求函数u=xyz在点M(3,4,5)处沿锥面z^2=x^2+y^2外法线方向的方向导数。 求函数u=xyz在点M(3,4,5)处沿锥面z^2=x^2+...
z=ln\u221a(x^2+y^2),\u6c42 z\u5728\u70b9(1,1)\u5904\u6cbf\u66f2\u7ebfx^2+y^2=2\u5916\u6cd5\u5411\u7684\u65b9\u5411\u5bfc\u6570\u5148\u6c42\u5207\u7ebf\u7684\u65b9\u5411\u5411\u91cf\uff0c\u66f2\u7ebf\u65b9\u7a0b\u5199\u4e3a\uff1af(x,y)=y²-x=0fx=-1\uff0cfy=2y\uff0c\u5219\u5207\u7ebf\u65b9\u5411\u5411\u91cf\u4e3a\uff1a(-1,2y)\uff0c\u5c06(1,1)\u4ee3\u5165\u5f97\uff1a(-1,2)\uff0c\u5355\u4f4d\u5316(-1/\u221a5,2/\u221a5)\uff0c\u5373cos\u03b1=-1/\u221a5,cos\u03b2=2/\u221a5\u3002
\u5bf9y\u6c42\u504f\u5bfc\uff1a\u044dz/\u044dy=2y/(1+x^2+y^2)\uff0c\u5219dz=2x/(1+x^2+y^2)dx+2y/(1+x^2+y^2)dy\uff0c\u5c06x=1,y=2\u5e26\u5165\uff0c\u5f97\u5230dz=1/3dx+2/3dy\uff0c\u044d\u4e3a\u504f\u5bfc\u7b26\u53f7\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u6ce8\u610f\u4e8b\u9879\uff1a
1\u3001\u5bf9\u4e8e\u591a\u53d8\u91cf\u51fd\u6570\uff0c\u81ea\u53d8\u91cf\u6709\u591a\u4e2a\uff0c\u8868\u793a\u81ea\u53d8\u91cf\u7684\u70b9\u5728\u4e00\u4e2a\u533a\u57df\u5185\u53d8\u52a8\uff0c\u4e0d\u4ec5\u53ef\u4ee5\u79fb\u52a8\u8ddd\u79bb\uff0c\u800c\u4e14\u53ef\u4ee5\u6309\u4efb\u610f\u7684\u65b9\u5411\u6765\u79fb\u52a8\u540c\u4e00\u6bb5\u8ddd\u79bb\u3002\u56e0\u6b64\u51fd\u6570\u7684\u53d8\u5316\u4e0d\u4ec5\u4e0e\u79fb\u52a8\u7684\u8ddd\u79bb\u6709\u5173\uff0c\u800c\u4e14\u4e0e\u79fb\u52a8\u7684\u65b9\u5411\u6709\u5173\u3002\u56e0\u6b64\u51fd\u6570\u7684\u53d8\u5316\u7387\u662f\u4e0e\u65b9\u5411\u6709\u5173\u7684\u3002\u8fd9\u4e5f\u624d\u6709\u4e86\u65b9\u5411\u5bfc\u6570\u7684\u5b9a\u4e49\uff0c\u5373\u67d0\u4e00\u70b9\u5728\u67d0\u4e00\u8d8b\u8fd1\u65b9\u5411\u4e0a\u7684\u5bfc\u6570\u503c\u3002
2\u3001\u5bf9\u4e8e\u5355\u53d8\u91cf\u51fd\u6570\uff0c\u81ea\u53d8\u91cf\u53ea\u6709\u4e00\u4e2a\uff0c\u5f53x\u8d8b\u8fd1\u4e8ex0\u65f6\u53ea\u80fd\u5728\u76f4\u7ebf\u4e0a\u53d8\u52a8\uff0c\u79fb\u52a8\u7684\u65b9\u5411\u53ea\u6709\u5de6\u53f3\u4e24\u65b9\u3002
3\u3001\u5f53L\u4e0en\u540c\u5411\u65f6\uff0c\u4fbf\u53d6\u5f97\u6700\u5927\u503c|n|\uff0c\u6211\u4eec\u79f0n\u4e3au\u5728\u8be5\u70b9\u7684\u68af\u5ea6\u3002\u53ef\u4ee5\u770b\u5230\u68af\u5ea6\u5373\u662f\u67d0\u4e00\u70b9\u6700\u5927\u7684\u65b9\u5411\u5bfc\u6570\uff0c\u6cbf\u68af\u5ea6\u65b9\u5411\u51fd\u6570\u6709\u6700\u5927\u7684\u53d8\u5316\u7387\uff08\u6b63\u5411\u589e\u52a0\uff0c\u9006\u5411\u51cf\u5c11\uff09\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u65b9\u5411\u5bfc\u6570
