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Parametric Surfaces and Their Areas Learning Goals Parametric Surfaces Tangent Planes Surface Area Review Math 213 - Parametric Surfaces and their Areas Peter A. Perry University of Kentucky November 22, 2019 Peter A. Perry University of Kentucky Math 213 - Parametric Surfaces and their Areas Learning Goals Parametric Surfaces Tangent Planes Surface Area Review Reminders • Homework D1 is due on Friday of this week • Homework D2 is due on Monday of next week • Thanksgiving is coming! Peter A. Perry University of Kentucky Math 213 - Parametric Surfaces and their Areas Learning Goals Parametric Surfaces Tangent Planes Surface Area Review Unit IV: Vector Calculus Fundamental Theorem for Line Integrals Green’s Theorem Curl and Divergence Parametric Surfaces and their Areas Surface Integrals Stokes’ Theorem, I Stokes’ Theorem, II The Divergence Theorem Review Review Review Peter A. Perry University of Kentucky Math 213 - Parametric Surfaces and their Areas Learning Goals Parametric Surfaces Tangent Planes Surface Area Review Goals of the Day This lecture is about parametric surfaces. You’ll learn: • How to define and visualize parametric surfaces • How to find the tangent plane to a parametric surface at a point • How to compute the surface area of a parametric surface using double integrals Peter A. Perry University of Kentucky Math 213 - Parametric Surfaces and their Areas Learning Goals Parametric Surfaces Tangent Planes Surface Area Review Parametric Curves and Parametric Surfaces Parametric Curve Parametric Surface A parametric curve in R3 is given by A parametric surface in R3 is given by r(t) = x(t)i + y(t)j + z(t)k r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k where a ≤ t ≤ b where (u, v) lie in a region D of the uv plane. There is one parameter, because a curve is a There are two parameters, because a surface is one-dimensional object a two-dimensional object There are three component functions, because There are three component functions, because the curve lives in three-dimensional space. the surface lives in three-dimensional space. Peter A. Perry University of Kentucky Math 213 - Parametric Surfaces and their Areas Learning Goals Parametric Surfaces Tangent Planes Surface Area Review You Are Living on a Parametric Surface Let u be your longitude (in radians, for this course) Let v be your latitude (in radians) Let R be the radius of the Earth Your position is r(u, v) = R cos(v) cos(u)i + R cos(v) sin(u)j + R sin(v)k v p/2 u 0 2p p/2 Peter A. Perry University of Kentucky Math 213 - Parametric Surfaces and their Areas Solution: Let r0 = h1, 0, 1i. Any point in the plane is given by r(s, t) = h1, 0, 1i + sh2, 0, 1i + th0, 2, 0i Now you try it: Find a parametric representation for the plane through the point (0, −1, 5) that con- tains the vectors h2, 1, 4i and h−3, 2, 5i. Learning Goals Parametric Surfaces Tangent Planes Surface Area Review More Parameterized Surfaces: Planes z Problem: Find a parametric representation for the plane through h1, 0, 1i that contains the vectors h2, 0, 1i and h0, 2, 0i y x Peter A. Perry University of Kentucky Math 213 - Parametric Surfaces and their Areas Now you try it: Find a parametric representation for the plane through the point (0, −1, 5) that con- tains the vectors h2, 1, 4i and h−3, 2, 5i. Learning Goals Parametric Surfaces Tangent Planes Surface Area Review More Parameterized Surfaces: Planes z Problem: Find a parametric representation for the plane through h1, 0, 1i that contains the vectors h2, 0, 1i and h0, 2, 0i Solution: Let r0 = h1, 0, 1i. Any point in the plane is given by r(s, t) = h1, 0, 1i + sh2, 0, 1i + th0, 2, 0i y x Peter A. Perry University of Kentucky Math 213 - Parametric Surfaces and their Areas Learning Goals Parametric Surfaces Tangent Planes Surface Area Review More Parameterized Surfaces: Planes z Problem: Find a parametric representation for the plane through h1, 0, 1i that contains the vectors h2, 0, 1i and h0, 2, 0i Solution: Let r0 = h1, 0, 1i. Any point in the plane is given by r(s, t) = h1, 0, 1i + sh2, 0, 1i + th0, 2, 0i y Now you try it: Find a parametric representation for the x plane through the point (0, −1, 5) that con- tains the vectors h2, 1, 4i and h−3, 2, 5i. Peter