scoresvideos
Calculus IV
Table of Contents
Unit 20 Overview: Green's Theorem
20.1 Green's theorem in the plane
20.2 Applications of Green's theorem
20.3 Simply and multiply connected regions

∞calculus iv review

20.2 Applications of Green's theorem

Citation:

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

  • 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

Key Terms to Review (17)

Divergence: Divergence is a mathematical operator that measures the magnitude of a vector field's source or sink at a given point, essentially indicating how much a field spreads out or converges in space. This concept is crucial in understanding the behavior of fluid flow and electromagnetic fields, as it relates to how quantities like mass or electric field lines are distributed over a region.
Flow across a curve: Flow across a curve refers to the quantitative measurement of a vector field's behavior as it interacts with a specific curve in a plane. This concept is essential when applying Green's theorem, which connects line integrals around a simple closed curve to double integrals over the plane region it encloses. Understanding flow across a curve helps to analyze circulation and flux, two fundamental aspects of vector fields in calculus.
Piecewise smooth boundary: A piecewise smooth boundary refers to a boundary of a region in space that is made up of a finite number of smooth curves or surfaces, each of which is differentiable except possibly at a finite number of points. This concept is important because it allows for the application of various integral theorems, ensuring that calculations involving line and surface integrals can be performed effectively over these boundaries.
Fundamental Theorem of Calculus: The Fundamental Theorem of Calculus connects differentiation and integration, showing that these two fundamental operations are essentially inverses of each other. It consists of two parts: the first part establishes that the integral of a function can be computed using its antiderivative, while the second part states that the derivative of an integral function is the original function. This theorem is crucial for understanding concepts like area under curves and relates directly to applications in vector fields.
Area Calculation: Area calculation refers to the process of determining the size of a two-dimensional region or shape. This involves integrating functions over specified regions, which can be rectangular or non-rectangular, and often requires techniques like double integrals or converting to polar coordinates for more complex shapes. Understanding area calculation is crucial in various fields, including physics, engineering, and computer graphics, as it allows for the quantification of space and resources.
Bounded region: A bounded region is a set of points in space that is enclosed within finite limits, often forming a closed shape. This concept is essential in multiple mathematical applications, as it determines the area or volume to be considered when integrating functions over that space. Recognizing a bounded region helps in applying various theorems and methods for evaluating integrals, such as changing the order of integration and utilizing the divergence theorem.
George Green: George Green was an English mathematician and physicist known for his contributions to potential theory and for formulating Green's Theorem, which relates a line integral around a simple closed curve to a double integral over the region it encloses. His work laid the groundwork for various important mathematical concepts and applications in physics, especially in understanding fluid flow and electromagnetism.
Work done by a force: Work done by a force is defined as the integral of the force vector along a path taken by an object, representing the energy transferred by the force in moving the object. This concept connects deeply with various principles, such as the idea of conservative vector fields, which are linked to potential energy, and how work can be represented through circulation in vector fields. Understanding this term also involves exploring line integrals of scalar fields and how they relate to calculating work in practical applications.
Continuity: Continuity is a property of functions that describes the behavior of a function at a point, ensuring that small changes in input result in small changes in output. It is crucial for understanding how functions behave, particularly when dealing with limits, derivatives, and integrals across multiple dimensions.
Line Integral: A line integral is a mathematical concept that allows us to integrate functions along a curve or path in a given space. It is particularly useful for calculating quantities like arc length, work done by a force field along a path, and evaluating circulations in vector fields. Line integrals can be used in both scalar and vector fields, connecting them to various important theorems and applications in physics and engineering.
Vector Field: A vector field is a function that assigns a vector to every point in a subset of space, representing quantities that have both magnitude and direction at each point. This concept is essential for understanding how physical quantities vary over a region, influencing calculations related to force, flow, and motion in various applications.
Green's theorem: Green's theorem states that the line integral around a simple closed curve in the plane is equal to the double integral of the divergence of a vector field over the region enclosed by the curve. This theorem connects the concepts of circulation around a curve to the behavior of vector fields in the area it encloses, illustrating important relationships between line integrals and double integrals.
Simple Closed Curve: A simple closed curve is a continuous curve in a plane that does not intersect itself and forms a closed loop, meaning it starts and ends at the same point. This concept is important as it serves as the boundary for a region in the plane, which is essential for applying certain mathematical theorems, like calculating areas and understanding properties of vector fields.
Circulation: Circulation refers to the line integral of a vector field around a closed curve, representing the total 'twisting' or 'rotation' of the field in that region. This concept is vital for understanding the behavior of fluid flow, electromagnetic fields, and various physical phenomena, showing how vector fields exhibit rotational characteristics through their paths.
Curl: Curl is a vector operator that measures the rotation of a vector field in three-dimensional space. It provides insight into the local spinning motion of the field, indicating how much and in which direction the field 'curls' around a point.
Flux: Flux is a measure of the flow of a field through a surface, typically quantified as the integral of a vector field across that surface. It is essential for understanding how physical quantities like fluid flow or electromagnetic fields behave in three-dimensional space, linking concepts like divergence, circulation, and surface integrals.
Conservative Fields: A conservative field is a vector field where the work done along a path is independent of the path taken and depends only on the initial and final points. This property implies that the line integral of the field around any closed loop is zero, indicating that the field can be represented as the gradient of a scalar potential function. Understanding conservative fields is essential for analyzing physical systems, particularly in mechanics and electromagnetism, where they relate closely to concepts like energy conservation and potential energy.