Line Integrals (Paper)

7/29/2019 Line Integrals (Paper)

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Line_Integrals.mth: Solv

problems of Line IntegralDerive 6

Dresden International Symposium on TechnoloIntegration into Mathematics Education 2

DES-TIME 2006

DRESDEN 20th-23th July 2006

Dpt. Applied Mathematic

University of Mlaga (Spain)

G. Aguilera C. Cielos J. L. Galn

M. A. Galn A. Glvez A. J. JimnezY. Padilla P. Rodrguez


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Index

z Introduction.

z Line_Integrals.mth.

z Final conclusions.


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Introduction

z In most cases, the use of CAS to using computers as poweperformance calculators.

z It is therefore necessary to cway people think about technologies in order to opt

opportunities they offer anencourage mathematical creativstudents.


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Introduction

z Math teachers that use CAS have totraditional uses given to these tools.

z It is a mistake to use CAS in teachinproblem-solving machines.

zThey should be used in ways that mopportunities that these technolpositively affecting student learning,

increasing opportunities for experimallowing students to construct mathematical knowledge.


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Introduction

z Math teachers must first set oof appropriate activities.

zThe use of CAS in Mathematireached optimum conditions. blackbox (showing the result inwithout teaching students ho

there) and should be whiteboxintermediate steps).


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Combining program

and CASz When students program, they

construct and refine strategiepreviously written programs and la

programs to solve problems. This mthe protagonists of their own learnin

zThe most appropriate approach invprogramming and CAS togethe

students to create the specificfunctions that will allow them toproblems involved in the subject mstudy.


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z Parametrization of curves.

z

Exact differential forms.z Potential function.

z Line Integrals.

z Line Integrals with respect to arcz Applications of Line Integrals.

Contents of

Line_Integrals.mt


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We prefer to use large names fo

instead of using shorthands becthink it is easier for pupils to remefull name and, on the other hand, name of the macro allows to guessolves


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Final Conclusion

z Our accumulated experience revCAS are computer tools which areuse and useful in Mathematics coEngineering.

zThe traditional uses given to CAteaching of Mathematics for Engineebe changed to maximize the oppoffered by CAS technologies. Op

should be aimed at improvingmotivation, autonomy and participatory and student-centred lea


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Final Conclusion

z One powerful idea involves combiresources with the flexibility of a proglanguage.

z

There exists reasonable evidencethat making programs with Derive learning and improves student motiv

z Although it would be desirable to dnot necessary to substantially m

traditional program of studies of Matfor Engineering to introduce the innhaving students make their own with Derive.


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Line_Integrals.mth: Solv

problems of Line IntegralDerive 6

Dresden International Symposium on TechnoloIntegration into Mathematics Education 2

DES-TIME 2006

DRESDEN 20th-23th July 2006

Dpt. Applied Mathematic

University of Mlaga (Spain)

G. Aguilera C. Cielos J. L. Galn

M. A. Galn A. Glvez A. J. JimnezY. Padilla P. Rodrguez


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_________________________________________________________

Line Integrals.dfwJuly, 2006

Jose Luis Galan Garcia ([email protected])Pedro Rodriguez Cielos ([email protected])M. Angeles Galan Garcia ([email protected])

Yolanda Padilla Dominguez ([email protected])

Department of Applied MathematicUniversity of Mlaga

http://www.satd.uma.es/matap/_________________________________________________________

The following functions have been developed in this utility/demo dfw file to deal with line integrals and some of theirapplications:

Parametrization of curveso Segment(p1,p2) parametrization of the segment which joins p1 with p2.o Circumference(x0,y0,r) parametrization of the circumference (x-x0) 2 +(y-y0) 2 =r 2.

