Math 213 - Parametric Surfaces And Their Areas

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Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewMath 213 - Parametric Surfaces and theirAreasPeter A. PerryUniversity of KentuckyNovember 22, 2019Peter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewReminders Homework D1 is due on Friday of this week Homework D2 is due on Monday of next week Thanksgiving is coming!Peter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewUnit IV: Vector CalculusFundamental Theorem for Line IntegralsGreen’s TheoremCurl and DivergenceParametric Surfaces and their AreasSurface IntegralsStokes’ Theorem, IStokes’ Theorem, IIThe Divergence TheoremReviewReviewReviewPeter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewGoals of the DayThis 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 usingdouble integralsPeter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewParametric Curves and Parametric SurfacesParametric CurveA parametric curve in R3 is given byr(t) x (t)i y(t)j z(t)kParametric SurfaceA parametric surface in R3 is given byr(u, v) x (u, v)i y(u, v)j z(u, v)kwhere a t bwhere (u, v) lie in a region D of the uv plane.There is one parameter, because a curve is aone-dimensional objectThere are two parameters, because a surface isa two-dimensional objectThere are three component functions, becausethe curve lives in three-dimensional space.There are three component functions, becausethe surface lives in three-dimensional space.Peter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewYou Are Living on a Parametric SurfaceLet u be your longitude (in radians, for this course)Let v be your latitude (in radians)Let R be the radius of the EarthYour position isr(u, v) R cos(v) cos(u)i R cos(v) sin(u)j R sin(v)kvπ/202πuπ/2Peter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewMore Parameterized Surfaces: PlanesProblem: Find a parametric representationfor the plane through h1, 0, 1i that containsthe vectors h2, 0, 1i and h0, 2, 0izyxPeter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewMore Parameterized Surfaces: PlanesProblem: Find a parametric representationfor the plane through h1, 0, 1i that containsthe vectors h2, 0, 1i and h0, 2, 0izSolution: Let r0 h1, 0, 1i. Any point in theplane is given byr(s, t) h1, 0, 1i sh2, 0, 1i th0, 2, 0iyxPeter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewMore Parameterized Surfaces: PlanesProblem: Find a parametric representationfor the plane through h1, 0, 1i that containsthe vectors h2, 0, 1i and h0, 2, 0izSolution: Let r0 h1, 0, 1i. Any point in theplane is given byr(s, t) h1, 0, 1i sh2, 0, 1i th0, 2, 0iyxPeter A. PerryMath 213 - Parametric Surfaces and their AreasNow you try it:Find a parametric representation for theplane through the point (0, 1, 5) that contains the vectors h2, 1, 4i and h 3, 2, 5i.University of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewMore Parameterized Surfaces: The Cylindervr(u, v) r cos(u)i r sin(u)j vkD {(u, v) : 0 u 2π, 0 v h}uPeter A. PerryMath 213 - Parametric Surfaces and their Areasparameterizes a cylinder of radius r andheight hUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewMore Parameterized Surfaces: The Cylindervr(u, v) r cos(u)i r sin(u)j vkD {(u, v) : 0 u 2π, 0 v h}uparameterizes a cylinder of radius r andheight hIf we fix v and vary u over the cylinder, wetrace out a circlePeter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewMore