Green's Theorem and Vector Fields

1. Instructions

This lab is due Mon. Nov 30 in lecture. Late labs are not accepted. You are encouraged to work in groups of no more than 3 people, though you can work alone.

On your title page, clearly mark

• Names
• Professor name and lecture section number
• Recitation TA name and recitation section number

 

Format is worth 20% of your grade. Please refer to the following writing guidelines for the expository sections of this report:

Writing Guidelines for Calculus Labs

You are required to know and follow the Writing Guidelines for all labs. For some sample "good" and "bad" labs see:

Sample Labs

 

2. Lab Goals

The purpose of this lab is to explore some of the material from chapters 13-14 in the context of two main problems. In the first problem we will examine gradient fields, and the flux, flow, divergence, and curl of vector fields. We will also look at Green's theorem and how it connects the ideas of flux with divergence, and flow with curl. In the second part of the lab, we will examine the first and second moments of a charged surface. Both of these problems should be

done in polar coordinates. Also, you may use Mathematica as you see fit, but note that all calculations can be done by hand. Please include these calculations in an appendix.

 

3. Background

In this lab we are going to investigate some of the fundamental properties of vector valued functions, also known as vector fields. In order to understand how a vector field changes, we must extend the ideas of calculus to vector-valued functions.

The mathematics of vector-valued functions has applictions in many fields. In fluid mechanics vector fields show up in the context of velocity fields and pressure gradients. The electric and magnetic fields are vector-valued. These fields are related in Maxwell's equations to the charge and current density by the divergence and curl operators. In the theory of elasticity, stress and strain tensors are related to an external field of force. The gravitational field is also a vector field related to the mass density by the divergence operator.

A fundamental idea in vector calculus is the differential vector operator . Using this vector operator and the definitions of vector-scalar multiplication, the dot product, and the cross product, we can define the differential vector operators: the gradient, divergence, and curl.

If we consider a vector-valued function , then in two dimensions the differential vector operators follow easily from the definitions:

Note that the cross-product in two-dimensions is a scalar valued funtion. In general, the divergence acts on a vector field but is scalar-valued. And the gradient acts on a scalar function and is vector-valued.

The vector fields we will be examining below are most easily expressed in polar coordinates. In this case, we can express the vector field in terms of radial and angular components . In polar coordinates these operators are:

In two dimensions we can also extend integration to vector-valued functions in the form of contour integration. In three dimensions vector-valued functions can also be integrated over surfaces, and in four dimensions integrated over volumes. In general, integrating a vector field over a contour or surface requires doting it with a vector. In two dimensions, Green's theorem provides a mathematical relationship between these differential operators and closed contour integration. This connection relates important concepts such as the flow of a vector field to its curl, and the flux of a vector field to its divergence.

In three dimensions the differential vector operator is . If we define a vector field in three dimensional space then the gradient, divergence, and curl operators become:

In three dimensions, the flux-divergence and circulation-curl forms of Green's theorem extend to what are known as the divergence theorem and Stokes' theorem. In fact, Green's theorem also extends to higher dimensions in the form of the generalized Stokes' theorem.

The purpose of this lab is to study some of the properties of vector-valued functions and differential vector operators, and the relationships of these operators as expressed in Green's theorem. Understanding vector fields and vector operators in two dimensions is essential to understanding them in three dimensions.

 

4. Problem Statement

We want to consider a surface given by the defined on the domain and . In the problems below we will construct the function and analyze its gradient field. Make sure you use the gradient, divergence, and curl operators in polar coordinates.

1.

A.) If the gradient field , what is the function that passes through the origin?

B.) Plot and its gradient field , for . In order to plot the gradient field in Mathematica you will need to use the function VectorPlot[] which plots the cartesian components of a vector field. Because is defined in terms of polar components, we have to transform the components before we can plot. Look up the function VectorPlot[] in the help browser. To do this transformation in Mathematica, we must first define functions and which transform cartesian coordinates to polar coordinates. In order to define you will need to use the ArcTan[] function. In Mathematica, the range of the ArcTan[] function is only , so will you will need to define as a piecewise function. To do this in Mathematica use the function Piecewise[]. Make sure that you define your function so that it works for all values of x and y in the domain. Then define the cartesian components of which transform as:

Finally, you can use these components to define . Plot this vector field in cartesian coordintes with and . Describe the figures.

