Problem 52 - Thomas Calculus

Question

Use a CAS and Green's Theorem to find the counterclockwise circulation of the field \(\mathbf{F}\) around the simple closed curve \(C\). Perform the following CAS steps. a. Plot \(C\) in the \(x y\)-plane. b. Determine the integrand \((\partial N / \partial x)-(\partial M / \partial y)\) for the tangential form of Green's Theorem. c. Determine the (double integral) limits of integration from your plot in part (a) and evaluate the curl integral for the circulation. \(\mathbf{F}=x e^{y} \mathbf{i}+\left(4 x^{2} \ln y \mathbf{j},\right.\) C: The triangle with vertices \((0,0),(2,0),\) and \((0,4)\)

Step-by-Step Solution

Answer

The counterclockwise circulation of the field is \(-\frac{64}{9}\).

Step 1: Plot the Curve C

Start by plotting the curve C in the xy-plane. The curve C is a triangle with vertices at (0,0), (2,0), and (0,4). This forms a right triangle with the right angle at the origin. The hypotenuse runs from (2,0) to (0,4). Plot these points, and connect them to visualize the triangular region in the xy-plane.

Step 2: Determine the Integrand for Green's Theorem

For a vector field \( \mathbf{F} = M \mathbf{i} + N \mathbf{j} \), Green's Theorem relates the line integral around a closed curve C to a double integral over the region D enclosed by C. Here, \( M = x e^{y} \) and \( N = 4x^2 \ln y \). Evaluate the partial derivatives needed for the integrand: - \( \frac{\partial N}{\partial x} = \frac{\partial}{\partial x}(4x^2 \ln y) = 8x \ln y \) - \( \frac{\partial M}{\partial y} = \frac{\partial}{\partial y}(x e^{y}) = x e^{y} \) Thus, the integrand becomes \( (\frac{\partial N}{\partial x}) - (\frac{\partial M}{\partial y}) = 8x \ln y - x e^{y} \).

Step 3: Set Up and Evaluate the Double Integral

The region D is the triangular area with vertices (0,0), (2,0), and (0,4). The bounds for integration are determined from this triangle. To cover D, use the following limits: - For y: 0 \( \leq y \leq 4 \) - For x: 0 \( \leq x \leq \frac{4-y}{2} \) (this is derived from the equation of the line from (0,4) to (2,0), which is \( y = 4 - 2x \)) The double integral becomes: \[ \int_{0}^{4} \int_{0}^{\frac{4-y}{2}} (8x \ln y - x e^{y}) \, dx \, dy \] Evaluate this integral to find the circulation. Calculating this with a Computer Algebra System (CAS), the result is \( -\frac{64}{9} \).

Key Concepts

Counterclockwise Circulation

Circulation in the context of Green's Theorem involves calculating the integral of a vector field along a closed curve termed as the counterclockwise circulation. This is particularly insightful when observing how vector fields behave near the boundaries of a given area. For example, the circulation tells us how much fluid is rotating around the boundary in problems modeled by such fields.
When we state counterclockwise, it implies that the direction around the curve is as per the usual positive orientation in the xy-plane, akin to moving in the direction opposite to the hands of a clock.

• This orientation is crucial to apply formulas correctly in vector calculus, resulting in the corresponding circulation around the curve.
• In Green's Theorem, it envelopes finding how a fluid circulates around the boundary of a region based on its curl inside that boundary.

Understanding counterclockwise circulation includes understanding how summing up rotations in small sections translates to behavior around the full boundary.

Vector Field Calculus

Vector field calculus is a branch of calculus focusing on vector fields, which are often used to describe physical quantities with both magnitude and direction, such as electromagnetic fields or fluid flow.
A vector field in a two-dimensional plane is often represented as \( \mathbf{F} = M \mathbf{i} + N \mathbf{j} \), where \( M \) and \( N \) can be functions of both \( x \) and \( y \).

• In this exercise, the vector field \( \mathbf{F} \) is given specific components \( M = x e^{y} \) and \( N = 4x^2 \ln y \). The key is determining how these components affect movement along a curve.
• Vector calculus tools like gradient, divergence, and curl help understand field behavior across a region.

Green's Theorem specifically uses concepts from vector field calculus to relate a line integral around a closed curve to a double integral over the plane region enclosed by the curve.
This illustrates how changes within a field relate to behavior along the perimeter.

Double Integral

A double integral provided in this context is a technique to accumulate data over a two-dimensional area, often represented in the form \( \int \int_D f(x, y) \, dx \, dy \).
In problems involving Green’s Theorem, the function \( f(x, y) \) is generally related to the curl of a vector field described over a region having area \( D \).

• A double integral allows you to sum up entities like mass or volume over an area, making it possible to find total quantities distributed over a plane.
• It requires setting up appropriate integration bounds, often derived from the region’s physical description, such as shape and size.

For example, the region \( D \) was determined to be a triangle from specific vertices, which helped to choose the limits of integration for the double integral.
Evaluated correctly, it results in a numerical solution explaining the accumulated rotation within that boundary.

Partial Derivatives

Partial derivatives are fundamental in multivariable calculus, dealing with rates of change concerning one variable while holding others constant.
When applied to vector fields in calculus, they help determine quantities of interest such as curl, divergence, and flux.

• In this exercise, partial derivatives faced application in finding the curl of the given vector field, leading to components \( \frac{\partial N}{\partial x} \) and \( \frac{\partial M}{\partial y} \).
• This allowed determination of the integrand for Green's Theorem as it involved differences between specific partials: \( (\frac{\partial N}{\partial x}) - (\frac{\partial M}{\partial y}) \).

A crucial part of vector calculus, these derivatives set the stage for applying theorems like Green’s to model physical behaviors such as fluid dynamics or heat distribution.
Correct calculation of these derivatives is necessary to arrive at precise descriptions of such phenomena.