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Tips and Techniques: Line Integrals for Work, Circulation, and Plane FluxRobert J. Lopez Emeritus Professor of Mathematics and Maple Fellow\251 Maplesoft, a division of Waterloo Maple Inc., 2005

In our last column, we discussed how to address the iterated integral in Maple 9.5. This month, we will continue discussing integrals that arise in the subject area of vector calculus. In particular, we will discuss line integrals of the tangential and normal components of a vector field.The line integral of the tangential component of a force field produces the work done by the force on a particle of unit mass, as the mass moves along a specified curve. For the tangential component of the velocity field of a planar fluid flow, the line integral around a closed curve - typically a circle - is the circulation of the flow. The line integral of the normal component of a vector field is the flux of the field through the curve, and is a measure of the net "flow" of the field in the direction of the normal.If F = NiMtSSJmRzYiNiRJInhHRiVJInlHRiU= i + NiMtSSJnRzYiNiRJInhHRiVJInlHRiU= j represents the vector field in Cartesian coordinates, and NiNJIkNHNiI= is a planar curve parametrized by the equationsNiMvSSJ4RzYiLUYkNiNJInRHRiU= NiMvSSJ5RzYiLUYkNiNJInRHRiU= NiMxSSJhRzYiSSJ0R0Yl NiMxSSFHNiJJImJHRiU= then work (or circulation), the line integral of the tangential component of F along NiNJIkNHNiI=, is given by NiMtSSRpbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkSSJmR0YoL0kieEdGKDtJIkNHRihJIUdGKA== + NiMqJkkiZ0c2IiIiIkkjZHlHRiVGJg== orNiMtSSRpbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkLUkhR0YoNiMsJiomLUkiZkdGKDYkLUkieEdGKDYjSSJ0R0YoLUkieUdGKEY0IiIiLUkjeCdHRihGNEY4RjgqJi1JImdHRihGMUY4LUkjeSdHRihGNEY4RjgvRjU7SSJhR0YoSSJiR0Yo Flux, the line integral of the normal component of F along NiNJIkNHNiI=, is given byNiMtSSRpbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkSSJmR0YoL0kieUdGKDtJIkNHRihJIUdGKA== NiMsJkkhRzYiIiIiKiZJImdHRiVGJkkjZHhHRiVGJiEiIg== orNiMtSSRpbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkLUkhR0YoNiMsJiomLUkiZkdGKDYkLUkieEdGKDYjSSJ0R0YoLUkieUdGKEY0IiIiLUkjeSdHRihGNEY4RjgqJi1JImdHRihGMUY4LUkjeCdHRihGNEY4ISIiL0Y1O0kiYUdGKEkiYkdGKA== (The mnemonic I always provided my students for the flux integral in the plane is that the form NiMsJiomSSJmRzYiIiIiSSNkeUdGJkYnRicqJkkiZ0dGJkYnSSNkeEdGJkYnISIi starts and ends with the same letters as the word "flux" and has a minus sign in the middle, thus determining where the letters NiNJInlHNiI= and NiNJImdHNiI= must go.)All of these physically meaningful quantities can be computed with the LineInt command in Maple's VectorCalculus package. Thus, the LineInt command computes line integral of the tangential component of the field F = NiNJImZHNiI= i + NiNJImdHNiI= j To compute the normal component (i.e., flux) of this field through a curve NiNJIkNHNiI=, apply the LineInt command to the fieldF* = NiMsJEkiZ0c2IiEiIg== i + NiNJImZHNiI= j (Incidentally, the mnemonic I always provided my students for the flux integral in the plane is that the form NiMsJiomSSJmRzYiIiIiSSNkeUdGJkYnRicqJkkiZ0dGJkYnSSNkeEdGJkYnISIi starts and ends with the same letters as the word "flux" and has a minus sign in the middle, thus determining where the letters NiNJInlHNiI= and NiNJImdHNiI= must go.)In addition to the LineInt command, Maple provides a PathInt command that we will discuss at the end of this column.

