<Text-field style="_cstyle260" layout="Heading 1"><Font bold="true">Work in Conservative and Non-Conservative Force Fields</Font></Text-field>Note: You may notice differences between this Maple worksheet and the equivalent Mathematica notebook. These differences were introduced to preserve the content of these modules and were necessary because of major functional differences between Maple and Mathematica.
<Text-field style="_cstyle259" layout="Heading 1">Introduction</Text-field>OBJECTIVE: Visualize and evaluate work integrals along different paths, and observe the effect of following different paths through conservative and non-conservative force fields.
You will explore integration over vector fields and experiment with both conservative and non-conservative force functions and different paths. These explorations should help you understand line integrals, as well as better appreciate situations when the work done is independent of the path taken.
<Text-field style="_cstyle261" layout="Heading 1"><Font size="18">Technology Guidelines</Font></Text-field>NOTE: If you have just finished a worksheet, restart Maple before executing a new worksheet.
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<Text-field style="Heading 1" layout="Heading 1">Part I: Examples in Two Dimensions</Text-field>
<Text-field style="Heading 1" layout="Heading 1">Conservative Force</Text-field>
<Text-field style="Heading 1" layout="Heading 1">Verifying that the Force is Conservative</Text-field>Consider the following force defined by 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, and verify that it is conservative.force:=[-x*cos(2*y), x^2*sin(2*y)];
my:=diff(force[1],y);
nx:=diff(force[2],x);
if (my=nx) then print(`The force is conservative.`) else print(`The force is not conservative.`) fi;
<Text-field style="Heading 1" layout="Heading 1">Visualizing the Force Field and Different Paths</Text-field>
To visualize the force field, you need to first load the plots package.with(plots):
The force field is plotted together with three paths between (0, 0) and (1, 1): y = x, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbXNxcnRHRiQ2Iy1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiM0YnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQUYrRkE=.pv:=fieldplot(force,x=0..1,y=0..1,arrows=SLIM):
pc:=plot({x,sqrt(x),x^3}, x=0..1, color=[red,cyan,blue]):
print(display({pv,pc}));
<Text-field style="Heading 1" layout="Heading 1">Writing Parametrizations and Computing Work Integrals</Text-field>Find the work done in traveling along the straight line, y = x. Choose an appropriate parameterization. We write the position and velocity vector to assist in computing the work integral. Note that by setting x and y equal to particular functions of t, the force function will reflect that parametrization when it appears in the line integral.x:=t:
y:=t:
r1:=[x,y]:
v1:=diff(r1,t):print(`velocity = `, v1);
print(`force function along curve = `,force);
w1:=int(linalg[dotprod](force,v1), t=0..1):
print(`work done along path = `, evalf(w1));Now find the work done in traveling along the lower curve 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. Choose an appropriate parametrization.x:=t:
y:=t^3:
r2:=[x,y]:
v2:=diff(r2,t):
print(`velocity = `, v2);
print(`force function along curve = `,force);
w2:=int(linalg[dotprod](force,v2), t=0..1):
print(`work done along path = `, evalf(w2));Now find the work done in traveling along the lower curve LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkmbXNxcnRHRiQ2Iy1GLDYlUSJ4RidGNEY3Rj5GK0Y+ . As before, choose an appropriate parametrization.x:=t:
y:=sqrt(t):
r3:=[x,y]:
v3:=diff(r3,t):
print(`velocity = `, v3);
print(`force function along curve = `,force);
w3:=int(linalg[dotprod](force,v3), t=0..1):
print(`work done along path = `, evalf(w3));Was the work done along each path the same?
<Text-field style="Heading 1" layout="Heading 1">Non-Conservative Force</Text-field>
Consider the a spinning force, and verify that it is not conservative.unassign('x','y','z','force');
force:=[-y/sqrt(x^2+y^2),x/sqrt(x^2+y^2)]:
my:=simplify(diff(force[1],y));
nx:=simplify(diff(force[2],x));
simplify(my=nx);You can see that for arbitrary values of x and y, this last equation will not be true. Therefore, by our definition, the force is not conservative.
