Line_Integrals.mth: Solving problems of Line Integrals with Derive 6

Transcription

1 Dresden International Symposium on Technology and its Integration into Mathematics Education 2006 DES-TIME 2006 DRESDEN 20th-23th July 2006 Line_Integrals.mth: Solving problems of Line Integrals with Derive 6 G. Aguilera C. Cielos J. L. Galán M. A. Galán A. Gálvez A. J. Jiménez Y. Padilla P. Rodríguez Dpt. Applied Mathematic University of Málaga (Spain)

2 Index Introduction. Line_Integrals.mth. Final conclusions.

3 Introduction In most cases, the use of CAS is reduced to using computers as powerful highperformance calculators. It is therefore necessary to change the way people think about information technologies in order to optimise the opportunities they offer and try to encourage mathematical creativity among students.

4 Introduction Math teachers that use CAS have to change the traditional uses given to these tools. It is a mistake to use CAS in teaching as simple problem-solving machines. They should be used in ways that maximize the opportunities that these technologies offer: positively affecting student learning, significantly increasing opportunities for experimentation and allowing students to construct their own mathematical knowledge.

5 Introduction Math teachers must first set out a series of appropriate activities. The use of CAS in Mathematics has not reached optimum conditions. Mostly are blackbox (showing the result in one step without teaching students how to get there) and should be whitebox (showing intermediate steps).

6 Combining programming and CAS When students program, they must read, construct and refine strategies, modify previously written programs and lastly, use the programs to solve problems. This makes them the protagonists of their own learning. The most appropriate approach involves using programming and CAS together to allow students to create the specific necessary functions that will allow them to solve the problems involved in the subject matter under study.

7 Contents of Line_Integrals.mth Parametrization of curves. Exact differential forms. Potential function. Line Integrals. Line Integrals with respect to arc length. Applications of Line Integrals.

8 We prefer to use large names for macros instead of using shorthands because we think it is easier for pupils to remember the full name and, on the other hand, the large name of the macro allows to guess what it solves

9 Final Conclusions Our accumulated experience reveals that CAS are computer tools which are easy to use and useful in Mathematics courses for Engineering. The traditional uses given to CAS in the teaching of Mathematics for Engineering must be changed to maximize the opportunities offered by CAS technologies. Optimal use should be aimed at improving student motivation, autonomy and achieving participatory and student-centred learning.

10 Final Conclusions One powerful idea involves combining CAS resources with the flexibility of a programming language. There exists reasonable evidence to show that making programs with Derive facilitates learning and improves student motivation. Although it would be desirable to do so, it is not necessary to substantially modify the traditional program of studies of Math courses for Engineering to introduce the innovation of having students make their own programs with Derive.

11 Dresden International Symposium on Technology and its Integration into Mathematics Education 2006 DES-TIME 2006 DRESDEN 20th-23th July 2006 Line_Integrals.mth: Solving problems of Line Integrals with Derive 6 G. Aguilera C. Cielos J. L. Galán M. A. Galán A. Gálvez A. J. Jiménez Y. Padilla P. Rodríguez Dpt. Applied Mathematic University of Málaga (Spain)

12 Line Integrals.dfw July, 2006 Jose Luis Galan Garcia Pedro Rodriguez Cielos M. Angeles Galan Garcia Yolanda Padilla Dominguez Department of Applied Mathematic University of Málaga The following functions have been developed in this utility/demo dfw file to deal with line integrals and some of their applications: Parametrization of curves o 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)].

