北邮高等数学英文版课件Lecture103.pptx

1、Section 10.312OverviewCURVExOyzr(),(),()P x ty tz tr()()i()j()ktx ty tz txyzSURFACE(,)0F x y z 1)Tangent line and normal plane2)Tangent planes and normal lines3The Parametric Equations of a Space CurveWe already know that a plane curve can be represented by a parametric0r()ra,R,tttby a parametric eq

2、uations,a line in space can be expressedequations(),(),()xx tyy ttor 000,xxltyymttzznt of the variable point P(x,y,z).r(,)x y z where is the position vectorLxyzO0rra4The Parametric Equations of a Space CurveSimilarly,a space curve may also be represented by parametricequations(),(),(),(),xx tyy tzz

3、ttr()(),(),()().tx ty tz ttor vector formxOyzris continuousr()tIf the vector valued function then is said to be a ,on the interval continuous curve;If is a continuous curve andholds for any 12r()r()tt and12,(,)t t 12,tt,then is said to be a simple curve.5The tangent line to The geometric meaning of

4、the derivative of the direction vector r(t)at t0 is that r(t0)is the direction vector of the tangent to the curve at the corresponding point P0.r(t0)is called the tangent vector to the curve at P0.:r()(),(),()tx ty tz tP0OxyzT0()r t0()r t The Vector equation of the tangent to the curve at P0 is00()(

5、)r ttr t 6The equation of the tangent line to curve 00()()r ttr t The Vector equation:The Parametric equation:000000()(),()(),()().x txtx ty tyty tz tztz t The Symmetric equation:000000()()()xxyyzzx ty tz t0()0r t 7The tangent line to A curve for which the direction of the tangent varies continuousl

6、y is called a smooth curve.0()0r t 322:r()(,)tttExample1:r()(cos,sin)tttOxy2yOx1piecewise smooth curve8The normal plane to We have seen that for a given space curve if r(t)is derivable at t0 and r(t0)0,then the tangent to at P0 exists and is unique.There is an infinite number of straight lines throu

7、gh the point P0,which are perpendicular to the tangent and lie in the same plane.The plane is called the normal plane to the curve at P0.through the point P0 perpendicular to the tangentthe equation of the normal plane9The normal plane to The equation of the normal plane to the curve at P0 is000000(

8、)()()()()()0 x txx ty tyy tz tzz t Example Find the equations of the tangent line and the normal plane to the following curve at point t=1.22:r()(,2,).tttt10Tangent line and normal plane to a space curveIf the equations of the curve is given in the general form(,)0,:(,)0,F x y zG x y z and the above

9、 equations of the curve determine two implicit functions of one variable x,y=y(x)and z=z(x)in the neighbourhood U(P0)and both y(x)and z(x)have continuous derivative.Thenthe symmetric equation of the tangent at P0(x0,y0,z0)is:000001xxxxyyzzdydzdxdx11Tangent line and normal plane to a space curveand t

10、he equation of the normal plane at P0(x0,y0,z0)is:00000()()()0 xxdydzxxyyzzdxdx Example Find the equations of the tangent line and the normal plane to the curve at point P0(-2,1,6).22222245,2xyzxyz 122.Tangent planes and normal lines of surfacesOyxz0000(,)P xy zNormal lineTangent plane13Parametrizin

11、g OyzxrAny space point can be imagined thatit lies on a sphere which is centered at(,)P x y zthe origin and the radius is 222.xyzIf the angle between the projection vector on the xOy plane and the positive OP of direction of x-axis is denoted by,and and the positive direction of z-axisOP the angle b

12、etween the vector is denoted bysincos,sinsin,cos,02,0.xryrzrthen the two coordinate system are related by,14Parametrizing Oyzxr(,)P x y zIf we denote222.xyrthe surface of the angle between the projection vector OP of on the xOy plane and the positive direction of x-axis is denoted by,and 220.rxyThen

13、 the coordinate canbe expressed by cos,sin,02.xryrzzlies onAnother way to parametrize is imagine that any point(,)P x y z(,)P x y zis also a point of a space curve or a space surface,then If we can parametrize the equation of the curve or surface.15Tangent Planes and Normal Lines to a SurfaceSuppose

14、 that the parametric equation of a surface S is 2rr(,)(,),(,),(,),(,)Ru vx u vy u v z u vu vDand the partial where r is continuous in D,the point00(,)u vD 0000r(,)r(,).uvu vu v 0000(,)r(,),vu vxyzu vvvv 0000(,)r(,),uu vxyzu vuuu exist,that is,derivatives of r at the point 00(,)u v00(,)u v,then the r

15、(,)u vwe can prove that if is differentiable at the point tangent plane of any smooth curve on the surface through the point r0,00(,)u vwith normal vector must lie in the plane which pass through is called a regular point).and (in this case,00(,)u v0000r(,)r(,)0uvu vu v16Tangent Planes and Normal Li

16、nes to a Surface 00r(,)u vrurvrruv xyOzS Therefore,the normal vector is0000r(,)r(,)uvu vu v 00(,)(,)(,)(,),(,)(,)(,)u vy zz xx yu vu vu v 00(,)ijkuuuvvvu vxyzxyz def ,.A B CThus the tangent plane is 0000.A xxB yyC zzThe normal line is000.xxyyzzABC17Tangent Planes and Normal Lines to a Surface Exampl

17、e Find the tangent plane and normal line to the right helicoid where the constant cos,sin,(0)xuvyuvazav at the point 2,.4uv 18Tangent Planes and Normal Lines to a Surface derivatives of F are all continuous and the vector ,0,xyzF F F say 0.zF which is determined by(,)zz x y Then,there exists a funct

18、ion,if all the first order partialIf the surface S is expressed by(,)0F x y z Thus,the surface(,)0F x y z and has continuous partial derivative.S can be repressed by r(,)(,(,)x yx y z x y It is easy to see that r(0,1,)(0,1,/),yyyzzFFr(1,0,)(1,0,/),xxxzzFFthen we haven(,).xyzF F F orrr(/,/,1)xyxzyzFF

19、 FF19Tangent Planes and Normal Lines to a Surface0(,)A surfaceF x y z 000int,(,),xyzPFor any poPa normal vector at P isFF F 000000000000 0 (,)(,()()()()()(),),.xyzxyzPSo the tangent plane at P to the surface isthF F Fxxyy zF PxxF PyyFatzzsziPThe normal line is000000.()()()xyzxxyyzzFPFPF P20Tangent P

20、lanes and Normal Lines to a Surface Example Given an ellipsoid2225:22Sxyzand a plane:40,xyz 1)Find the tangent plane to the ellipsoid at the point P(x0,y0,z0)parallel to the plane.2)Find the points on the ellipsoid with minimum and maximum distance to the plane.21ReviewTangent line and normal plane to a space curve ParametrizingThe tangent plane and the norm line of a surface