If F is a vector field, then along a curve specified parametrically in Cartesian coordinates, the line integral of the tangential component of F is given by the syntaxLineInt(F, Path(<NiM2JC1JInhHNiI2I0kidEdGJi1JInlHRiZGJw==>, t = a .. b)) where LineInt(F, Path(<NiM2JC1JInhHNiI2I0kidEdGJi1JInlHRiZGJw==, ... >, t = a .. b)) is the obvious generalization to higher dimensions.

restart;interface(warnlevel=0): with(VectorCalculus): with(Student[Precalculus]): with(plots): BasisFormat(false):

The work done by the vector field F = NiMtSSJmRzYiNiRJInhHRiVJInlHRiU= i + NiMtSSJnRzYiNiRJInhHRiVJInlHRiU= j wheref := x*y; g := x+y;so thatF := VectorField(<f,g>, cartesian[x,y]);as it acts on a unit mass moving along the curveX := 1+t-t^2; Y := 3*t^2-5*t-7; t0 := 0; tf := 1;is given by the line integralWork := LineInt(F, Path(<X,Y>, t=t0..tf), inert);whose value can be obtained from eithervalue(Work);or LineInt(F, Path(<X,Y>, t=t0..tf));The positive value of this line integral indicates that the field does work on the particle, that is, the field causes the particle to move along the curve NiNJIkNHNiI=.

Special cases in which the curve can be specified directly include1) LineInt(F, Line(<NiM2JEkiYUc2IkkiYkdGJQ==>, <NiM2JEkiY0c2IkkiZEdGJQ==>)) 2) LineInt(F, LineSegments(<NiM2JCZJInhHNiI2IyIiIiZJInlHRiZGJw==>, ..., <NiM2JCZJInhHNiI2I0kibkdGJiZJInlHRiZGJw==>))3) LineInt(F, Circle(<NiM2JEkiaEc2Ikkia0dGJQ==>, r)) 4) LineInt(F, Circle3D(<NiM2JUkiYUc2IkkiYkdGJUkiY0dGJQ==>, NiNJInJHNiI=, <NiM2JUkidUc2IkkidkdGJUkid0dGJQ==>))5) LineInt(F, Ellipse( ( NiMsKComLCZJInhHNiIiIiJJImhHRichIiIiIiMqJEkiYUdGJ0YrRipGKComLCZJInlHRidGKEkia0dGJ0YqRisqJEkiYkdGJ0YrRipGKEYoRio= ))6) LineInt(F, Arc(Circle(<NiM2JEkiaEc2Ikkia0dGJQ==>, r), NiM2JEkmYWxwaGFHNiJJJWJldGFHRiU=))7) LineInt(F, Arc(Ellipse( ( NiMsKComLCZJInhHNiIiIiJJImhHRichIiIiIiMqJEkiYUdGJ0YrRipGKComLCZJInlHRidGKEkia0dGJ0YqRisqJEkiYkdGJ0YrRipGKEYoRio= ), NiNJJmFscGhhRzYi, NiNJJWJldGFHNiI=))In (1), the curve NiNJIkNHNiI= consists of the line segment connecting the points NiMtSSFHNiI2JEkiYUdGJUkiYkdGJQ== and NiMtSSFHNiI2JEkiY0dGJUkiZEdGJQ==. This form generalizes to the NiNJIm5HNiI=-dimensional case in the obvious way.In (2), the curve NiNJIkNHNiI= consists of the polygonal line sequentially connecting the NiNJIm5HNiI= points NiM2JS1JIUc2IjYkJkkieEdGJjYjIiIiJkkieUdGJkYqSSQuLi5HRiYtRiU2JCZGKTYjSSJuR0YmJkYtRjI=, with an obvious generalization to the NiNJIm5HNiI=-dimensional case. For a closed polygonal path, the first point should be repeated as the last point. In (3), the curve NiNJIkNHNiI= is a circle with center NiMtSSFHNiI2JEkiaEdGJUkia0dGJQ== and radius NiNJInJHNiI=.In (4), the curve NiNJIkNHNiI=, a circle in NiMqJEkiUkc2IiIiJA==, lies in the plane containing (NiM2JUkiYUc2IkkiYkdGJUkiY0dGJQ==), the center of the circle, and having the vector u i + v j + w k as its normal. The radius of the circle is NiNJInJHNiI=.In (5), the curve NiNJIkNHNiI= is the ellipse determined by the equation NiMvLCgqJiwmSSJ4RzYiIiIiSSJoR0YoISIiIiIjKiRJImFHRihGLEYrRikqJiwmSSJ5R0YoRilJImtHRihGK0YsKiRJImJHRihGLEYrRilGKUYrIiIh. (To enter the equation of the ellipse, type it as (x-h)^2 / a^2 + (y-k)^2 / b^2 - 1.)In (6) and (7), the curve NiNJIkNHNiI= is an arc of the respective circle or ellipse. The parameters NiNJJmFscGhhRzYi and NiNJJWJldGFHNiI= are the angles subtending the required arc. The angles are measured in radians from the positive NiNJInhHNiI=-axis.The LineInt command in any of its forms computes line integral of the tangential component of the field F = NiNJImZHNiI= i + NiNJImdHNiI= j To compute the normal component (i.e., flux) of this field through a curve NiNJIkNHNiI=, apply the LineInt command to the fieldF* = NiMsJEkiZ0c2IiEiIg== i + NiNJImZHNiI= j

