曲面方程的切平面方程问题 高数，求曲面在某点处的切平面方程
\u7b2c\u5341\u9898\u6c42\u8be6\u89e3\u8fc7\u7a0b\uff0c\u66f2\u9762\u5e73\u884c\u4e8e\u5e73\u9762\u7684\u5207\u5e73\u9762\u65b9\u7a0b\u3002\uff1f\uff1f\uff1f\uff1f\u8bbe\u66f2\u9762\u65b9\u7a0b\u4e3a F\uff08X\uff0cY\uff0cZ\uff09\u3002
\u5176\u5bf9X Y Z\u7684\u504f\u5bfc\u5206\u522b\u4e3a Fx\uff08X\uff0cY\uff0cZ\uff09\uff0cFy\uff08X\uff0cY\uff0cZ\uff09 \uff0cFz\uff08X\uff0cY\uff0cZ\uff09\u3002
\u5c06\u70b9\uff08a\uff0cb\uff0cc\uff09\u4ee3\u5165\u5f97 n\uff1d\uff3bFx\uff0cFy\uff0cFz\uff3d \uff08\u5207\u5e73\u9762\u6cd5\u5411\u91cf\uff09\u3002
\u518d\u5c06\u5207\u70b9\uff08a\uff0cb\uff0cc\uff09\u4ee3\u5165\u5f97\u3002
\u5207\u5e73\u9762\u65b9\u7a0bFx\uff0a\uff08X\uff0da\uff09\uff0bFy\uff0a\uff08Y\uff0db\uff09\uff0bFz\uff08Z\uff0dc\uff09\uff1d0\u3002
\uff08\u6c42\u5207\u5e73\u9762\u65b9\u7a0b\u7684\u5173\u952e\u662f\u901a\u8fc7\u6c42\u504f\u5bfc\u6570\u5f97\u5230\u5207\u5e73\u9762\u6cd5\u5411\u91cf\uff09\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u5e73\u9762\u7684\u6cd5\u5411\u91cfN\uff0c N \uff1d \uff08A\uff0c B\uff0c C\uff09\uff0c M M\u7684\u4efb\u610f\u4e24\u70b9\u7684\u98de\u673a\uff0c\u6709N\u3002\u6beb\u7c73\uff1d 0\uff0cMM\u201d\uff1d \uff08x0 x \uff0d y \uff0d y0\uff0c z \uff0d z0\uff09\uff0c\u4e00\u4e9b\u6cd5\u56fd\u5e73\u9762\u65b9\u7a0b\uff1aB \uff08x \uff0d x0\uff09 \uff0b \uff08y \uff0d y0\uff09 \uff0b C \uff08z \uff0d z0\uff09 \uff1d 0\u3002
\u4e09\u4e2a\u70b9\u7ec4\u6210\u7684\u5e73\u9762\u53ef\u4ee5\u7528\u65b9\u5411\u7684\u4e58\u79ef\u6765\u5f52\u4e00\u5316\u3002