<Text-field style="Heading 1" layout="Heading 1">Defining and Visualizing the Force Field</Text-field>pv:=fieldplot(force, x=0.01..1, y=0..1,arrows=SLIM):
pc:=plot({x,sqrt(x), x^3},x=0..1, color=[red,cyan,blue]):
print(display({pc,pv}));
<Text-field style="Heading 1" layout="Heading 1">Writing Parametrizations and Computing Work Integrals</Text-field>As you have learned from hand computation, line integrals can be difficult to evaluate. Even Maple has problems with some line integrals. When you experiment with your own force or path functions, sometimes you may have to use the integration type=numerical option, and sometimes you may have to use symbolic integration (default). You will also get error messages at times, warning you about problems associated with the convergence of the integration technique. In these cases, you will frequently, but not always, be given an answer that is reasonably accurate. In this example, we have a problem at t = 0, because the force function has a 0 denominator for the parametrizations given.
Find the work done in traveling along the straight line y = x. Choose an appropriate parametrization. We write the position and velocity vector to assist in computing the work integral. Note that by setting x and y equal to particular functions of t, the force function will reflect that parametrization when it appears in the line integral.x:=t:
y:=t:
r1:=[x,y]:
v1:=diff(r1,t):
print(`velocity = `, v1);
print(`force function along curve = `,force);
w1:=int(linalg[dotprod](force,v1), t=0..1):
print(`work done along path = `, evalf(w1));Could you have predicted this answer by looking at the graph?
Now find the work done in traveling along the lower curve 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. Choose an appropriate parametrization.x:=t:
y:=t^3:
r2:=[x,y]:
v2:=diff(r2,t):
print(`velocity = `, v2);
print(`force function along curve = `,force);
w2:=int(linalg[dotprod](force,v2), t=0..1):
print(`work done along path = `, evalf(w2));You can ignore the imaginary portion of the answer; it is near 0 and due to round off error. The result is very different from our previous value. Now find the work done in traveling along the upper curve LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkmbXNxcnRHRiQ2Iy1GLDYlUSJ4RidGNEY3Rj5GK0Y+. As before, choose an appropriate parametrization. x:=t:
y:=sqrt(t):
r3:=[x,y]:
v3:=diff(r3,t):
print(`velocity = `, v3);
print(`force function along curve = `,force);
w3:=int(linalg[dotprod](force,v3), t=0..1):
print(`work done along path = `, evalf(w3));What does it mean when our work done went from 0 on the first path to a positive number on the second path and now to a negative value? Could you have predicted that from the graph showing the force field? This example demonstrates how for non-conservative forces, the work done in getting from one point to another is not independent of the path taken.
<Text-field style="Heading 1" layout="Heading 1">You Try It: Part I</Text-field>Try a different path in going from (0, 0) to (1, 1). Suppose you go from (0, 0) to (1, 0) and then to (1, 1), all along parallel and perpendicular lines. The following commands specify appropriate parameterizations. You can execute them for any two-dimensional force. Begin by entering any force you wish by replacing the terms in newforce.unassign('x','y','t');
newforce:=[cos(5*x),-3*x*y]:
First, go from (0,0) to (1,0).
x:=t:
y:=0:
r1:=[x,y]:
dr1:=diff(r1,t):
w4a:=int(linalg[dotprod](newforce,dr1), t=0..1):
print(`work done along path =`, evalf(w4a));
Next, go from (1,0) to (1,1).