2.

Compute and .

3.

Is conservative? Is it exact? What can you say about the conservation of the gradient field of a continuously differentiable function ?

4. We will now verify that the flux of is zero across the boundary of the domain.

A.) Consider the boundary of the domain defined by a closed contour C. This contour is defined by the upper half of the unit circle and the x-axis with . Partition this contour into three parts in polar coordinates:

Define the first contour by and , the second contour by and , and the third contour by and . Plot the contour C, and label each of the three sections. To do this it may be useful to plot each of the sections separately and in different coordinate systems. If you use the same axes in each of your plots, you can use the Mathematica function Show[] to superimpose them. Look this function up in the help browser. You can also use Show[] to superimpose the image of the vector field you found in (1B) with the contours found here.

B.) Compute the flux of across each of these contours. What can you conclude about the flux of across the contour enclosing the domain? (Remember to integrate in the counterclockwise direction.)

5.

What is the flux-divergence form of Green's theorem? Explain the meaning of the terms in this theorem. What does this double integral express? Evaluate the double integral in Green's theorem over the domain, and show that it is the same as your answer to question (4).

6.

A.) What is the circulation-curl form of Green's theorem? Calculate the flow of around the contour described in question (4).

B.) Show that the flow of around C is the same as the integral of the curl of over the domain.

C.) What does the double integral in this form of Green's theorem express? Show that the in cartesian coordinates for an arbitrary twice differentiable function . How does your answer relate to the conservation of ?

7. Now, let's show that the flux and flow of are zero around any closed contour C in the domain.

A.) First find the flux and flow of around an arbitrary wedge in the domain of ranging from to , and to , where and . To do this, you will need to partition the closed contour into four pieces. Plot an example of such a contour within the given domain.

B.) Next, construct an argument that this implies the flux and flow of are zero around any closed curve in the domain. What can you conclude about the conservation of ?

8. Consider the vector-valued function defined on the domain and .

A.) Plot the vector field . What is the flow of around the perimeter of the domain? What does this tell you about ? What can you conclude about the conservation of ? Can be written in terms of a potential? If so, what is it?

B.) What is the divergence of from the domain? Verify that it is the same as the flux through the boundary of the domain. Plot , , and .

(Questions 9-11) In the remaining questions we will only consider the function , with n=2 . What are and in this case?

9.

What is the area of the domain, and what is the surface area of over the domain? Write the surface area in terms of the field . Determine and explain its significance. (Hint: define a function )

10.

A.) Now, let the surface have a charge per unit area given by the charge density function , where is a constant. Write the total charge in terms of of and .

B.) What is the total charge of the surface? Remember, the charge is distributed over the surface , not the domain. How does this affect your computation of ? What is the average charge per unit surface area?

 

5. Lab Report

This report should have the look and feel of a technical paper. Write your report in an organized and logical fashion, and include the following sections: introduction, results, conclusion, and appendix.

Remember the Following:

• Always label plots and refer to them in the text.
• The main body of your paper should NOT include lengthy calculations. These should be included in an appendix, and referred to in the main body.
• Labs must be typed. Part of your duty is to learn to use an equation editor, so the equations in the main body must be typed or copied in as figures. Lengthy calculations in the appendix may be hand written, but must be neat and clean.
• Your report does not have to be long. You need quality, not quantity of work. Of course you cannot omit any important piece of information, but you need not add any extras.
• Section headers such as Introduction, Background, Problem Statement, Calculations, Results, Conclusion, Appendix, etc... are not mandatory, but are highly recommended. They help you write your report, help the reader navigate the paper, and give it a cleaner look.
• Your report should begin with an introduction. This should briefly describe what you plan to say in the body of your report. You should also provide a brief list of the mathematical concepts that you will use to make your arguments and perform your calculations.
• The details of your work should be described in the body of your report.
• Finally, you should summarize what you have accomplished in a conclusion. No new information or new results should appear in your conclusion. You should only review the highlights of what you wrote about in body. Briefly, what were you investigating? What were the overall results? Are there any suggestions to better analyze/describe the same problem? 

Be sure to:

• Explain the purpose of the paper.
• State which mathematical tools were used while working on the lab.
• Define the equations and all variables used.
• Answer all the questions and include all of the requested computer generated contour and vector field, and surface plots.
• Interpret your results.