The work done by the fieldF;on a unit mass moving along the line segment connecting the pointsP1 := <1,2>; P2 := <3,7>;is given by the line integralWork := LineInt(F, :-Line(P1,P2),inert);whose value can be obtained either byvalue(Work);orLineInt(F, :-Line(P1,P2));(In each call to the LineInt command containing the parameter Line, we have used the syntax :-Line. This forces Maple to use the Line parameter from the VectorCalculus package, and not the Line command from the Precalculus subpackage of the Student package. This subpackage was loaded in anticipation of its use below, for obtaining the equation of the line connecting the given points.)To obtain this result by first principles, parametrize the line segment asP := P1+t*(P2-P1): <x,y> = P;and writeNiMtSSRpbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkLUkhR0YoNiMsJiomLUkiZkdGKDYkLUkieEdGKDYjSSJ0R0YoLUkieUdGKEY0IiIiLUkjeCdHRihGNEY4RjgqJi1JImdHRihGMUY4LUkjeSdHRihGNEY4RjgvRjU7SSJhR0YoSSJiR0Yo asInt(eval(convert(F,Vector).diff(P,t), [x=P[1],y=P[2]]), t=0..1);which compares favorably withWork;The equationline := Line(P1,P2)[1];is an alternate representation of the line, so the required line integral becomesNiMtSSRpbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkLUkhR0YoNiMsJi1JImZHRig2JEkieEdGKC1JInlHRig2I0YxIiIiKiYtSSJnR0YoRjBGNS1JI3knR0YoRjRGNUY1L0YxOyZGMTYjIiIhJkYxNiNGLw== that is,work := Int(eval(convert(F,Vector).<1,diff(rhs(line),x)>, line), x=P1[1]..P2[1]): work = value(work);

The work done by the fieldF;on a unit mass moving counterclockwise along the boundary of the triangle whose vertices are P1 := <1,3>; P2 := <4,-7>; P3 := <8, 5>;is given by the line integralWork := LineInt(F, LineSegments(P1,P2,P3,P1),inert);whose value can be obtained either byvalue(Work);orLineInt(F, LineSegments(P1,P2,P3,P1));The negative value of the work done by the field on the particle indicates that energy must be put into the system for the particle to move along the curve NiNJIkNHNiI=. Thus, an external agent must push the particle against the resistive force exerted by the field.An equivalent solution from first principles would require the tedious repetition of the techniques demonstrated in Example 2.