x:=1:
y:=t:
r2:=[x,y]:
dr2:=diff(r2,t):
w4b:=int(linalg[dotprod](newforce,dr2), t=0..1):
print(`work done along path =`, evalf(w4b));
Add your results, and contrast them to what you would get with the paths used earlier. Do this for both conservative and non-conservative forces.w4:=evalf(w4a+w4b);Compute the work done along the line y = x.x:=t:
y:=t:
r1:=[x,y]:
v1:=diff(r1,t):
print(`velocity = `, v1);
print(`force function along curve = `,newforce);
w1:=int(linalg[dotprod](newforce,v1), t=0..1):
print(`work done along path = `, evalf(w1));Compute the work done along the curve 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.x:=t:
y:=t^3:
r2:=[x,y]:
v2:=diff(r2,t):
print(`velocity = `, v2);
print(`force function along curve = `,newforce);
w2:=int(linalg[dotprod](newforce,v2), t=0..1):
print(`work done along path = `, evalf(w2));Compute the work done along the curve LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkmbXNxcnRHRiQ2Iy1GLDYlUSJ4RidGNEY3Rj5GK0Y+ .x:=t:
y:=sqrt(t):
r3:=[x,y]:
v3:=diff(r3,t):
print(`velocity = `, v3);
print(`force function along curve = `,newforce);
w3:=int(linalg[dotprod](newforce,v3), t=0..1):
print(`work done along path = `, evalf(w3));Is the work done in going from (0, 0) to (1, 1) the same for all the paths?evalf([w1,w2,w3,w4]);if (evalf(w1)=evalf(w2) and evalf(w3) =evalf(w4)) then print(`true`) else print(`false`) fi;
<Text-field style="Heading 1" layout="Heading 1">Part II: Example in Three Dimensions</Text-field>
<Text-field style="Heading 1" layout="Heading 1">Conservative Force</Text-field>Given the force with components: {LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW9HRiQ2LVEvJkV4cG9uZW50aWFsRTtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRictSSNtaUdGJDYlUSN5ekYnLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMg==, 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, 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 }, find the work done in going from (1, 0, 1) to (1, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGNy1GIzYkLUkjbW5HRiQ2JFEiMkYnRjdGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRS8lKWJldmVsbGVkR0Y2Rjc= , 0) by traveling along three different paths. The parameterizations for each of these paths are detailed below and the work is computed for each. The results from the three paths are then compared and visualized.unassign('x','y','z'):
force:=[exp(y*z), x*z*exp(y*z)+z*cos(y), x*y*exp(y*z)+sin(y)];
<Text-field style="Heading 1" layout="Heading 1">Verifying that the Force is Conservative</Text-field>
By computing and comparing the appropriate first partial derivatives, verify that the force defined is conservative. my:=diff(force[1],y);
nx:=diff(force[2],x);
mz:=diff(force[1],z);
px:=diff(force[3],x);
nz:=diff(force[2],z);
py:=diff(force[3],y);
if (my=nx) then print(`true`) else print(`false`) fi;
if (mz=px) then print(`true`) else print(`false`) fi;
if (nz=py) then print(`true`) else print(`false`) fi;
<Text-field style="Heading 1" layout="Heading 1">Writing the Parameterization and Computing the Work Integral- Path A</Text-field>Find the work done in traveling along the straight line from (1, 0, 1) to (1, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGNy1GIzYkLUkjbW5HRiQ2JFEiMkYnRjdGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRS8lKWJldmVsbGVkR0Y2Rjc=, 0). Choose an appropriate parameterization. Write the position and velocity vector to assist in computing the work integral. Set x, y, and z equal to particular functions of t, the force function will reflect that parametrization when it appears in the line integral.x:=1:
y:=Pi*t/2:
z:=1-t:
r1:=[x,y,z]:
v1:=diff(r1,t):
print(`velocity`, v1);
print(`force function along curve =`, force);
w1:=int(linalg[dotprod](force, v1), t=0..1):
print(`work done along path = `,evalf(w1));
<Text-field style="Heading 1" layout="Heading 1">Writing Parameterizations and Computing Work Integrals - Path B</Text-field>Find the work done in the two straight line paths. Choose appropriate parametrizations. First, go from (1, 0, 1) to the origin.
x:=1-t:
y:=0:
z:=1-t:
r21:=[x,y,z]:
v2:=diff(r21,t):
print(`velocity`, v2);
print(`force function along curve =`, force);
w21:=int(linalg[dotprod](force, v2), t=0..1):
print(`work done along path = `,evalf(w21));
Next, go from the origin to (1, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGNy1GIzYkLUkjbW5HRiQ2JFEiMkYnRjdGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRS8lKWJldmVsbGVkR0Y2Rjc=, 0).
