高数-d86几何中应用

PAGEPAGE7一、引例:的参数方程zMryt[,Oy rxyz),f(t(t),rf(t),t[ f:[,]R3,称此 对上的动点M显然rOM即是r的终点M的轨迹,此轨迹称为向量值函数的终端曲线.ff(t0)f(tt)f(tΔ定义:给定数集DR,称 f:DRn为一元向量值函数（简称向量值函数）,记为 rf(t),t因变连续和导数密切相关,因此下面仅以n=3的情形为代表 f(t)f1t),f2t),f3t)),tD,极限：limf(tlimf1(tlimf2(tlim连续：limf(t)f(t0导数：f(tf1(t),f2(t),自变向量值函数的导数运算法则设uv是可导向量值函数C是常向量,c是任一常数(t是可导函数,(1)dCd(2)d[cu(t)]cud(3)d[u(t)v(t)]u(t)vd(4)d[(t)u(t)](t)u(t)(t)ud(5)d[u(t)v(t)]u(t)v(t)u(t)vd(6)d[u(t)v(t)]u(t)v(t)u(t)vd(7)d(t)ud向量向量值函数导数的几何意义R3中rf(ttD的终端曲线为OMf(t),ONf(tΔ0Δrf(t0Δt)f(t00f(tz ΔΔrNΔrlimΔrf(tOyΔ0xf(t00,f(t0表示终端曲线在t0处的切向量其指向与t的增长方向一致向量值函数导数的物理意义设rf(t)表示质点沿光滑曲线运动的位置向量,则有速度向量：v(t)f(t)加速度向量：av(t)f例设f(tcostisint)jtk求limflimf(t)(limcostilimsint)jlimt2i2jπk(f(π) 4例设空间曲线例设空间曲线rf(t)(t21,4t3,2t26t) t求曲线上对应于t02的点处的单位切向量. f(t)(2t,4,4t6),tRf(2)(4,4,f(2)4242 =f(,,2233tt的增长方向相反的单位切向量为221 fM处的切线为此点处割线的极限位置.M与切线垂直的平面称为曲线在该点的法平面：f(t)((t),点M(x,y,z)处的切向量及法平面的f(t)((t),(t),利TM1.1.曲线方程为参数方程的给定光滑曲线：x(t),y(t),z(t),t[]设上的点M(x0y0z0对应tt0(t0(t0(t0不全为0,则在点M的导向量为f(t0)((t0),(t0),(t0因此曲线M处切线方xx0yMf(t0(t0z(t0法平面(t0)(xx0)(t0)(yy0)(t0)(zz0)例求曲线xt,yt2,zt3M(111)处的切方程与法平面方程x1,y2t,z3t2,点(11,1对应t0故点M处的切向量为T12因此所求切线方程x1y1z 法平面方程(x1)2(y1)3(z1)思考:光滑曲:y的切向量有何特点:y切向量T1答即x2y3z2.2.曲线为一般式的光滑F(x,y,z)G(x,y,z)当JF,G)0可表示为y(x且(y,zdxJ(z, J(x,曲线上M(x0y0z0处的切向量dy1(F,G),dz1(F,G)x T1,(x),(x)1,1(F J(z,x)MJ(x,,1(FM或T(F ,(F ,(F (y, (z,x) (x,MT(F ,(F ,(F (y, (z, (x, 则在点M(x0y0z0)M切线方x yy0 zz0 (F (F (F(y,z) (z,x) (x,y)(F法平面方 (y,(xx0) (z,(F(yy0M(F(x,y)(zz0)((F,(y,z)(xx)(F0(z,x)(yy0(F,(x,M(zz)0x y zFx(M)Fy(M)Fz(M)Gx(M)Gy(M)Gz(M例x2y2z26xyz0M(1,–2,1)处的切线方程与法平面方程解法1Fx2y2z26,Gxyz(F(y,z)(F2y 2(yMM(z,(F(x,y)xMyz切向 T(6,0,6切线方程x1y2z 6xz2即y2法平法平面方 6(x1)0(y2)6(z1) xzd解法2方程x求导 dydz x1y1yzx,dzyz11xy1y曲线M(1,–21)处有切向T1,d dx,dz(1,0,M切线方x1y2z 即xz2法平面方程1x10y21z1 xzM(1,–2,1)处的切向T(1,0,光滑曲面F(xyz通过其上定点M(x0y0z0任意引一条光滑x(ty(tz(ttt0对应点M,(t0(t0(t0)不全为0在点M的切向量为TM切线方(t0)(t0 (txx0yy0z下面证明:M的任何曲线在该点在同一平面上.