The work done by the fieldF;on a unit mass moving counterclockwise around a circle with center NiMtSSFHNiI2JCIiIyIiJA== and radius 5, is given by the line integralWork := LineInt(F, Circle(<2,3>,5),inert): Work = value(Work);To compute the flux of this field in the direction of the outward normal along this circle, form the vector fieldF_flux := VectorField(<-g,f>, cartesian[x,y]);and write the line integralflux := LineInt(F_flux, Circle(<2,3>,5), inert);whose value isvalue(flux);The positive value of the flux indicates that the net flow of the field is along the normal. But in what direction does the normal Maple used point?Since the circle is a closed curve, we expect the normal to be the outward normal. Traversing the circle in the counterclockwise direction with the interior of the circle to the left means the exterior of the circle, and hence the outward normal, is to the right.Parametrizing the circle withX := 2 + 5*cos(t); Y := 3 + 5*sin(t);means the circle is traversed counterclockwise as NiNJInRHNiI= increases. Moreover, a unit vector tangent to the circle and pointing in the direction of motion is given by T = NiMqJiIiIkYkLUklc3FydEc2IjYjLCYqJC1JIUdGJzYjSSN4J0dGJyIiI0YkKiQtRiw2I0kjeSdHRidGL0YkISIi NiMtSSdNQVRSSVhHNiI2IzckNyMtSSN4J0dGJTYjSSJ0R0YlNyMtSSN5J0dGJUYr so that a unit vector orthogonal and to the right of T isN = NiMqJiIiIkYkLUklc3FydEc2IjYjLCYqJC1JIUdGJzYjSSN4J0dGJyIiI0YkKiQtRiw2I0kjeSdHRidGL0YkISIi NiMtSSdNQVRSSVhHNiI2IzckNyMtSSN5J0dGJTYjSSJ0R0YlNyMsJC1JI3gnR0YlRishIiI=Thus, we can writeF . N NiNJI2RzRzYi = NiMtSSdNQVRSSVhHNiI2IzckNyNJImZHRiU3I0kiZ0dGJQ== . NiMqJiIiIkYkLUklc3FydEc2IjYjLCYqJC1JIUdGJzYjSSN4J0dGJyIiI0YkKiQtRiw2I0kjeSdHRidGL0YkISIi NiMtSSdNQVRSSVhHNiI2IzckNyMtSSN5J0dGJTYjSSJ0R0YlNyMsJC1JI3gnR0YlRishIiI= NiNJI2R0RzYi = NiMsJiomSSJmRzYiIiIiSSNkeUdGJkYnRicqJkkiZ0dGJkYnSSNkeEdGJkYnISIi = NiMtSSdNQVRSSVhHNiI2IzckNyMsJEkiZ0dGJSEiIjcjSSJmR0Yl . NiMqJiIiIkYkLUklc3FydEc2IjYjLCYqJC1JIUdGJzYjSSN4J0dGJyIiI0YkKiQtRiw2I0kjeSdHRidGL0YkISIi NiMtSSdNQVRSSVhHNiI2IzckNyMtSSN4J0dGJTYjSSJ0R0YlNyMtSSN5J0dGJUYr NiNJI2R0RzYi = F* . T NiNJI2RzRzYi so that the line integralFLUX := Int(eval(convert(F_flux,Vector).diff(<X,Y>,t), [x=X,y=Y]), t=0..2*Pi): FLUX = value(FLUX);is the flux integral in which the normal is clearly outward. Since it the same line integral as Maple's, we see that Maple did indeed choose the outward normal.

