14.1 Parametric Equations of Curves14.2 Parametric Equations of Surfaces14.3 Velocity and Acceleration Vectors in Space14.4 Vector Fields14.5 Line Integrals with Maple V14.6 Divergence and Curl in Maple V14.7 Surface Integrals with Maple V8214 Vector Calculus14.1 Parametric Equations of CurvesIn this section we will see how Maple V can be used to plot interesting curves using parametric representation.First of all consider the circle centered at the origin with radius 2. The equation for this curve isx2+ y2= 4If you wish plot this curve then you soon become aware that the set of points represented by the above equationcannot satisfy a relationship of the form y = f (x) where f is a single valued function (the vertical line test fails).The graph of the functionf1( x ) =√4 − x2is the upper semi-circle of radius 2.> P1 := plot(sqrt(4-x^2),x=-2..2):";Whereas the following plot gives the bottom half:> P2 := plot(-sqrt(4-x^2),x=-2..2):";We can now use display to put both plots together. See Figure 70.> with(plots):> display({P1,P2},scaling =constrained);We can obtain a parameterization for the entire circle. Since the following identity is true for all real values of t-2-1012-2 -1 0 1 2xFigure 70: Circle of Radius 2-20-1001020-3 -2 -1 0 1 2 3Figure 71: Parabola x = t, y = t3− 2tsin( t )2+ cos( t )2= 1it follows that x = 2sin(t), y = 2cos(t) satisfy the equationx2+ y2= 4for all values of t. Moreover, the image of the mapping defined by the parametric equations x = 2cos(t), y =2sin(t) generate the entire circle, which is shown in Figure 70, as t varies for 0 to 2π with one plot statement usingthe parametric plot syntax.> plot([2*cos(t),2*sin(t),t=0..2*Pi],scaling=constrained);14 VECTOR CALCULUS83If a curve is given as an explicit function of x or y then it is easy to write a parameterization: if a curve isdescribed by the relation y = f (x), for a ≤ x ≤ b, then a parameterization for this curve is x = t, y = f (t),for a ≤ t ≤ b, and, similarly, if a curveis described as x = g(y), then the parameterization can be of the formx = g(t), y = t. For example, supposey = x3− 2 xis given. Then either of the following plot commands will give the Figure 71. for any range, say, −3 ≤ x ≤ 3.> plot(x^3-2*x,x=-3..3);> plot([t,t^3-2*t,t=-3..3]);Similarly if x = y2− y is the function you wish to plot then the following command does the trick. See Fig-ure 72.> plot([t^2-t,t,t=-2..2]);-2-101201 2 3 456Figure 72: Parabola x = t2− t, y = t-202468-1 0 1 2 3Figure 73: Line x = 1 + 2t, y = 3 + 5tStraight lines are very easy to parameterize. For example if y = mx + b is the equation of the line then x =t, y = mt +b gives a parameterization. But you can sometimes write the parametric equations directly from theirgeometric description. For example, a portion of the straight line through the point (1,3) parallel to the vector <2, 5 > has parametric equations x = 1+2t, y = 3+5t, is shown in Figure 73 and can be plotted by the command:> plot([1+2*t,3+5*t,t=-1..1]);On occasion it possible to plot curves that would be very complicated to express in even implicit algebraic terms.For example, the following curve is known as a Lissajous Figure and is shown in Figure 74.> plot([cos(3*t),sin(5*t),t=0..2*Pi],scaling=constrained);Solutions of differential equations are usually represented in parametric form. For example x = cos(2t), y =−2sin(2t) is a solution of the initial value problem:ddtx(t ) = x( t )ddty(t ) =−4x(t)x(0) = 1, y(0) =0as can easily be verified. A plot of this curve is given by> plot([cos(2*t),-2*sin(2*t),t=0..Pi],scaling = constrained);14 VECTOR CALCULUS84-1-0.500.51-1 -0.5 0 0.5 1Figure 74: Lissajous Figure x = cos(3t), y = sin(5t)-2-1012-1 -0.5 0 0.5 1Figure 75: Ellipse x= cos(2t), y= -2sin(2t)See Figure 75. When plotting curves in three dimensions using Maple V we need to use the command spacecurvewhich is part of the plots package. Consider the curve written parametrically asx = cos t, y = sin t, z = t.It’s plot, Figure 76 over the range [0, 2π] is given by> spacecurve([cos(t),sin(t),t],t=0..2*Pi);Figure 76: Cylindrical Helix: x = cos(t), y = sin(t), z=tFigure 77: A Closed Curve in Three SpaceThe complicated curve shown in Figure 77, is actually a simple closed curve, and can be obtained as follows:> spacecurve([(4+sin(3*t))*cos(2*t),(4+sin(3*t))*sin(2*t),cos(3*t)],> t=0..2*Pi);Exercises 14.11. Make a Maple V plot of the conical helix given by the parametric equationsx = t cos(3t), y = t sin(3t), z = t, −∞ < t < ∞.14 VECTOR CALCULUS852. Write parametric equations for first octant portion of the curve of intersection of the sphere and cylinderx2+ y2+ z2= 64, x2+ (y −4)2= 16.Make a Maple V plot of this curve.3. Show that x = sin(2t) cos(t), y = sin(2t) sin(t), 0 ≤ t ≤ 2π is a parametric representation for the curvegiven implicitly by the equation(x2+ y2)3= 4x2y2.Plot the graph of this curve using Maple V using the parametric representation and also using the explicitrepesentation. Is it worthwhile to obtain the parametric representation?14 VECTOR CALCULUS8614.2 Parametric Equations of SurfacesAs with the implicit representation of a circle, the implicit representation of a sphere may require a piece mealapproach when making plots. In order to plot the graph of the spherex2+ y2+ z2= 4we could solve for z and obtain a parameterization for the lower half withz = f1(x, y) =−p4−x2−y2, 0≤x2+y2≤4and obtain a plot with plot3d.> P1 := plot3d(-sqrt(4-x^2-y^2),x=-2..2,y=-sqrt(4-x^2)..sqrt(4-x^2),> style=patchnogrid,numpoints=2500):";In order to get the top half we type in the following:> P2 := plot3d(sqrt(4-x^2-y^2),x=-2..2,y=-sqrt(4-x^2)..sqrt(4-x^2),> style=patchnogrid,numpoints=2500):";Finally, to put both plots, as shown in Figure 78, at the same time we write.> with(plots):> display({P1,P2},axes=normal,scaling=constrained);> with(plots):> display({P1,P2},scaling =constrained);-2-1012-2-1012-2-1012Figure 78: Sphere of Radius 2Figure 79: Cone Plot with implicitplot3dThis is analogous to the procedure that was use in plotting a circle when its implicit equation is given in rect-angular coordinates. Just as with the situation with a circle, there is a parameterization which enables us to plotthe graph with just one set of defining equations. From a well known trigonometric identity you can see thatx = 2sin( u ) cos(v),y=2sin( u ) sin(v),z=2cos( u )provides a parametric representation for the entire sphere if 0 ≤ u ≤ π, 0 ≤ v ≤ 2π. The sphere can