices or eddies in the fluid
• Incompressible fluids have a divergence-free velocity field: $\nabla \cdot \mathbf{v} = 0$
  • By the divergence theorem, the net flux through any closed curve is zero for an incompressible fluid