此平面称为在该点的切平面证x(t),y(tz(t上,F((t),(t),(t))两边在tt0处求导,注意tt0对应点M得TMT(t0(t0(t0n(Fx(x0,y0,z0),Fy(x0,y0,z0),Fz(x0,y0,z0切向量T由于表明这些切线都在以为法向量的平面上,从而切平面存在.Fx(x0,y0,z0)(t0)Fy(x0,y0,z0)(t0Fz(x0,y0,z0)(t0)n(Fx(x0,y0,z0),Fy(x0,y0,z0),Fz(x0,y0,z0Fx(x0,y0,z0)(xx0)Fy(x0,y0,z0)(yy0Fz(x0,y0,z0)(zz0)T过MnMFx(x0,y0,z0 x y zz0 Fy(x0,y0,z0 Fz(x0,y0,z0特别当光滑曲面zf(xy时,F(x,y,z)f(x,y)(xyzFxfxFyfy,Fzf(xy在点x0y0有连续偏导数时Σ在点x0y0z0法向 n(fx(x0,y0),fy(x0,y0),zz0fx(x0,y0)(xx0)fy(x0,y0)(yy0法线方程x y zfx(x0,y0 fy(x0,y0 用用,表示法向量的方向角,并假定法向量方向向上,则为锐角.法向量nfxx0,y0fyx0,y0fx(x0y0fy(x0y0分别记为fx,fy,法向量的方向cos , cos ,1fx2fy 1fx2fycos11fx2fy例求球面x2y2z214在点(123处的切解法向F(x,y,z)x2y2z2n(2x,2y,2n(1,2,3)(2,4,所以球面在点(1,2,3)处有切平面方 2(x1)4(y2)6(z3) x2y3z14法线方x1y2z 即12xy3(可见法线经过原点，即球心例例确定正数使曲面xyz 与球面x2y2za2M(x0y0z0相切解二曲M点的法向量分别n1(y0z0,x0z0,x0y0),n2(x0,y0,z0二曲面M相切,n1n2x0y0z0x0y0z0x0y0x0y0z0x02y2z M在球面上,x0y02 z03于是 x0y03旋转抛物面z2x22y24在点(1,1,0)处法线为（B(A)x1y1 x1y1 x1y1x1y11414解:由题旋转抛z2x22y24的法向量为在点(1,1,0)处法线的点向式方程x1y1 (07-08,一 平面 平面xyz3与直线x1y1z 的位置关系（D(08-09,一(A) 垂(C) 解：平面法向量(1,1,1),直线方向向量111(2110,则平面与直线平行但平面过点(1,1,1),则直线在平面上例求曲面D:ezzxy3在点(2,1,0)解令F(xyzezz(08-09,三则Fy,FFez 则点(2,1,0)则切平面方程为x22(y1)x2y4则法线方程为x2y1 例例旋转抛物面ezzxy0在点(2,1,0)处切平面为 2xy4 (B)2xyz4x2y4 2xy5解:由题旋转抛物面ezzxy0的法向(yx,e在点(2,1,0)处切平面方程为1x22y10z0 x2y4(09-10,一 求曲面x22y23z221平行于平面x4y6z的各切平面方 (09-10,三解:设(x0y0z0)为曲面上的切点，则该点的切平2x0(xx0)4y0(yy0)6z0(zz0)且由平x4y6z02x04y0 即2x0y0z0,且满足方程x2y3zx01,得切点(1,2,2),(得切平面方程2(x18y212(z2即x4y6z 或2(x18y212(z20即x4y6z 求曲面zylnx在点(1,1,1)处的切平面和法线方z解:由题曲面方程为F(x,yzylnxzz)n则切平面方程为(x1y12(z1)法线方x1y1z 1.空间曲线的切线与法平1参数式情况空间光滑曲线y切向 T((t0),(t0),(t0x切线方xx0yy0z (t0法平面(t0)(xx0)(t0)(yy0)(t0)(zz0)2一般式情况2一般式情况.F(xyzG(x,y,z)T(F(y, (z,x) (x,,(F ,(FMx yy0 z (F,(y,z)(F, (F,(z,x) (x,y)(F(y, (xx0)(z,(FMM(yy0(F(x,y)(zz0)21F(xyzn(Fx(x0,y0,z0),Fy(x0,y0,z0),Fz(x0,y0,z0Fx(x0,y0,z0)(xx0)Fy(x0,y0,z0)(yy0Fz(x0,y0,z0)(zz0) x y z Fx(x0,y0,z0 Fy(x0,y0,z0 Fz(x0,y0,z022显式情况空间光滑:zf(x法向 n(fx,fy法线的方向余弦cosx,cos1fx2fyfy 1fx2fy2cos11fx2fy切平面zz0fx(x0,y0)(xx0)fy(x0,y0)(yy0法线方xx0yy0zfx(x0,y0 fy(x0,y0 27,28,11如果平面3xy3z160与椭3x2z216相切,提示:设切点为M(x0y0z06x02y0(二法向量平行33x0y03z016 (切点在平面上3x2y2