Copyright © Cengage Learning. All rights reserved. 16 Vector CalculusCopyright © Cengage Learning. All rights reserved. 16.6 Parametric Surfaces and Their Areas3 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 function4 Parametric Surfaces5 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.6 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 such that x = x(u, v) 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!7 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. Figure 1 A parametric surface8 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.9 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 410 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.11 Example Find a parametric representation of the sphere x2 + y2 + z2 = a2 Solution: The sphere has a simple representation ρ = a in spherical coordinates, so let’s choose the angles φ and θ in spherical coordinates as the parameters.12 Example – Solution Then, putting ρ = a in the equations for conversion from spherical to rectangular coordinates, we obtain x = a sin φ cos θ y = a sin φ sin θ z = a cos φ as the parametric equations of the sphere. The corresponding vector equation is r(φ, θ ) = a sin φ cos θ i + a sin φ sin θ j + a cos φ k continued13 Example – Solution We have 0 ≤ φ ≤ π and 0 ≤ θ ≤ 2π, so the parameter domain is the rectangle D = [0, π] × [0, 2π]. The grid curves with φ constant are the circles of constant latitude (including the equator). continued14 Example – Solution The grid curves with θ constant are the meridians (semi-circles), which connect the north and south poles (see Figure 7). Figure 7 continued15 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.16 Surfaces of Revolution17 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 θ be the angle of rotation as shown in Figure 10. Figure 1018 Surfaces of Revolution If (x, y, z) is a point on S, then x = x y = f (x) cos θ z = f (x) sin θ Therefore we take x and θ as parameters and regard Equations 3 as parametric equations of S. The parameter domain is given by a ≤ x ≤ b, 0 ≤ θ ≤ 2π .19 Example Find parametric equations for the surface generated by rotating the curve y = sin x, 0 ≤ x ≤ 2π, 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 θ z = sin x sin θ and the parameter domain is 0 ≤ x ≤ 2π , 0 ≤ θ ≤ 2π.20 Example – Solution Using a computer to plot these equations and rotate the image, we obtain the graph in Figure 11. Figure 11 continued21 Tangent Planes22 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).23 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 1224 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.25 Surface Area26 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.27 Surface Area Let’s choose to be the lower left corner of Rij. (See Figure 14.) Figure 14 The image of the subrectangle Rij is the patch Sij.28 Surface Area The part Sij of the surface that corresponds to …
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