\u4efb\u4f55\u4e09\u5143\u7ebf\u6027\u65b9\u7a0b\u7684\u56fe\u5f62\u90fd\u662f\u4e00\u4e2a\u5e73\u9762\uff0c\u5176\u4e2dx\uff0cy\uff0c z\u7684\u7cfb\u6570\u90fd\u662f\u5e73\u9762\u6cd5\u5411\u91cf\u7684\u5750\u6807\u3002
\u5b83\u4eec\u76f8\u4e92\u5782\u76f4\u5b83\u4eec\u5c31\u7b49\u4e8eA1A2 \uff0b B1B2 \uff0b C1C2 \uff1d 0\u3002
\u5e73\u884c\u6216\u91cd\u53e0\u7684\u5e73\u9762\u4e5f\u5c31\u662fA1\uff0fA2\uff1dB1\uff0fB2\uff1dC1\uff0fC2\u3002
\u70b9\u5230\u5e73\u9762\u7684\u8ddd\u79bb\uff1dabs\uff08Ax0\uff0bBy0\uff0bCz0\uff0bD\uff09\uff0f\u221a\uff08A\u65b9\uff0bB\u65b9\uff0bC\u65b9\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u5e73\u9762\u65b9\u7a0b
则 f 对 x、y、z 的偏导数分别为 2y、2x、1-e^z ，
由此得点（1，2，0）处的切平面的法向量为 n=（4，2，0），
因此切平面方程为 4(x-1)+2(y-2)=0 ，即 2x+y-4=0 ，
法线方程为 (x-1)/4=(y-2)/2=(z-0)/0 。
绛旓細鏇查潰鐨勫垏骞抽潰鐨勬柟绋嬫槸Fx锛圶-a锛+Fy锛圷-b锛+Fz锛圸-c锛=0锛屾眰鍒囧钩闈㈡柟绋嬬殑鍏抽敭鏄氳繃姹傚亸瀵兼暟寰楀埌鍒囧钩闈㈡硶鍚戦噺锛屾洸闈㈠彲浠ョ湅浣滄槸涓鏉″姩绾垮湪绌洪棿杩炵画杩愬姩鎵褰㈡垚鐨勮建杩广傛瘝绾垮湪鏇查潰涓鐨勪换涓浣嶇疆绉颁负鏇查潰鐨勭礌绾匡紝鐢ㄦ潵鎺у埗姣嶇嚎杩愬姩鐨勯潰銆佺嚎鍜岀偣绉颁负瀵奸潰銆佸绾垮拰瀵肩偣銆傛瘝绾胯繍鍔ㄦ椂鎵鍙楃殑绾︽潫锛岀О涓鸿繍鍔...
绛旓細瀵逛簬鏇查潰鍦ㄦ煇鐐鐨勫垏骞抽潰鍜屾硶绾鏂圭▼鐨姹傝В锛屽彲浠ラ噰鍙栦互涓嬫楠わ細1銆侀鍏堬紝璁惧畾鏇查潰鐨勬柟绋涓簓^2+z^2=2x銆傝嫢浠ヨ鏂圭▼涓哄熀纭锛屽洿缁昘杞存棆杞竴鍛紝鎵褰㈡垚鐨勬棆杞鏇查潰鏂圭▼涓篎=0锛寉=0銆傚悓鐞嗭紝鍥寸粫Z杞存棆杞竴鍛紝鎵褰㈡垚鐨勬棆杞洸闈㈡柟绋嬩负F=卤鈭(0)銆2銆佸湪鏃嬭浆杩囩▼涓紝鍥哄畾涓涓彉閲忥紝鑰屽皢鍙︿竴涓彉閲忕殑骞...
绛旓細鍥犳鍒囧钩闈㈡柟绋涓 4(x-1)+2(y-2)=0 锛屽嵆 2x+y-4=0 锛屾硶绾挎柟绋嬩负 (x-1)/4=(y-2)/2=(z-0)/0 銆
绛旓細璁鏇查潰鏂圭▼涓 F锛圶锛孻锛孼锛夈傚叾瀵筙 Y Z鐨勫亸瀵煎垎鍒负 Fx锛圶锛孻锛孼锛夛紝Fy锛圶锛孻锛孼锛 锛孎z锛圶锛孻锛孼锛夈傚皢鐐癸紙a锛宐锛宑锛変唬鍏ュ緱 n锛濓蓟Fx锛孎y锛孎z锛 锛堝垏骞抽潰娉曞悜閲忥級銆傚啀灏嗗垏鐐癸紙a锛宐锛宑锛変唬鍏ュ緱銆傚垏骞抽潰鏂圭▼Fx锛婏紙X锛峚锛夛紜Fy锛婏紙Y锛峛锛夛紜Fz锛圸锛峜锛夛紳0銆傦紙姹鍒囧钩闈㈡柟绋嬬殑...