o Ellipse(x0,y0,a,b) parametrization of the ellipse (x-x0) 2/a 2 +(y-y0) 2/b 2 =1.o Lemniscata(a) parametrization of the lemniscata (x 2+y 2) 2 =a 2 (x 2-y 2).o Astroid(a) parametrization of the astroid x (2/3) +y (2/3) =a (2/3).o EllipticalAstroid(a,b) parametrization of the elliptical astroid (x/a) (2/n) +(y/b) (2/n) =1.o Cardioid(a) parametrization of the cardioid (x 2+y 2-2ax) 2 =4a 2(x 2+y 2).o Catenary(a) parametrization of the catenary y =a cosh(x/a).o Cicloid(r,h) parametrization of the cicloid [rt - h sin(t) , r - h cos(t)].o Cisoid(a) parametrization of the cisoid x 3+xy 2 =2ay 2.o DescartesFolium(a) parametrization of Descartes' folium x 3+y 3 =3axy.o EightFigure(a) parametrization of the eight figure x 4 =a 2(x 2-y 2).o PascalSnail(a,b) parametrization of Pascal's snail (x 2+y 2-2ax) 2 =b 2(x 2+y 2).o Rosacea(a,k) parametrization of Rosacea [a sin(kt)cos(t), a sin(kt)sin(t)].o Folium(a,b) parametrization of the folium (x 2+y 2) (x 2+y 2+xb) =4axy 2.o Hipocicloid(a,b) parametrization of the hipocicloid

[(a-b)cos(t)+bcos((a-b)t/b),(a-b)sin(t)-bsin((a-b)t/b)].o Trisectriz(a) parametrization of the trisectriz y 2(a+x) =x 2(3a-x).o Tractriz(a) parametrization of the tractriz [a sin(t), a (cos(t) +ln(tan(t/2)))].o ArchimedesSpiral(a) parametrization of Archimedes' spiral [at cos(t), at sin(t)].o Astroid3D(a,b,n) parametrization of the astroid [a cos n(t), b sin n(t), cos(2t)].o SphericalSpiral(a,n,m) parametrization of the spherical spiral

[a cos(nt)cos(mt), a cos(nt)sin(mt), a sin(nt)].

o VivianiCurve(a) parametrization of Viviani's curve [a(1+cos(t)), a sin(t), 2a sin(t/2)].o HelicoidalCurve(curve,c) parametrization of the helicoidal curve [x(t), y(t), ct] where curve=[x(t),y(t)].


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Exact differentialo ExactDifferential2(p,q) to check if Pdx +Qdy is an exact differential.o ExactDifferential3(p,q,r) to check if Pdx +Qdy +Rdz is an exact differential.o ExactDifferential(F) to check if Fd is an exact differential.

Potential functiono Potential2(p,q) to calculate, if there exists, the potential function of P dx +Q dy.o Potential3(P,Q,R) to calculate, if there exists, the potential function of P dx +Q dy +R dz.o Potential(F) to calculate, if there exists, the potential function of Fd.

Line Integralso LineIntegral2(p,q,c1,c2,a,b) to integrate the field (P,Q) along the curve (c1,c2) which parameter t

belongs to [a,b].o Lineintegral3(p,q,r,c1,c2,c3,a,b) to integrate the field (P,Q,R) along the curve (c1,c2,c3) which parameter t

belongs to [a,b].o Lineintegral(F,C,a,b) to integrate the field F along the curve C which parameter t belongs to

[a,b].

Line Integrals with respect to arc lengtho ArcLineIntegral2(f,c1,c2,a,b) to integrate the scalar field f along the curve (c1,c2) which parameter t

belongs to [a,b].o ArcLineintegral3(f,c1,c2,c3,a,b) to integrate the scalar field f along the curve (c1,c2,c3) which parameter t

belongs to [a,b].o ArcLineintegral(f,C,a,b) to integrate the scalar field f along the curve C which parameter t

belongs to [a,b].

Applications of Line Integralso AreaInsideCurve(C,a,b) to compute the area inside the curve C which parameter t belongs to [a,b].

o CurveLength(C,a,b) to compute the length of the curve C which parameter t belongs to [a,b].

o WireMass(,C,a,b) to compute the mass of a wire of equation C which parameter t belongs to [a,b],being its density function.

o WireMx(,C,a,b) to compute the static moment with respect to X-axe of a wire of equation Cwhich parameter t belongs to [a,b], being its density function.

o WireMy(,C,a,b) to compute the static moment with respect to Y-axe of a wire of equation C

which parameter t belongs to [a,b], being its density function.o WireMxy(,C,a,b) to compute the static moment with respect to XY-plane of a wire of equation C

which parameter t belongs to [a,b], being its density function.o WireMxz(,C,a,b) to compute the static moment with respect to XZ-plane of a wire of equation C

which parameter t belongs to [a,b], being its density function.o WireMyz(,C,a,b) to compute the static moment with respect to YZ-plane of a wire of equation C

which parameter t belongs to [a,b], being its density function.o WireMassCenter(,C,a,b) to compute the mass of a wire of equation C which parameter t belongs to [a,b],

being its density function.o WireInertiaMoment(,C,a,b)to compute the inertial moment with respect to an axe of distance to the wire

of equation C which parameter t belongs to [a,b] it, being its density function.

o MediumValue(f,C,a,b) to compute the medium value of the scalar field f along the curve of equation Cwhich parameter t belongs to [a,b].


