Vector Fields and Line Integrals

Oliver Knill and Dale Winter, Harvard University

Part 1. Vector valued functions

Problem 1

A vector valued function is entered as in the following example.

f[x_, y_] := {x + y, x - y};

You can evaluate the function at a specific point like P=(1,1) with

f[1, 1]

Problem 2

We can plot the field with

<< Graphics`PlotField`
PlotVectorField[
      f[x, y], {x, -10, 10}, {y, -10, 10}]

A three dimensional example:

<< Graphics`PlotField3D`
PlotVectorField3D[{2  x, 2  y, 1}, {x, -1, 1}, {y, -1, 
        1}, {z, -1, 1}]

Problem 3

List your matchings between graphical and symbolic representations of vector field here:

A:                                                 B:                                                    C:                                                  D:                                            E:

Problem 4

Experiment here to find a vector valued function which matches the vector field  given in the text.

Part 2. The Concept of Work in Physics

Problem 1

The constant force field is given by

V = {2, 3, 1};

The work this field does to go from (1,3,4) to (2,8,9) is equal to

P1 = {1, 3, 4};
P2 = {2, 8, 9}; 
V . (P2 - P1)

Problem 2

How much work does the force field F(x,y)=(1,1) do when moving from (-3/2,3/2) to (3/2,-3/2)?

Problem 3

Describe a path which would make no work on the force field  F(x,y)=(2,-1)

Problem 4

The work done on the individual paths is

Work1 = {1, 1} . {3, 0}
Work2 = {1, 1} . {0, 1}
Work3 = {1, 1} . {-2, 0}
Work4 = {1, 1} . {0, -2}

The work on the entire path is

TotalWork = {1, 
          1} . ({3, 0} + {0, 1} + {-1, 1} + {-2, 
              0} + {0, -2})

Problem 5

Record your conjecture here.

Problem 6

Test your conjecture here.

Part 3: Experiments with Work and Vector Fields

Problem 1

Under what conditions is it possible for an object to move and no work to be done.

Problem 2

Describe the curves (if any) in the applet along which it would be possible for an object to move and no work be done:

Vector field  1:                 

Vector field  2:

Vector field  3:

Vector field  4:

Vector field  5:

Vector field  6:

Vector field  7:

Problem 3

What is the geometrical relationship between the curve(s) that you have found and the vectors in the vector fields?

Problem 4

Next,select a particular vector field from the seven you have available.You can use any of the vector fields that you like,but it important to choose one and stick with it for this question.Using the applet either as a tool for exploration (or as a way of checking your answers),describe (in words) closed curves thatgive:
    
    A large, positive amount of work: 
   
    A small,positive amount of work:
    
    A small,negative amount of work:
    
    A large negative amount of work:

Problem 5

For two of the curves that you have found,explain why the shape of the curve and the arrangement of vectors in the vector field combine to give the amount of work (large positive,small positive,small negative or large negative) that you predict.

Problem 6

.Next, choose one of the other vector fields in the applet. Working from the appearance of the vector field alone, describe (in words) the appearance of closed curves that will give:

A large, positive amount of work: 
   
    A small,positive amount of work:
    
    A small,negative amount of work:
    
    A large negative amount of work:

Problem 7

Use the applet to test your predictions.

Part 4: Approximating the work done along a curved path

Problem 1

Calculate the work done if the circular arc is approximated by one line segment.

F[{x_, y_}] := {x + y, x - y};
r[t_] := {0, 1} + t*{1, -1};
F[r[1/2]] . {1, 1}

Problem 2

Calculate the work done if the circular arc is approximated by four line segments

F[{x_, y_}] := {x + y, x - y};
P[k_] := {Cos[(Pi/2)*k/4], 
          Sin[(Pi/2)*k/4]};
r[k_, t_] := 
        P[k - 1] + t*(P[k - 1] - P[k]);
N[Sum[F[r[k, 1/2]] . (P[k - 1] - P[k]), {k, 1, 4}]]

F[{x_, y_}] := {x + y, x - y};
P[k_] := {Cos[(Pi/2)*k/40], 
          Sin[(Pi/2)*k/40]};
r[k_, t_] := 
        P[k - 1] + t*(P[k - 1] - P[k]);
N[Sum[F[r[k, 1/2]] . (P[k - 1] - P[k]), {k, 1, 40}]]

Problem 3

Refine your estimates further by approximating the circular arc with a greater number of line segments.

Part 5: Line Integrals

Problem 1

How can you alter the sum to make it a better approximation of the work done?How could you express the amount of work done symbolically in a way that is consistent with the relationship between Riemann sums and definite integrals for areas under curves?

Problem 2

Evaluate the integral  of  the function  f(t)=(r1[t]+r2[t])r'[t], where r[t]=1-t, r2[t]=(1-(1-t)^2)^(1/2) from t=0 to t=1:

r1[t_] := 1 - t; 
r2[
          t_] := (1 - (1 - t)^2)^(1/
              2);
NIntegrate[(r1[t] + r2[t])*r1'[t], {t, 0, 
        1}]
Integrate[(r1[t] - r2[t])*r2'[t], {t, 0, 1}]

What is the sum of these integrals?

Problem 3

How are the sums and integrals related?

Problem 4

Adapt the commands when the force  acts on an object moving along the path  in the direction shown in Figure 16. Refine your
approximation until you are confident that you have found the amount of work done.

F[{x_, y_}] := {x + y, x - y};
r[t_] := {Cos[Pi/2 - t], Sin[Pi/2 - t]};
NIntegrate[F[r[t]] . r'[t], {t, 0, Pi/2}]

Part 6: Summary

Problem 1

Which of the following three vector fields represents the Planimeter vector field?

A:                                                 B:                                                    C:                                                  D:

Problem 2

Explain how vectors and dot products can be used to calculate the work done when a force moves an object along a path described by a position vector.Under what circumstances can movement take place without doing any work?

Problem 3

Consider the two dimensional vector field

F[x_, y_] := {x/Sqrt[x^2 + y^2], y/Sqrt[x^2 + y^2]}

Describe (in words) closed curves that will give positive and negative work if represents the force acting,and the curve you describe represents the path that the object moves along.

Problem 4

Summarize what you have learned about the relationship between the curve,the vector field and the sign of the work done.Give plausibility arguments for your conclusions using what you have learned about approximating work using collections of straight line segments and sums.

Problem 5

Evaluate the following  procedure for calculating the work done by a force F(x,y) acting on an object that moves along a path parametrized by t -> r(t)=(x(t),y(t)) with t in [a,b].

F[x_, y_] = {Sin[x], Cos[x*y]};
a = 0;
b = 1;
x[t_] := Cos[t];
y[t_] := Sin[t];
r[t_] := {x[t], y[t]};
v[t_] := {x'[t], y'[t]};
 TotalWork = NIntegrate[F[x[t], y[t]] . v[t], {t, a, b}]

7/13/2001, Oliver Knill and Dale Winter, Harvard University, Copyright CCP and the Authors