13 Exact differential o ExactDifferential2(p,q) o ExactDifferential3(p,q,r) o ExactDifferential(F) Potential function o Potential2(p,q) o Potential3(P,Q,R) o Potential(F) to check if Pdx + Qdy is an exact differential. to check if Pdx + Qdy + Rdz is an exact differential. to check if Fdα is an exact differential. to calculate, if there exists, the potential function of P dx + Q dy. to calculate, if there exists, the potential function of P dx + Q dy + R dz. to calculate, if there exists, the potential function of Fdα. Line Integrals o LineIntegral2(p,q,c1,c2,a,b) o Lineintegral3(p,q,r,c1,c2,c3,a,b) o Lineintegral(F,C,a,b) to integrate the field (P,Q) along the curve (c1,c2) which parameter t belongs to [a,b]. to integrate the field (P,Q,R) along the curve (c1,c2,c3) which parameter t belongs to [a,b]. to integrate the field F along the curve C which parameter t belongs to [a,b]. Line Integrals with respect to arc length o 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 Integrals o AreaInsideCurve(C,a,b) o CurveLength(C,a,b) to compute the area inside the curve C which parameter t belongs to [a,b]. to compute the length of the curve C which parameter t belongs to [a,b]. o WireMass(ρ,C,a,b) to compute the mass of a wire of equation C which parameter t belongs to [a,b], being ρ its density function. o WireMx(ρ,C,a,b) to compute the static moment with respect to X-axe of a wire of equation C which parameter t belongs to [a,b], being ρ its density function. o WireMy(ρ,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 ρ its density function. o WireMxy(ρ,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 ρ its density function. o WireMxz(ρ,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 ρ its density function. o WireMyz(ρ,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 ρ its density function. o WireMassCenter(ρ,C,a,b) to compute the mass of a wire of equation C which parameter t belongs to [a,b], being ρ its density function. o WireInertiaMoment(ρ,C,a,b)to compute the inertial moment with respect to an axe of distance δ to the wire of equation C which parameter t belongs to [a,b] it, being ρ its density function. o MediumValue(f,C,a,b) to compute the medium value of the scalar field f along the curve of equation C which parameter t belongs to [a,b].

14 Parametrization of curves Parametrization of a segment Segment(p1,p2) 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) + c t, b (1 - t) + d t] Parametrization of a circumference Circumference(x0,y0,r) Circumference(1,-2,4) to parametize the circumference (x-1)^2 + (y+2)^2 = 4^2 #3: Circumference(1, -2, 4) #4: [4 COS(t) + 1, 4 SIN(t) - 2] #5: Circumference(x0, y0, r) #6: [r COS(t) + x0, r SIN(t) + y0]

15 Parametrization of an ellipse Ellipse(x0,y0,a,b) 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: [2 COS(t) + 1, 3 SIN(t) - 2] #9: Ellipse(x0, y0, a, b) #10: [a COS(t) + x0, b SIN(t) + y0] Parametrization of a lemniscata Lemniscata(a) Lemniscata(3) to parametize the lemniscata (x^2+y^2)^2 = 3^2 (x^2-y^2)

16 #11: Lemniscata(3) #12: [3 COS(t) (COS(2 t)), 3 SIN(t) (COS(2 t))] #13: Lemniscata(a) #14: [a COS(t) (COS(2 t)), a SIN(t) (COS(2 t))] Parametrization of an astroid Astroid(a) Astroid(3) to parametize the astroid x^(2/3) + y^(2/3) = 3^(2/3) #15: Astroid(3) 3 3 #16: 3 COS(t), 3 SIN(t) #17: Astroid(a) 3 3 #18: a COS(t), a SIN(t)

17 Parametrization of an elliptical astroid EllipticalAstroid(a,b,n) 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: 2 COS(t), SIN(t) #21: EllipticalAstroid(a, b, n) n n #22: a COS(t), b SIN(t) Parametrization of a cardioid Cardioid(a) 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 COS(t) 3 COS(t) 3 SIN(t) COS(t) 3 SIN(t) #24: +, #25: Cardioid(a)

18 2 #26: 2 a COS(t) + 2 a COS(t), 2 a SIN(t) COS(t) + 2 a SIN(t) Parametrization of a catenary Catenary(a) Catenary(1) to parametize the catenary y = 1 * cosh(x/1) #27: Catenary(1) t -t e e #28: t, #29: Catenary(a) t/a - t/a a e a e #30: t, + 2 2