The work done by the fieldF;on a unit mass moving counterclockwise around the ellipseE := (x-2)^2/9 + (y-3)^2/16 - 1: E = 0;is given by the line integralWork := LineInt(F, Ellipse(E),inert): Work = value(Work);To compute the flux of this field in the direction of the outward normal along this ellipse, form the vector fieldF_flux := VectorField(<-g,f>, cartesian[x,y]);and write the line integralflux := LineInt(F_flux, Ellipse(E), inert);whose value isvalue(flux);The positive value of the flux indicates that the net flow of the field is along the normal. But in what direction does the normal Maple used point?Since the ellipse is a closed curve, we expect the normal to be the outward normal. Traversing the ellipse in the counterclockwise direction with the interior of the ellipse to the left means the exterior of the ellipse, and hence the outward normal, is to the right.Parametrizing the ellipse withX := 2 + 3*cos(t); Y := 3 + 4*sin(t);means the ellipse is traversed counterclockwise as NiNJInRHNiI= increases. Moreover, a vector tangent to the ellipse and pointing in the direction of motion is given by T = NiMtSSdNQVRSSVhHNiI2IzckNyMtSSN4J0dGJTYjSSJ0R0YlNyMtSSN5J0dGJUYr so that a vector orthogonal and to the right of T isN = NiMtSSdNQVRSSVhHNiI2IzckNyMtSSN5J0dGJTYjSSJ0R0YlNyMsJC1JI3gnR0YlRishIiI=Thus, F . N = NiMtSSdNQVRSSVhHNiI2IzckNyNJImZHRiU3I0kiZ0dGJQ== . NiMtSSdNQVRSSVhHNiI2IzckNyMtSSN5J0dGJTYjSSJ0R0YlNyMsJC1JI3gnR0YlRishIiI= NiNJI2R0RzYi = NiMsJiomSSJmRzYiIiIiSSNkeUdGJkYnRicqJkkiZ0dGJkYnSSNkeEdGJkYnISIi = NiMtSSdNQVRSSVhHNiI2IzckNyMsJEkiZ0dGJSEiIjcjSSJmR0Yl . NiMtSSdNQVRSSVhHNiI2IzckNyMtSSN4J0dGJTYjSSJ0R0YlNyMtSSN5J0dGJUYr NiNJI2R0RzYi = F* . T Thus, the line integralFLUX := Int(eval(convert(F_flux,Vector).diff(<X,Y>,t), [x=X,y=Y]), t=0..2*Pi): FLUX = value(FLUX);is the flux integral in which the normal is clearly outward. Since it the same line integral as Maple's, we see that Maple did indeed choose the outward normal.

Suppose the center of a circle is (3, 2, 5), its radius is 2, and it lies in the plane whose normal is the vector N = -3 i + 2 j + k. Find the work done by the fieldF := VectorField(<z*x^2,x*y,y^2*z>, cartesian[x,y,z]);on a unit mass moving counterclockwise around a circle in NiMqJEkiUkc2IiIiJA==. Write the center, normal vector, and position vector asC := <3,2,5>: N := <-3,2,1>: X := <x,y,z>: C,N,X;then use Maple's LineInt command to computeL := LineInt(F, Circle3D(C,2,N));It is surprising to see that the unevaluated integral isLineInt(F, Circle3D(C,2,N), inert);At first glance, an iterated double integral for a line integral appears to be a grave error. However, Stokes' theorem has been invoked, so the line integral of the tangential component of F is equivalent to the flux of the curl of F through a capping surface. We can verify this calculation by parametrizing the circle and actually computing the line integral along this space curve. To this end, we write S1 := DotProduct(X-C,X-C) = 4; S2 := DotProduct(X-C,N) = 0;which are, respectively, the equations of the sphere with center (3, 2, 5) and radius 2, and the plane containing (3, 2, 5) and having normal N. The intersection of these surfaces is the prescribed circle, the equations for which are given parametrically byq := allvalues(solve({S1,S2},{y,z})): C1 := eval([x,y,z],q[1]); C2 := eval([x,y,z],q[2]);The parameter along each branch of the circle is NiNJInhHNiI=, and we find the domain for NiNJInhHNiI= as the zeros of d := op([2,3,2,1],C1);the quantity under the radicals in the parametric form for each branch. These bounds are then found to bea,b := solve(d,x): 'a' = a; 'b' = b;The circle must be traversed counterclockwise. The following animation shows how the projection of the branches to the NiNJI3h5RzYi-plane are traced as NiNJInhHNiI= varies from NiNJImFHNiI= to NiNJImJHNiI=. The projection of branch NiMmSSJDRzYiNiMiIiI= is traced in black, while for NiMmSSJDRzYiNiMiIiM=, it is traced in red.p1 := animatecurve([x,C1[2],x=a..b], frames=20, color=black, numpoints=500): p2 := animatecurve([x,C2[2],x=a..b], frames=20, color=red, numpoints=500): display([p1,p2], scaling=constrained);The animation shows that to traverse the circle in a counterclockwise direction, the parameter NiNJInhHNiI= must advance from NiNJImFHNiI= to NiNJImJHNiI= along branch NiMmSSJDRzYiNiMiIiM=, but from NiNJImJHNiI= to NiNJImFHNiI= along branch NiMmSSJDRzYiNiMiIiI=. Hence, the line integral along F is given by the sum LineInt(F, Path(Vector(C1),x=b..a)) + LineInt(F, Path(Vector(C2),x=a..b)); which compares favorably withL;Figure 6.1 shows N, the circle, and the plane in which it lies.p3 := implicitplot3d(S2, x=1.5..4.5, y=0..4,z=2.5..7.5, style=patchnogrid): p4 := spacecurve(C1, x=a..b, color=black, thickness=2, numpoints=1000): p5 := spacecurve(C2, x=a..b, color=black, thickness=2, numpoints=1000): p6 := arrow(C, N, color=red): display([p||(3..6)], scaling=constrained, axes=box, view=2.5..7.5, orientation=[-165,75], title="Figure 6.1");

