高等数学英文版课件Lecture-10-3

Section10.3ApplicationofDifferentialCalculusofMultivariableFunctioninGeometry12OverviewCURVESURFACE1)Tangentlineandnormalplane2)Tangentplanesandnormallines3TheParametricEquationsofaSpaceCurveWealreadyknowthataplanecurvecanberepresentedbyaparametricbyaparametricequations,alineinspacecanbeexpressedequationsorofthevariablepointP(x,y,z).whereisthepositionvector4TheParametricEquationsofaSpaceCurveSimilarly,aspacecurveΓ

mayalsoberepresentedbyparametricequationsorvectorformiscontinuousIfthevectorvaluedfunctionthenΓissaidtobeaontheintervalcontinuouscurve;IfΓisacontinuouscurveandholdsforanyand,thenΓissaidtobeasimplecurve.5ThetangentlinetoΓThegeometricmeaningofthederivativeofthedirectionvectorr(t)att0isthatr′(t0)isthedirectionvectorofthetangenttothecurveΓatthecorrespondingpointP0.r′(t0)iscalledthetangentvectortothecurveΓatP0.P0OxyzTTheVectorequationofthetangenttothecurveΓatP0

is6TheequationofthetangentlinetocurveΓTheVectorequation:TheParametricequation:TheSymmetricequation:7ThetangentlinetoΓAcurveforwhichthedirectionofthetangentvariescontinuouslyiscalledasmoothcurve.ExampleOxyΓ2yOxΓ1piecewisesmoothcurve8ThenormalplanetoΓWehaveseenthatforagivenspacecurveΓ

if

r(t)isderivableatt0andr′(t0)≠0,thenthetangenttoΓatP0existsandisunique.ThereisaninfinitenumberofstraightlinesthroughthepointP0

,whichareperpendiculartothetangentandlieinthesameplane.TheplaneiscalledthenormalplanetothecurveΓatP0.

throughthepointP0

perpendiculartothetangenttheequationofthenormalplane9ThenormalplanetoΓTheequationofthenormalplanetothecurveΓatP0isExample

Find

theequationsofthetangentlineandthenormalplane

tothefollowingcurveΓatpointt=1.10TangentlineandnormalplanetoaspacecurveIftheequationsofthecurveΓisgiveninthegeneralformandtheaboveequationsofthecurveΓdeterminetwoimplicitfunctionsofonevariablex,y=y(x)andz=z(x)intheneighbourhoodU(P0)andbothy(x)andz(x)havecontinuousderivative.Thenthesymmetricequationofthe

tangent

atP0(x0,y0,z0)is:11Tangentlineandnormalplanetoaspacecurveandtheequationofthenormalplane

atP0(x0,y0,z0)is:

ExampleFindtheequationsofthetangentlineandthenormalplanetothecurveatpointP0(-2,1,6).122.TangentplanesandnormallinesofsurfacesNormallineTangentplane13ParametrizingAnyspacepointcanbeimaginedthatitliesonaspherewhichiscenteredattheoriginandtheradiusisIftheanglebetweentheprojectionvectoronthexOyplaneandthepositiveofdirectionofx-axisisdenotedbyθ,andandthepositivedirectionofz-axistheanglebetweenthevectorisdenotedbythenthetwocoordinatesystemarerelatedby14ParametrizingIfwedenotethesurfaceoftheanglebetweentheprojectionvectorofonthexOyplaneandthepositivedirectionofx-axisisdenotedbyθ,andThenthecoordinatecanbeexpressedbyliesonAnotherwaytoparametrizeisimaginethatanypointisalsoapointofaspacecurveoraspacesurface,thenIfwecanparametrizetheequationofthecurveorsurface.15TangentPlanesandNormalLinestoaSurfaceSupposethattheparametricequationofasurfaceSisandthepartialwhereriscontinuousinD,thepointexist,thatis,derivativesofratthepoint,thenthewecanprovethatifisdifferentiableatthepointtangentplaneofanysmoothcurveonthesurfacethroughthepointr0,withnormalvectormustlieintheplanewhichpassthroughiscalledaregularpoint).and(inthiscase,16TangentPlanesandNormalLinestoaSurfaceTherefore,thenormalvectorisThusthetangentplaneisThenormallineis17TangentPlanesandNormalLinestoaSurfaceExample

Findthetangentplaneandnormallinetotherighthelicoidatthepoint18TangentPlanesandNormalLinestoaSurfacederivativesofFareallcontinuousandthevectorsaywhichisdeterminedbyThen,thereexistsafunction,ifallthefirstorderpartialIfthesurfaceSisexpressedbyThus,thesurfaceandhascontinuouspartialderivative.ScanberepressedbyItiseasytoseethatthenwehaveor19TangentPlanesandNormalLinestoaSurfaceThenormallineis20TangentPlanesandNormalLinestoaSurface

ExampleGivenanellipsoidandaplane1)Findthetangentplanetotheellipsoidatthepo