Parameterized Surfaces: The Cylindervr(u, v) r cos(u)i r sin(u)j vkD {(u, v) : 0 u 2π, 0 v h}uparameterizes a cylinder of radius r andheight hIf we fix v and vary u over the cylinder, wetrace out a circleIf we fix u and vary v, we trace out a verticallinePeter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewMore Parameterized Surfaces: The Cylindervr(u, v) r cos(u)i r sin(u)j vkD {(u, v) : 0 u 2π, 0 v h}uparameterizes a cylinder of radius r andheight hIf we fix v and vary u over the cylinder, wetrace out a circleIf we fix u and vary v, we trace out a verticallineEach of these curves has a tangent vector:ru (u, v) r sin(u)i r cos(u)jrv (u, v) kPeter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewMore Parameterized Surfaces: The Cylindervr(u, v) r cos(u)i r sin(u)j vkD {(u, v) : 0 u 2π, 0 v h}uparameterizes a cylinder of radius r andheight hThe two tangent vectorsru (u, v) r sin(u)i r cos(u)jrv (u, v) kspan the tangent plane to the cylinder at thegiven pointPeter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewThe Tangent Vectors ru and rvSupposer(u, v) x (u, v)i y(u, v)j z(u, v)k,(u, v) Dis a parameterized surface.At a point P0 r(u0 , v0 ), the vectorsr u ( u0 , v0 ) x y z( u0 , v0 ) i ( u0 , v0 ) j ( u0 , v0 ) k u u ur v ( u0 , v0 ) x y z( u0 , v0 ) i ( u0 , v0 ) j ( u0 , v0 ) k v v vare both tangent to the surface.Peter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewThe Tangent Planer u ( u0 , v0 ) x y z( u0 , v0 ) i ( u0 , v0 ) j ( u0 , v0 ) k u u ur v ( u0 , v0 ) x y z( u0 , v0 ) i ( u0 , v0 ) j ( u0 , v0 ) k v v vThe tangent plane to a parameterized surface at P0 r(u0 , v0 ) is the plane passingthrough P0 and perpendicular to ru (u0 , v0 ) rv (u0 , v0 ).Find the equation of the tangent plane to the surfacer(u, v) u2 i 2u sin vj u cos vkat u 1, v 0.Peter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewThe Tangent Plane202 20024Peter A. PerryMath 213 - Parametric Surfaces and their Areas 2University of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewThe Tangent Planer(u, v) hu2 , 2u sin v, u cos viru (u, v) h2u, 2 sin v, cos virv (u, v) h0, 2u cos v, u sin vi202 20024Peter A. PerryMath 213 - Parametric Surfaces and their Areas 2University of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewThe Tangent Planer(u, v) hu2 , 2u sin v, u cos viru (u, v) h2u, 2 sin v, cos virv (u, v) h0, 2u cos v, u sin vi2r(1, 0) h1, 0, 1i02 20024Peter A. PerryMath 213 - Parametric Surfaces and their Areasru (1, 0) h2, 0, 1irv (1, 0) h0, 2, 0i 2University of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewThe Tangent Planer(u, v) hu2 , 2u sin v, u cos viru (u, v) h2u, 2 sin v, cos virv (u, v) h0, 2u cos v, u sin vi2r(1, 0) h1, 0, 1i02 20024ru (1, 0) h2, 0, 1irv (1, 0) h0, 2, 0i 2The normal to the plane isru rv h 1, 0, 2iPeter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewThe Tangent Planer(u, v) hu2 , 2u sin v, u cos viru (u, v) h2u, 2 sin v, cos virv (u, v) h0, 2u cos v, u sin vi2r(1, 0) h1, 0, 1i02 20024ru (1, 0) h2, 0, 1irv (1, 0) h0, 2, 0i 2The normal to the plane isru rv h 1, 0, 2iso the equation of the plane is( 1)( x 1) 2(z 1) 0The tangent plane to the surface at (1, 0, 1) is parameterized byh1 2s, 2t, 1 siPeter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewThe Sphere Revisitedr(u, v) sin(v) cos(u)i sin(v) sin(u)j cos(v)k0 u 2π, 0 v πru sin(v) sin(u)i sin(v) cos(u)jrv cos(v) cos(u)i cos(v) sin(u)j sin(v)kFind the tangent plane to the sphere at (u, v) (π/4, π/4)Peter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewThe Sphere Revisitedr(u, v) sin(v) cos(u)i sin(v) sin(u)j cos(v)k0 u 2π, 0 v πru sin(v) sin(u)i sin(v) cos(u)jrv cos(v) cos(u)i cos(v) sin(u)j sin(v)kFind the tangent plane to the sphere at (u, v) (π/4, π/4) 112i j k22211ru (π/4, π/4) i j22 112rv (π/4, π/4) i j k222r(π/4, π/4) Peter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewThe Sphere Revisitedr(u, v) sin(v) cos(u)i sin(v) sin(u)j cos(v)k0 u 2π, 0 v πru sin(v) sin(u)i sin(v) cos(u)jrv cos(v) cos(u)i cos(v) sin(u)j sin(v)kFind the tangent plane to the sphere at (u, v) (π/4, π/4) 112i j k22211ru (π/4, π/4) i j22 112rv (π/4, π/4) i j k222r(π/4, π/4) Peter A. PerryMath 213 - Parametric Surfaces and their Areas 11 i j k2211110 ( x ) (y )2222 2 (z )2n ru rv 12 University of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewSneak PreviewParametric Curves - Arc Lengthr(t) Parametric Surfaces - Arear(u, v) x (t)i y(t)j z(t)kr0 (t) x 0 (t)i y0 (t)j z0 (t)kr0 (t) qx (u, v)i y(u, v)j z(u, v)kru (u, v) r(u, v) urv (u, v) r(u, v) vx 0 ( t )2 y 0 ( t )2 z 0 ( t )2dA ru rv du dvds r0 (t) dtS L bZr0 (t) dtZZD ru rv du dvaPeter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewSurface AreaFind the area A of a small patch of surfacevThe map (u, v) 7 r(u, v) takes the square to aparallelogram with sides ru u and rv vThe area of the parallelogram is v ru u rv v ru rv u vu uThe area of the surface is approximatelyzA ru (ui , vi ) rv (ui , vi ) u vi,jand exactlyZZD ru (ui , vi ) rv (ui , vi ) du dvyxPeter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewSurface Area of a Spherer(u, v) a sin(v) cos(u)i a sin(v) sin(u)j a cos(v)k0 u 2π, 0 v πru a sin(v) sin(u)i a sin(v) cos(u)jrv a cos(v) cos(u)i a cos(v) sin(u)j sin(v)kru rv a2 sin2 (v) cos(u)i a2 sin2 (v) sin(u)j a2 cos(v) sin(v)k ru rv a2 sin2 (v)HenceS πZ0Peter A. PerryMath 213 - Parametric Surfaces and their Areas2πZ0a2 sin2 v du dv 4πa2University of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewSurfaces Area of a GraphThe graph of a function z f ( x, y) is also a parameterized surface:r( x, y) xi yj f ( x, y)k fk x fkry ( x, y) j yr x ( x, y) i f fi j k x ys 2 2 f f 1 x yr x ry r x ryHence, the surface area of the graph over a domain D in the xy plane iss 2 2ZZ f f1 dAS x yDPeter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewSurface Area of a GraphThe surface area of the graph over a domain D in the xy plane iss 2 2ZZ f f dAS 1 x yDFind the area of the graph of z x2 y2 that lies inside the cylinder x2 y2 4Peter A. PerryMath 213 - Parametric Surfaces and their AreasUniversity of Kentucky

Learning GoalsParametric SurfacesTangent PlanesSurface AreaReviewCurves and tionr(u, v) x (u, v)i y(u, v)jr(t) x (t)i y(t)j z(t)k z(u, v)kTangentTangentsr0 (t) x 0 (t)i y0 (t)j z0 (t)k r(u, v) u r(u, v)rv (u, v) vru (u, v) Tangent line at t aL(s) r( a) sr0 ( a)Arc length differentialNormaln ru rvArea Differentialds qx 0 (t)2 y0 (t)2 z0 (t)2 dtPeter A. PerryMath 213 - Parametric Surfaces and their AreasdA ru rv du dvUniversity of Kentucky

Learning Goals Parametric Surfaces Tangent Planes Surface Area Review Parametric Curves and Parametric Surfaces Parametric Curve A parametric curve in R3 is given by r(t) x(t)i y(t)j z(t)k where a t b There is one parameter, because a curve is a one-dimensional object There are three component functions, because the curve lives in three .

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