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Parametrization of curves

Parametrization of a segment

o Syntax: Segment(p1,p2)

o Example: Segment([a,b],[c,d])

to parametize the segment which join [a,b] with [c,d]#1: Segment([a, b], [c, d])

#2: [a(1 - t) + ct, b(1 - t) + dt]

Parametrization of a circumference

o Syntax: Circumference(x0,y0,r)

o Example: Circumference(1,-2,4)

to parametize the circumference (x-1) 2 +(y+2) 2 =4 2#3: Circumference(1, -2, 4)

#4: [4COS(t) + 1, 4SIN(t) - 2]

#5: Circumference(x0, y0, r)

#6: [rCOS(t) + x0, rSIN(t) + y0]


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Parametrization of an ellipse

o Syntax: Ellipse(x0,y0,a,b)

o Example: Ellipse(1,-2,2,3)

to parametize the ellipse (x-1) 2/2 2 +(y+2) 2/3 2 =1#7: Ellipse(1, -2, 2, 3)

#8: [2COS(t) + 1, 3SIN(t) - 2]

#9: Ellipse(x0, y0, a, b)

#10: [aCOS(t) + x0, bSIN(t) + y0]

Parametrization of a lemniscata

o Syntax: Lemniscata(a)

o Example: Lemniscata(3)

to parametize the lemniscata (x 2+y 2) 2 =3 2 (x 2-y 2)


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#11: Lemniscata(3)

#12: [3COS(t)(COS(2t)), 3SIN(t)(COS(2t))]

#13: Lemniscata(a)

#14: [aCOS(t)(COS(2t)), aSIN(t)(COS(2t))]

Parametrization of an astroid

o Syntax: Astroid(a)

o Example: Astroid(3)

to parametize the astroid x (2/3) +y (2/3) =3 (2/3)#15: Astroid(3)

3 3#16: 3COS(t) , 3SIN(t)

#17: Astroid(a)

3 3

#18: aCOS(t) , aSIN(t)


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Parametrization of an elliptical astroid

o Syntax: EllipticalAstroid(a,b,n)

o Example: EllipticalAstroid(2,1,3)

to parametize the elliptical astroid x (2/3)/2 2 +y (2/3)/1 1 =1 (2/3)#19: EllipticalAstroid(2, 1, 3)

3 3#20: 2COS(t) , SIN(t)

#21: EllipticalAstroid(a, b, n) n n

#22: aCOS(t) , bSIN(t)

Parametrization of a cardioid

o Syntax: Cardioid(a)

o Example: Cardioid(3/4)

to parametize the cardioid (x 2+y 2- 2 * 3/4x) 2 =4 * (3/4) 2(x 2+y 2) 3

#23: Cardioid 4

2 3COS(t) 3COS(t) 3SIN(t)COS(t) 3SIN(t)

#24: + , + 2 2 2 2

#25: Cardioid(a)


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2 #26: 2aCOS(t) + 2aCOS(t), 2aSIN(t)COS(t) + 2aSIN(t)

Parametrization of a catenary

o Syntax: Catenary(a)

o Example: Catenary(1)

to parametize the catenary y =1 * cosh(x/1)#27: Catenary(1)

t -t e e

#28: t, + 2 2

#29: Catenary(a)

t/a - t/a ae ae

#30: t, + 2 2


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Parametrization of a cicloid

o Syntax: Cicloid(r,h)

o Example: Cicloid(1,1)

to parametize the cicloid [t - sin(t) , 1 - cos(t)]#31: Cicloid(1, 1)

#32: [t - SIN(t), 1 - COS(t)]

#33: Cicloid(r, h)

#34: [rt - hSIN(t), r - hCOS(t)]

Parametrization of a cisoid

o Syntax: Cisoid(a)

o Example: Cisoid(3/4)

to parametize the cisoid x 3+xy 2 =2 * 3/4 * y 2 3

#35: Cisoid 4

2 3 3SIN(t) 3SIN(t)

#36: , 2 2COS(t)