x:=t:
y:=Pi*t/2:
z:=0:
r22:=[x,y,z]:
v2:=diff(r22,t):
print(`velocity`, v2);
print(`force function along curve =`, force);
w22:=int(linalg[dotprod](force, v2), t=0..1):
print(`work done along path = `,evalf(w22));So the total work done is as follows.w2:=w21+w22;
<Text-field style="Heading 1" layout="Heading 1">Writing Parameterizations and Computing Work Integrals - Path C</Text-field>
Find the work done for the two straight line paths plus the parabolic path. Choose appropriate parametrizations. First, we will go from (1, 0, 1) to (1, 0, 0).x:=1:
y:=0:
z:=1-t:
r31:=[x,y,z]:
v3:=diff(r32,t):
print(`velocity`, v3);
print(`force function along curve =`, force);
w31:=int(linalg[dotprod](force, v3), t=0..1):
print(`work done along path = `,evalf(w31));Now follow the x axis in going from (1,0,0) to the origin.x:=1-t:
y:=0:
z:=0:
r32:=[x,y,z]:
v3:=diff(r32,t):
print(`velocity`, v3);
print(`force function along curve =`, force);
w32:=int(linalg[dotprod](force, v3), t=0..1):
print(`work done along path = `,evalf(w32));Now follow a parabola from the origin to (1, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGNy1GIzYkLUkjbW5HRiQ2JFEiMkYnRjdGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRS8lKWJldmVsbGVkR0Y2Rjc=, 0).x:=t:
y:=Pi*t^2/2:
z:=0:
r33:=[x,y,z]:
v3:=diff(r33,t):
print(`velocity`, v3);
print(`force function along curve =`, force);
w33:=int(linalg[dotprod](force, v3), t=0..1):
print(`work done along path = `,evalf(w33));So the total work done alon Path C is as follows.w3:=w31+w32+w33;Was the work done along each path the same?if (w1=w2 and w2=w3) then print(`true`) else print(`false`) fi;
<Text-field style="Heading 1" layout="Heading 1">Visualizing the Force Field and Different Paths</Text-field>Once the graph is displayed, view it from different angles by clicking and dragging the plot.pp1:=spacecurve(r1, t=0..1):
pp21:=spacecurve(r21, t=0..1):
pp22:=spacecurve(r22, t=0..1):
pp31:=spacecurve(r31, t=0..1):
pp32:=spacecurve(r32, t=0..1):
pp33:=spacecurve(r33, t=0..1):
unassign('x','y','z');
forceplot:=fieldplot3d(force,x=0..1, y=0..1,z=0..1, axes=BOXED,arrows=THICK, grid=[5,5,5]):
print(display({pp1,pp21,pp22,pp31,pp32,pp33, forceplot}));
<Text-field style="Heading 1" layout="Heading 1">You Try It: Part II</Text-field>
Redefine your force in the previous example, and re-execute all the cells. Only the terms in the force function need to be changed. This time try a non-conservative force. You could begin by checking out the function given; we have followed the same paths as in Part II. Be careful to use correct terminolgy.
unassign('x','y','z');
force:=[x^2*cos(y), exp(y*z)*(1-x), x*y*sin(z)];
<Text-field style="Heading 1" layout="Heading 1">Is the Force Conservative or Not?</Text-field>my:=diff(force[1],y):
nx:=diff(force[2],x):
mz:=diff(force[1],z):
px:=diff(force[3],x):
nz:=diff(force[2],z):
py:=diff(force[3],y):
my=nx;
mz=px;
nz=py;Are the corresponding partial derivatives equal to one another?