绛旓細璇烽棶鎮ㄩ棶鐨勬槸鈥鏇查潰鐨勫垏骞抽潰鍜屾硶绾挎柟绋嬧濇槸浠涔堣繖涓闂鍚楋紵涓ょ鐨勬柟绋嬪叕寮忓涓嬶細1銆鍒囧钩闈㈡柟绋=F_x锛坸0锛寉0锛寊0锛*锛坸-x0锛+F_y锛坸0锛寉0锛寊0锛*锛坹-y0锛+F_z锛坸0锛寉0锛寊0锛*锛坺-z0锛=0銆傚叾涓紝F_x锛孎_y锛孎_z鏄疐瀵箈锛寉锛寊鐨勫亸瀵兼暟銆2銆佹硶绾挎柟绋=-F_x锛坸0锛寉0锛寊0锛*锛...
绛旓細鏇查潰鐨勫垏骞抽潰鏂圭▼鍜屾硶绾挎柟绋嬪涓嬶細绌洪棿鏇查潰鐨勫垏骞抽潰鍜屾硶绾.璁剧┖闂鏇查潰鐨勬柟绋涓 锛孎(x,y,z)=0锛岃岃孧(x0,y0,z0)鏄洸闈⑽ｄ笂鐨勪竴鐐.娉曞悜閲忥細(Fx(x0,y0,z0),Fy(x0,y0,z0),Fz(x0,y0,z0)).娉曠嚎鏂圭▼锛歺−x0Fx(x0,y0,z0)=y−y0Fy(x0,y0,z0)=z−z0Fz(x0,y0...
绛旓細鏇插钩闈㈠湪鐐瑰鐨勫垏骞抽潰鏂圭▼ 璁鏇查潰鏂圭▼涓 F锛圶,Y,Z锛夊叾瀵筙 Y Z鐨勫亸瀵煎垎鍒负 Fx锛圶,Y,Z锛,Fy锛圶,Y,Z锛 ,Fz锛圶,Y,Z锛夊皢鐐癸紙a锛宐锛宑锛変唬鍏ュ緱 n=[Fx,Fy,Fz] 锛堝垏骞抽潰娉曞悜閲忥級鍐嶅皢鍒囩偣锛坅锛宐锛宑锛変唬鍏ュ緱 鍒囧钩闈㈡柟绋婩x*锛圶-a锛+Fy*锛圷-b锛+Fz锛圸-c锛=0 锛堟眰鍒囧钩闈...
绛旓細姹傝В鏇查潰鐨勫垏骞抽潰鏂圭▼鐨鏂规硶 1銆佺‘瀹氭洸闈㈢殑绫诲瀷锛氶鍏堬紝鎴戜滑闇瑕佺‘瀹氱粰瀹氱殑鏇查潰鏄粈涔堢被鍨嬬殑鏇查潰銆傚父瑙佺殑鏇查潰绫诲瀷鏈夌悆闈佸渾鏌遍潰銆佸渾閿ラ潰绛夈備笉鍚岀被鍨嬬殑鏇查潰鏈変笉鍚岀殑鍒囧钩闈㈡柟绋嬨2銆佸紩鍏ュ弬鏁帮細瀵逛簬鏌愪簺鏇查潰锛屾垜浠彲鑳介渶瑕佸紩鍏ヤ竴浜涘弬鏁版潵鎻忚堪鍏跺舰鐘躲備緥濡傦紝鐞冮潰鐨勫垏骞抽潰鏂圭▼鍙互閫氳繃鍙傛暟r琛ㄧず銆傚渾鏌遍潰鐨勫垏骞抽潰...
绛旓細绠鍗曞垎鏋愪竴涓嬶紝璇︽儏濡傚浘鎵绀
绛旓細绠鍗曞垎鏋愪竴涓嬶紝绛旀濡傚浘鎵绀