The PathInt command computes "the path integral of a function from NiMpSSJSRzYiSSJuR0Yl to NiNJIlJHNiI=" where the definition of a "path integral" is inferred from the following example.The path integral of the functionf := x^2 + y^2;along the path NiM2JC9JInlHNiIqJEkieEdGJiIiJDEiIiFGKA== NiMxSSFHNiIiIiI=, is given byPIf := PathInt(f, [x,y]=Path(<x,x^3>, x=0..1), inert): PIf = evalf(PIf);In the integrand, the function NiMtSSJmRzYiNiRJInhHRiVJInlHRiU= has been evaluated along the path, then multiplied by the element of arc length along the path. In an attempt to understand the origins of this construct, we hypothesize that it arises from the definition of the line integral of the tangential component of the field F = NiNJImZHNiI= i + NiNJImdHNiI= j. Thus, if T is a unit tangent vector along the path NiNJIkNHNiI=, one computes the line integral ofF . T NiNJI2RzRzYi where NiNJI2RzRzYi is the element of arc length along NiNJIkNHNiI=. If NiNJIkNHNiI= is expressed in radius-vector form and parametrized by NiNJInNHNiI=, the arc length, its derivative with respect to NiNJInNHNiI= is the unit tangent vector T. Thus, T = NiMqJkkiZEc2IiIiIkkjZHNHRiUhIiI= NiMtSSdNQVRSSVhHNiI2IzckNyMtSSJ4R0YlNiNJInNHRiU3Iy1JInlHRiVGKw== = NiMtSSdNQVRSSVhHNiI2IzckNyMqJkkjZHhHRiUiIiJJI2RzR0YlISIiNyMqJkkjZHlHRiVGK0YsRi0= andT . NiNJI2RzRzYi = NiMtSSdNQVRSSVhHNiI2IzckNyMqJkkjZHhHRiUiIiJJI2RzR0YlISIiNyMqJkkjZHlHRiVGK0YsRi0= NiNJI2RzRzYi = NiMtSSdNQVRSSVhHNiI2IzckNyNJI2R4R0YlNyNJI2RzR0Yl so that F . T NiNJI2RzRzYi becomes NiMsJiomSSJmRzYiIiIiSSNkeEdGJkYnRicqJkkiZ0dGJkYnSSNkeUdGJkYnRic=. Consequently, the integrand of Maple's PathInt is not just one component of the work, circulation, or flux integrals.

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