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u5916\u6cd5\u5411\u91cf
z^2=x^2+y^2
\u4ee4
F\uff08x,y,z\uff09=x²+y²-z²
Fx=2x
Fy=2y
Fz=-2z
\u6240\u4ee5
\u6cd5\u5411\u91cf\u4e3a\uff1an=1/(5\u221a2)(3,4,-5) (\u56e0\u4e3a\u5916\u6cd5\u7ebf)
ux=yz=20
uy=xz=15,
uz=xy=12
\u6240\u4ee5
\u65b9\u5411\u5bfc\u6570\u4e3a\uff1a1/(5\u221a2)\u00d7\u30103\u00d720+4\u00d715-5\u00d712\u3011=12/(\u221a2)=6\u221a2\u3002
则球面的法向量为（fx,fy,fz)=（2x,2y,2z)
fx
表示f对x的偏导
则在点m(0,0,1)处球面的法向量（0，0，2）
则与这个法向量方向相同的单位向量为（0，0，1）
这个方向导数为
偏u/偏l=1*0+1抚梗掂妓郾幻淀潍丢璃*0+1*1=1
绛旓細Fz=-2z 鎵浠 娉曞悜閲忎负锛歯=1/(5鈭2)(3,4,-5) (鍥犱负澶栨硶绾)ux=yz=20 uy=xz=15,uz=xy=12 鎵浠 鏂瑰悜瀵兼暟涓猴細1/(5鈭2)脳銆3脳20+4脳15-5脳12銆=12/(鈭2)=6鈭2銆
绛旓細fx 琛ㄧずf瀵箈鐨勫亸瀵 鍒欏湪鐐筸(0,0,1)澶勭悆闈㈢殑娉曞悜閲忥紙0锛0锛2锛夊垯涓庤繖涓硶鍚戦噺鏂瑰悜鐩稿悓鐨勫崟浣嶅悜閲忎负锛0锛0锛1锛夎繖涓柟鍚戝鏁颁负 鍋弖/鍋弆=1*0+1鎶氭鎺傚閮惧够娣娼嶄涪鐠*0+1*1=1
绛旓細鏂瑰悜瀵兼暟 ∂u/∂L = (∂u/∂x)cosa + (∂u/∂y)cosb + (∂u/∂z)cosc = yzcosa + xzcosb + xycosc 銆 ∂u(1,1,1)/∂L = cosa + cosb + cosc.gradu(x,y,z) = (∂u/∂x)i + (∂...
绛旓細鑻ユ湁涓嶆噦璇疯拷闂,濡傛灉瑙ｅ喅闂璇风偣涓嬮潰鐨勨滈変负婊℃剰绛旀鈥.绗竴姝ラ鍏堟眰鏇查潰2z-xy=0鍦(2,3,3)澶勭殑娉曞悜閲忚F(x,y,z)=2z-xy,鍒欐硶鍚戦噺涓猴細(Fx,Fy,Fz)=(-y,-x,2)鐢变簬鎴戜滑闇瑕佺殑鏄悜涓嬬殑娉曞悜閲,鍥犳绗笁涓垎閲忛渶涓鸿礋,鍒欏悜...
绛旓細U=xy(1-x-y)浠=x+y 鍒欐湁t²=(x+y)²>=4xy xy<=t²/4 U=xy(1-t)<=(t²/4)(1-t)=-t³/4+(t²/4)浠'=-3t²/4+t/2=0 瑙ｅ緱,t=0,t=2/3 缁忔楠,褰搕=0鏃禪鍙栨瀬灏忓,褰搕=2/3鏃,U鍙栨瀬澶у,姝ゆ瀬澶у煎嵆鏄渶澶у U(...
绛旓細u鍦(1锛1锛1)澶勫x,y,z鐨勫亸瀵兼暟閮芥槸1 璇ユ柟鍚戠殑鍗曚綅鍚戦噺鏄(2/鈭14锛-1鈭14锛3/鈭14)鎵浠ヨ鏂瑰悜鐨勬柟鍚戝鏁版槸1脳2/鈭14-1脳1/鈭14锛1脳3鈭14=(2鈭14)/7
绛旓細鍦u=xyz鏃讹紝鏄剧劧寰楀埌 鈻絝=(f'x,f'y,f'z)=(yz,xz,xy)浠ｅ叆x=5,y=1,z=2锛屽嵆涓(2,10,5)鑰屽崟浣嶆柟鍚戝悜閲弉=(4,3,12)/鏍瑰彿(4^2+3^2+12^2)=(4,3,12)/13 浜庢槸浜岃呯浉涔(2,10,5)(4,3,12)/13 寰楀埌鏂瑰悜瀵兼暟涓98/13
绛旓細娉曞悜閲忎负 (2x,2y,-1)=(2,2,-1)e=(2/3,2/3,-1/3)ux=yz=2 uy=xz=2 uz=xy=1 鎵浠 鏂瑰悜瀵兼暟=2脳2/3+2脳2/3-1脳1/3 =7/3
绛旓細5)/鏍瑰彿(1^2+2^2+5^2)=(1,2,5)/鏍瑰彿30 鈻絝=(fx,fy,fz)=(yz,xz,xy)|(x=1,y=-1,z=0)=(0,0,-1)鎵浠ユ柟鍚戝=(0,0,-1)路(1,2,5)/鏍瑰彿30 =-5/鏍瑰彿30 =-鏍瑰彿30/6 鍑芥暟鍦鏌愮偣鐨勬搴︽槸杩欐牱涓涓悜閲忥紝鏂瑰悜涓庡彇寰楁渶澶ф柟鍚戝鏁扮殑鏂瑰悜涓鑷达紝鑰屾ā涓烘柟鍚戝鏁扮殑鏈澶у笺
绛旓細z=x^2-y^2 n=锛-2x,2y,1锛墊(1,2,-3)=(-2,4,1)ux=yz|(1,2,-3)=-6 uy=xz|(1,2,-3)=-3 uz=xy|(1,2,-3)=2 鎵浠 鏂瑰悜瀵兼暟au/an=-2脳(-6)+4脳(-3)+1脳2=12-12+2=2