19 Parametrization of a cicloid Cicloid(r,h) 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: [r t - h SIN(t), r - h COS(t)] Parametrization of a cisoid Cisoid(a) Cisoid(3/4) to parametize the cisoid x^3+xy^2 = 2 * 3/4 * y^2 3 #35: Cisoid SIN(t) 3 SIN(t) #36:, 2 2 COS(t) #37: Cisoid(a)

20 3 2 2 a SIN(t) #38: 2 a SIN(t), COS(t) Parametrization of Descartes' folium DescartesFolium(a) DescartesFolium(1.5) to parametize the Descartes' folium x^3+y^3 = 3 * 1.5 * xy #39: DescartesFolium(1.5) 2 9 t 9 t #40:, (t + 1) 2 (t + 1) #41: DescartesFolium(a) 2 3 a t 3 a t #42:, 3 3 t + 1 t + 1

21 Parametrization of EightFigure EightFigure(a) EightFigure(1.5) to parametize the eight figure x^4 = 1.5^2(x^2-y^2) #43: EightFigure(1.5) 3 (COS(2 t)) 3 SIN(t) (COS(2 t)), #44: 2 COS(t) 2 2 COS(t) #45: EightFigure(a) a (COS(2 t)) a SIN(t) (COS(2 t)), #46: COS(t) 2 COS(t) Parametrization of Pascal's Snail PascalSnail(a,b) 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: PascalSnail 1, COS(t) 3 SIN(t) #48: 2 COS(t) +, 2 SIN(t) COS(t) + 2 2

22 #49: PascalSnail(a, b) 2 #50: 2 a COS(t) + b COS(t), 2 a SIN(t) COS(t) + b SIN(t) Parametrization of Rosacea Rosacea(a,k) Rosacea(3,5) to parametize the rosacea [3 sin(5t)cos(t), 3 sin(5t)sin(t)] #51: Rosacea(3, 5) #52: [3 COS(t) SIN(5 t), 3 SIN(t) SIN(5 t)] #53: Rosacea(a, k) #54: [a COS(t) SIN(k t), a SIN(t) SIN(k t)]

23 Parametrization of Folium Folium(a,b) 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) #56: COS(t) (4 SIN(t) - 1), COS(t) (4 SIN(t) - SIN(t)) #57: Folium(a, b) #58: COS(t) (4 a SIN(t) - b), COS(t) (4 a SIN(t) - b SIN(t)) Parametrization of Hipocicloid Hipocicloid(a,b) 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: Hipocicloid 2, 4 5 t 5 t 3 COS 3 SIN #60: 3 5 COS(t) 5 SIN(t) 3 +, #61: Hipocicloid(a, b)

24 a t a t #62: b COS - t + (a - b) COS(t), (a - b) SIN(t) - b SIN - b b t Parametrization of Trisectriz Trisectriz(a) Trisectriz(1) to parametize the trisectriz y^2(1+x) = x^2(3*1-x) #63: Trisectriz(1) 2 #64: 4 COS(t) - 1, 2 SIN(2 t) - TAN(t) #65: Trisectriz(a) 2 #66: 4 a COS(t) - a, 2 a SIN(2 t) - a TAN(t)

25 Parametrization of Tractriz Tractriz(a) Tractriz(3) to parametize the tractriz [3 sin(t), 3 (cos(t) + ln(tan(t/2)))] #67: Tractriz(3) t #68: 3 SIN(t), 3 LN TAN + 3 COS(t) 2 #69: Tractriz(a) t #70: a SIN(t), a LN TAN + a COS(t) 2 Parametrization of Archimedes' spiral ArchimedesSpiral(a) ArchimedesSpiral(0.03) to parametize the Archimedes' spiral [at cos(t), at sin(t)] #71: ArchimedesSpiral(0.03) 3 t COS(t) 3 t SIN(t) #72:, #73: ArchimedesSpiral(a) #74: [a t COS(t), a t SIN(t)]