<Text-field style="Heading 1" layout="Heading 1">Writing the Parameterization and Computing the Work Integral for Path A</Text-field>Find the work done in traveling along the straight line from (1, 0, 1) to (1, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGNy1GIzYkLUkjbW5HRiQ2JFEiMkYnRjdGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRS8lKWJldmVsbGVkR0Y2Rjc=, 0). Choose an appropriate parameterization. Write the position and velocity vector to assist in computing the work integral. When you set x, y, and z equal to particular functions of t, the force function will reflect that parametrization when it appears in the line integral.x:=1:
y:=Pi*t/2:
z:=1-t:
r1:=[x,y,z]:
v1:=diff(r1,t):
print(`velocity`, v1);
print(`force function along curve =`, force);
w1:=int(linalg[dotprod](force, v1), t=0..1):
print(`work done along path = `,evalf(w1));
<Text-field style="Heading 1" layout="Heading 1">Writing Parameterizations and Computing Work Integrals for Path B</Text-field>
Now find the work done in the two straight line paths specified for path B. Choose appropriate parametrizations. First, we go from (1, 0, 1) to the origin.x:=1-t:
y:=0:
z:=1-t:
r21:=[x,y,z]:
v2:=diff(r21,t):
print(`velocity`, v2);
print(`force function along curve =`, force);
w21:=int(linalg[dotprod](force, v2), t=0..1):
print(`work done along path = `,evalf(w21));Next, go from the origin to (1, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGNy1GIzYkLUkjbW5HRiQ2JFEiMkYnRjdGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRS8lKWJldmVsbGVkR0Y2Rjc= , 0).x:=t:
y:=Pi*t/2:
z:=0:
r22:=[x,y,z]:
v2:=diff(r22,t):
print(`velocity`, v2);
print(`force function along curve =`, force);
w22:=int(linalg[dotprod](force, v2), t=0..1):
print(`work done along path = `,evalf(w22));So the total work done is as follows.w2:=evalf(w21+w22);
<Text-field style="Heading 1" layout="Heading 1">Writing Parameterizations and Computing Work Integrals for Path C</Text-field>Find the work done for the two straight line paths plus the parabolic path. Choose appropriate parametrizations. First, go from (1, 0, 1) to (1, 0, 0).
x:=1:
y:=0:
z:=1-t:
r31:=[x,y,z]:
v3:=diff(r31,t):
print(`velocity`, v3);
print(`force function along curve =`, force);
w31:=int(linalg[dotprod](force, v3), t=0..1):
print(`work done along path = `,evalf(w31));Now follow the x axis in going from (1,0,0) to the origin.x:=1-t:
y:=0:
z:=0:
r32:=[x,y,z]:
v3:=diff(r32,t):
print(`velocity`, v3);
print(`force function along curve =`, force);
w32:=int(linalg[dotprod](force, v3), t=0..1):
print(`work done along path = `,evalf(w32));Now follow a parabola from the origin to (1, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGNy1GIzYkLUkjbW5HRiQ2JFEiMkYnRjdGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRS8lKWJldmVsbGVkR0Y2Rjc=, 0,).x:=t:
y:=Pi*t^2/2:
z:=0:
r33:=[x,y,z]:
v3:=diff(r33,t):
print(`velocity`, v3);
print(`force function along curve =`, force);
w33:=int(linalg[dotprod](force, v3), t=0..1):
print(`work done along path = `,evalf(w33));So the total work done alon path C is as follows:w3:=evalf(w31+w32+w33);Was the work done along each path the same?if (w1=w2 and w2=w3) then print(`true`) else print(`false`) fi;
<Text-field style="Heading 1" layout="Heading 1">Visualizing the Force Field and Different Paths</Text-field>Once the graph is displayed, view it from different angles by clicking and dragging the plot.pp1:=spacecurve(r1, t=0..1):
pp21:=spacecurve(r21, t=0..1):
pp22:=spacecurve(r22, t=0..1):
pp31:=spacecurve(r31, t=0..1):
pp32:=spacecurve(r32, t=0..1):
pp33:=spacecurve(r33, t=0..1):
unassign('x','y','z');
forceplot:=fieldplot3d(force,x=0..1, y=0..1,z=0..1, axes=BOXED,arrows=THICK, grid=[5,5,5]):
print(display({pp1,pp21,pp22,pp31,pp32,pp33, forceplot}));