#37: Cisoid(a)


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3 2 2aSIN(t)

#38: 2aSIN(t) , COS(t)

Parametrization of Descartes' folium

o Syntax: DescartesFolium(a)

o Example: DescartesFolium(1.5)

to parametize the Descartes' folium x 3+y 3 =3 * 1.5 * xy#39: DescartesFolium(1.5)

2 9t 9t

#40: , 3 3 2(t + 1) 2(t + 1)

#41: DescartesFolium(a)

2 3at 3at

#42: ,

3 3 t + 1 t + 1


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Parametrization of EightFigure

o Syntax: EightFigure(a)

o Example: EightFigure(1.5)

to parametize the eight figure x 4 =1.5 2(x 2-y 2)#43: EightFigure(1.5)

3(COS(2t)) 3SIN(t)(COS(2t)) ,

#44: 2COS(t) 2

2COS(t) #45: EightFigure(a)

a(COS(2t)) aSIN(t)(COS(2t)) ,

#46: COS(t) 2 COS(t)

Parametrization of Pascal's Snail

o Syntax: PascalSnail(a,b)

o Example: PascalSnail(1,3/2)

to parametize the Pascal's snail (x 2+y 2-2*1*x) 2 =(3/2) 2(x 2+y 2) 3

#47: PascalSnail1, 2

2 3COS(t) 3SIN(t)

#48: 2COS(t) + , 2SIN(t)COS(t) + 2 2


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#49: PascalSnail(a, b)

2 #50: 2aCOS(t) + bCOS(t), 2aSIN(t)COS(t) + bSIN(t)

Parametrization of Rosacea

o Syntax: Rosacea(a,k)

o Example: Rosacea(3,5)

to parametize the rosacea [3 sin(5t)cos(t), 3 sin(5t)sin(t)]#51: Rosacea(3, 5)

#52: [3COS(t)SIN(5t), 3SIN(t)SIN(5t)]

#53: Rosacea(a, k)

#54: [aCOS(t)SIN(kt), aSIN(t)SIN(kt)]


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Parametrization of Folium

o Syntax: Folium(a,b)

o Example: Folium(1,1)

to parametize the folium (x 2+y 2) (x 2+y 2+x*1) =4*1*xy 2

#55: Folium(1, 1)

2 2 3 #56: COS(t) (4SIN(t) - 1), COS(t)(4SIN(t) - SIN(t))

#57: Folium(a, b)

2 2 3 #58: COS(t) (4aSIN(t) - b), COS(t)(4aSIN(t) - bSIN(t))

Parametrization of Hipocicloid

o Syntax: Hipocicloid(a,b)

o Example: Hipocicloid(1,1)

to parametize the hipocicloid [(a-b)cos(t)+bcos((a-b)t/b),(a-b)sin(t)-bsin((a-b)t/b)]

3 #59: Hipocicloid2,

4

5t 5t 3COS 3SIN

#60: 3 5COS(t) 5SIN(t) 3 + , - 4 4 4 4

#61: Hipocicloid(a, b)


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at at#62: bCOS - t + (a - b)COS(t), (a - b)SIN(t) - bSIN -

b b

t

Parametrization of Trisectriz

o Syntax: Trisectriz(a)

o Example: Trisectriz(1)

to parametize the trisectriz y 2(1+x) =x 2(3*1-x)#63: Trisectriz(1)

2 #64: 4COS(t) - 1, 2SIN(2t) - TAN(t)

#65: Trisectriz(a)

2 #66: 4aCOS(t) - a, 2aSIN(2t) - aTAN(t)


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Parametrization of Tractriz

o Syntax: Tractriz(a)

o Example: Tractriz(3)

to parametize the tractriz [3 sin(t), 3 (cos(t) +ln(tan(t/2)))]#67: Tractriz(3)

t #68: 3SIN(t), 3LNTAN + 3COS(t)

2

#69: Tractriz(a)

t #70: aSIN(t), aLNTAN + aCOS(t)

2

Parametrization of Archimedes' spiral

o Syntax: ArchimedesSpiral(a)

o Example: ArchimedesSpiral(0.03)to parametize the Archimedes' spiral [at cos(t), at sin(t)]

#71: ArchimedesSpiral(0.03)

3tCOS(t) 3tSIN(t) #72: ,

100 100

#73: ArchimedesSpiral(a)

