16 Vector Calculus Copyright Cengage Learning All rights reserved 16 6 Parametric Surfaces and Their Areas Copyright Cengage Learning All rights reserved Parametric Surfaces and Their Areas Learning objectives parametrize a surface that is given by a geometric description or the graph of a function of two variables match a surface with its parametric equations find the surface area of a surface that is given by parametric equations or is the graph of a function 3 Parametric Surfaces 4 Parametric Surfaces In much the same way that we describe a space curve by a vector function r t of a single parameter t we can describe a surface by a vector function r u v of two parameters u and v We suppose that r u v x u v i y u v j z u v k is a vector valued function defined on a region D in the uv plane 5 Parametric Surfaces So x y and z the component functions of r are functions of the two variables u and v with domain D The set of all points x y z in x x u v such that y y u v z z u v and u v varies throughout D is called a parametric surface S and Equations 2 are called parametric equations of S Both the parametric equations in Equation 2 or the vector equation in Equation 1 and the domain D are needed to describe the parametric surface S 6 Parametric Surfaces Each choice of u and v gives a point on S by making all choices we get all of S By restricting the parameters u and v to a subset of the domain D we obtain only a part of the surface S In other words the surface is traced out by the tip of the position vector r u v as u v moves throughout the region D A parametric surface Figure 1 7 Parametric Surfaces If a parametric surface S is given by a vector function r u v then there are two useful families of curves that lie on S one family with u constant and the other with v constant These families correspond to vertical and horizontal lines in the uv plane 8 Parametric Surfaces If we keep u constant by putting u u0 then r u0 v becomes a vector function of the single parameter v and defines a curve C1 lying on S See Figure 4 Figure 4 9 Parametric Surfaces Similarly if we keep v constant by putting v v0 we get a curve C2 given by r u v0 that lies on S We call these curves grid curves In fact when a computer graphs a parametric surface it usually depicts the surface by plotting these grid curves 10 Example Find a parametric representation of the sphere x2 y2 z2 a2 Solution The sphere has a simple representation a in spherical coordinates so let s choose the angles and in spherical coordinates as the parameters 11 Example Solution continued Then putting a in the equations for conversion from spherical to rectangular coordinates we obtain x a sin cos y a sin sin z a cos as the parametric equations of the sphere The corresponding vector equation is r a sin cos i a sin sin j a cos k 12 Example Solution continued We have 0 and 0 2 so the parameter domain is the rectangle D 0 0 2 The grid curves with constant are the circles of constant latitude including the equator 13 Example Solution continued The grid curves with constant are the meridians semi circles which connect the north and south poles see Figure 7 Figure 7 14 Parametric Surfaces We saw in the example that the grid curves for a sphere are curves of constant latitude and longitude For a general parametric surface we are really making a map and the grid curves are similar to lines of latitude and longitude Describing a point on a parametric surface like the one in Figure 5 by giving specific values of u and v is like giving the latitude and longitude of a point 15 Surfaces of Revolution 16 Surfaces of Revolution Surfaces of revolution can be represented parametrically and thus graphed using a computer For instance let s consider the surface S obtained by rotating the curve y f x a x b about the x axis where f x 0 Let be the angle of rotation as shown in Figure 10 Figure 10 17 Surfaces of Revolution If x y z is a point on S then x x y f x cos z f x sin Therefore we take x and as parameters and regard Equations 3 as parametric equations of S The parameter domain is given by a x b 0 2 18 Example Find parametric equations for the surface generated by rotating the curve y sin x 0 x 2 about the x axis Use these equations to graph the surface of revolution Solution From Equations 3 the parametric equations are x x y sin x cos z sin x sin and the parameter domain is 0 x 2 0 2 19 Example Solution continued Using a computer to plot these equations and rotate the image we obtain the graph in Figure 11 Figure 11 20 Tangent Planes 21 Tangent Planes We now find the tangent plane to a parametric surface S traced out by a vector function r u v x u v i y u v j z u v k at a point P0 with position vector r u0 v0 22 Tangent Planes If we keep u constant by putting u u0 then r u0 v becomes a vector function of the single parameter v and defines a grid curve C1 lying on S See Figure 12 The tangent vector to C1 at P0 is obtained by taking the partial derivative of r with respect to v Figure 12 23 Tangent Planes Similarly if we keep v constant by putting v v0 we get a grid curve C2 given by r u v0 that lies on S and its tangent vector at P0 is If ru rv is not 0 then the surface S is called smooth it has no corners For a smooth surface the tangent plane is the plane that contains the tangent vectors ru and rv and the vector ru rv is a normal vector to the tangent plane 24 Surface Area 25 Surface Area Now we define the surface area of a general parametric surface given by Equation 1 For simplicity we start by considering a surface whose parameter domain D is a rectangle and we divide it into subrectangles Rij 26 Surface Area Let s choose See Figure 14 to be the lower left corner of Rij The image of the subrectangle Rij is the patch Sij Figure 14 27 Surface Area The part Sij of the surface that corresponds to the subrectangle Rij is called a patch and it has the point Pij with position vector as one of its corners Let and be the tangent vectors at Pij as given by Equations 5 and …
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