A. Perry University of Kentucky Math 213 - Parametric Surfaces and their Areas If we fix v and vary u over the cylinder, we trace out a circle If we fix u and vary v, we trace out a vertical line Each of these curves has a tangent vector: ru(u, v)= −r sin(u)i + r cos(u)j rv(u, v)= k Learning Goals Parametric Surfaces Tangent Planes Surface Area Review More Parameterized Surfaces: The Cylinder v r(u, v) = r cos(u)i + r sin(u)j + vk D = f(u, v) : 0 ≤ u ≤ 2p, 0 ≤ v ≤ hg parameterizes a cylinder of radius r and height h u Peter A. Perry University of Kentucky Math 213 - Parametric Surfaces and their Areas If we fix u and vary v, we trace out a vertical line Each of these curves has a tangent vector: ru(u, v)= −r sin(u)i + r cos(u)j rv(u, v)= k Learning Goals Parametric Surfaces Tangent Planes Surface Area Review More Parameterized Surfaces: The Cylinder v r(u, v) = r cos(u)i + r sin(u)j + vk D = f(u, v) : 0 ≤ u ≤ 2p, 0 ≤ v ≤ hg parameterizes a cylinder of radius r and height h u If we fix v and vary u over the cylinder, we trace out a circle Peter A. Perry University of Kentucky Math 213 - Parametric Surfaces and their Areas Each of these curves has a tangent vector: ru(u, v)= −r sin(u)i + r cos(u)j rv(u, v)= k Learning Goals Parametric Surfaces Tangent Planes Surface Area Review More Parameterized Surfaces: The Cylinder v r(u, v) = r cos(u)i + r sin(u)j + vk D = f(u, v) : 0 ≤ u ≤ 2p, 0 ≤ v ≤ hg parameterizes a cylinder of radius r and height h u If we fix v and vary u over the cylinder, we trace out a circle If we fix u and vary v, we trace out a vertical line Peter A. Perry University of Kentucky Math 213 - Parametric Surfaces and their Areas Learning Goals Parametric Surfaces Tangent Planes Surface Area Review More Parameterized Surfaces: The Cylinder v r(u, v) = r cos(u)i + r sin(u)j + vk D = f(u, v) : 0 ≤ u ≤ 2p, 0 ≤ v ≤ hg parameterizes a cylinder of radius r and height h u If we fix v and vary u over the cylinder, we trace out a circle If we fix u and vary v, we trace out a vertical line Each of these curves has a tangent vector: ru(u, v)= −r sin(u)i + r cos(u)j rv(u, v)= k Peter A. Perry University of Kentucky Math 213 - Parametric Surfaces and their Areas Learning Goals Parametric Surfaces Tangent Planes Surface Area Review More Parameterized Surfaces: The Cylinder v r(u, v) = r cos(u)i + r sin(u)j + vk D = f(u, v) : 0 ≤ u ≤ 2p, 0 ≤ v ≤ hg parameterizes a cylinder of radius r and height h u The two tangent vectors ru(u, v)= −r sin(u)i + r cos(u)j rv(u, v)= k span the tangent plane to the cylinder at the given point Peter A. Perry University of Kentucky Math 213 - Parametric Surfaces and their Areas Learning Goals Parametric Surfaces Tangent Planes Surface Area Review The Tangent Vectors ru and rv Suppose r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k, (u, v) 2 D is a parameterized surface. At a point P0 = r(u0, v0), the vectors ¶x ¶y ¶z r (u , v ) = (u , v )i + (u , v )j + (u , v )k u 0 0 ¶u 0 0 ¶u 0 0 ¶u 0 0 ¶x ¶y ¶z r (u , v ) = (u , v )i + (u , v )j + (u , v )k v 0 0 ¶v 0 0 ¶v 0 0 ¶v 0 0 are both tangent to the surface. Peter A. Perry University of Kentucky Math 213 - Parametric Surfaces and their Areas Learning Goals Parametric Surfaces Tangent Planes Surface Area Review The Tangent Plane ¶x ¶y ¶z r (u , v ) = (u , v )i + (u , v )j + (u , v )k u 0 0 ¶u 0 0 ¶u 0 0 ¶u 0 0 ¶x ¶y ¶z r (u , v ) = (u , v )i + (u , v )j + (u , v )k v 0 0 ¶v 0 0 ¶v 0 0 ¶v 0 0 The tangent plane to a parameterized surface at P0 = r(u0, v0) is the plane passing through P0 and perpendicular to ru(u0, v0) × rv(u0, v0). Find the equation of the tangent plane to the surface r(u, v) = u2i + 2u sin vj + u cos vk at u = 1, v = 0. Peter A. Perry University of Kentucky Math 213 - Parametric Surfaces and their Areas r(u, v) = hu2, 2u sin v, u cos vi ru(u, v) = h2u, 2 sin v, cos vi rv(u, v) = h0, 2u cos v, −u sin vi r(1, 0) = h1, 0, 1i ru(1, 0) = h2, 0, 1i rv(1, 0) = h0, 2, 0i The normal to the plane is ru × rv = h−1, 0, 2i so the equation of the plane is (−1)(x − 1) + 2(z − 1) = 0 The tangent plane to the surface at (1, 0, 1) is parameterized by h1 + 2s, 2t, 1 + si Learning Goals Parametric Surfaces Tangent Planes Surface Area Review The Tangent Plane 2 0 2 0 −2 0 2 −2 4 Peter A.
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