#74: [atCOS(t), atSIN(t)]


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Parametrization of Astroid in R 3

o Syntax: Astroid3D(a,n)

o Example: Astroid3D(1,1,1)

to parametize the astroid [cos(t), sin(t), cos(2t)]#75: [COS(t), SIN(t), COS(2t)]

#76: Astroid3D(a, b, n)

n n

#77: aCOS(t) , bSIN(t) , COS(2t)

#78: Astroid3D(1, 1, 1)

Parametrization of spherical spiral

o Syntax: SphericalSpiral(a,n,m)

o Example: SphericalSpiral(5,1,25)

to parametize the spherical spiral [5 cos(1*t)cos(25*t), 5 cos(1*t)sin(25*t), 5sin(1*t)]


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#79: SphericalSpiral(5, 1, 25)

#80: [5COS(t)COS(25t), 5COS(t)SIN(25t), 5SIN(t)]

#81: SphericalSpiral(a, n, m)

#82: [aCOS(mt)COS(nt), aSIN(mt)COS(nt), aSIN(nt)]

Parametrization of Viviani's curve

o Syntax: VivianiCurve(a)

o Example: VivianiCurve(2)

to parametize the Viviani's curve [2(1+cos(t)), 2 sin(t), 4 sin(t/2)]#83: VivianiCurve(2)

t #84: 2COS(t) + 2, 2SIN(t), 4SIN

2

#85: VivianiCurve(a)

t

#86: aCOS(t) + a, aSIN(t), 2aSIN 2


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Parametrization of helicoidal curve

o Syntax: HelicoidalCurve(curve,c)

o Example: HelicoidalCurve(Circumference(0,0,r))

to parametize the helicoidal curve [r cos(t), r sin(t), ct]#87: HelicoidalCurve(Circumference(0, 0, r))

#88: [rCOS(t), rSIN(t), ct]

#89: HelicoidalCurve(ArchimedesSpiral(a))

#90: [atCOS(t), atSIN(t), ct]

#91: HelicoidalCurve(Astroid(a))

3 3 #92: aCOS(t) , aSIN(t) , ct


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Exercises

Exercise 1Compute the line integral of [xy^2+x+1, x^2y-2] along curve C given by:

The ellipse (x-x_0)^2/a^2+(y-y_0)^2/b^2 = 1.The portion of the curve given by y=(x+2)\ln x + (1-x)^2 which joins thepoint (1,0) with (-2,9).

2 2 #93: ExactDifferential(xy + x + 1, x y - 2)

#94: This is an exact differential

2 2 #95: U(x, y) Potential(xy + x + 1, x y - 2)

2 2 y 1

#96: U(x, y) x + + x - 2y 2 2

#97: U(-2, 9) - U(1, 0)

285#98:

2

Exercise 2Compute the line integral of [e^x+1,x+z,xy+x+y+2 e^z] along the segmentwhich joins (0,1,2) with (2,-1,6).

x z#99: ExactDifferential(e + 1, x + z, xy + x + y + 2e)

#100: This is not an exact differential

x z#101: LineIntegral(e + 1, x + z, xy + x + y + 2e, Segment([0, 1,

2], [2, -1, 6]), 0, 1)

6 2 19#102: 2e - e -

3


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Exercise 3Compute the area inside the cardiod (x 2+y 2- 2 * 3/4x) 2 =4 * (3/4) 2(x 2+y 2).Compute its length.

3 #103: AreaInsideCurveCardioid, 0, 2

4

27#104:

8

3 #105: CurveLengthCardioid, 0, 2

4

#106: 12

Exercise 4Let H be a wire which shape is the curve (cost,sint,t) ; t in [0,2pi] withdensity function(x,y,z)=x^2+y^2+z^2. Compute the length, mass, mass center and media densityof the wire.

#107: CurveLength([COS(t), SIN(t), t], 0, 2)

#108: 22

2 2 2#109: WireMass(x + y + z , [COS(t), SIN(t), t], 0, 2)

382

#110: + 223

2 2 2#111: WireMassCenter(x + y + z , [COS(t), SIN(t), t], 0, 2)

2 6 6 3(2 + 1)

#112: , - , 2 2 2 4 + 3 4 + 3 4 + 3

2 2 2#113: MediumValue(x + y + z , [COS(t), SIN(t), t], 0, 2)

24 + 3

#114: 3

Line Integrals (Paper)

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