26 Parametrization of Astroid in R^3 Astroid3D(a,n) Astroid3D(1,1,1) to parametize the astroid [cos(t), sin(t), cos(2t)] #75: [COS(t), SIN(t), COS(2 t)] #76: Astroid3D(a, b, n) n n #77: a COS(t), b SIN(t), COS(2 t) #78: Astroid3D(1, 1, 1) Parametrization of spherical spiral SphericalSpiral(a,n,m) SphericalSpiral(5,1,25) sin(1*t)] to parametize the spherical spiral [5 cos(1*t)cos(25*t), 5 cos(1*t)sin(25*t), 5

27 #79: SphericalSpiral(5, 1, 25) #80: [5 COS(t) COS(25 t), 5 COS(t) SIN(25 t), 5 SIN(t)] #81: SphericalSpiral(a, n, m) #82: [a COS(m t) COS(n t), a SIN(m t) COS(n t), a SIN(n t)] Parametrization of Viviani's curve VivianiCurve(a) VivianiCurve(2) to parametize the Viviani's curve [2(1+cos(t)), 2 sin(t), 4 sin(t/2)] #83: VivianiCurve(2) t #84: 2 COS(t) + 2, 2 SIN(t), 4 SIN 2 #85: VivianiCurve(a) t #86: a COS(t) + a, a SIN(t), 2 a SIN 2

28 Parametrization of helicoidal curve HelicoidalCurve(curve,c) HelicoidalCurve(Circumference(0,0,r)) to parametize the helicoidal curve [r cos(t), r sin(t), ct] #87: HelicoidalCurve(Circumference(0, 0, r)) #88: [r COS(t), r SIN(t), c t] #89: HelicoidalCurve(ArchimedesSpiral(a)) #90: [a t COS(t), a t SIN(t), c t] #91: HelicoidalCurve(Astroid(a)) 3 3 #92: a COS(t), a SIN(t), c t

29 Exercises Exercise 1 Compute 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 the point (1,0) with (-2,9). 2 2 #93: ExactDifferential( x y + x + 1, x y - 2 ) #94: This is an exact differential 2 2 #95: U(x, y) Potential( x y + x + 1, x y - 2 ) 2 2 y 1 #96: U(x, y) x + + x - 2 y 2 2 #97: U(-2, 9) - U(1, 0) 285 #98: 2 Exercise 2 Compute the line integral of [e^x+1,x+z,xy+x+y+2 e^z] along the segment which joins (0,1,2) with (2,-1,6). x z #99: ExactDifferential( e + 1, x + z, x y + x + y + 2 e ) #100: This is not an exact differential x z #101: LineIntegral( e + 1, x + z, x y + x + y + 2 e, Segment([0, 1, 2], [2, -1, 6]), 0, 1) #102: 2 e - e - 3

30 Exercise 3 Compute 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: AreaInsideCurve Cardioid, 0, 2 π 4 27 π #104: 8 3 #105: CurveLength Cardioid, 0, 2 π 4 #106: 12 Exercise 4 Let H be a wire which shape is the curve (cost,sint,t) ; t in [0,2pi] with density function ρ(x,y,z)=x^2+y^2+z^2. Compute the length, mass, mass center and media density of the wire. #107: CurveLength([COS(t), SIN(t), t], 0, 2 π) #108: 2 2 π #109: WireMass(x + y + z, [COS(t), SIN(t), t], 0, 2 π) π #110: π #111: WireMassCenter(x + y + z, [COS(t), SIN(t), t], 0, 2 π) π 3 π (2 π + 1) #112:, -, π π π #113: MediumValue(x + y + z, [COS(t), SIN(t), t], 0, 2 π) 2 4 π + 3 #114: 3