切平面与法线 求切平面和法线
\u66f2\u7ebf\u7684\u5207\u5e73\u9762\u4e0e\u6cd5\u7ebf\u6c42\u504f\u5bfc
zx=2x
zy=6y
\u6240\u4ee5,(1,1,3)\u5904\u7684\u6cd5\u5411\u91cf\u4e3a
(zx,zy,-1)=(2,4,-1)
\u5207\u5e73\u9762\u65b9\u7a0b\u4e3a
2(x-1)+4(x-1)-(x-3)=0
\u5373\u4e3a
2x+4y-z-3=0
\u6cd5\u7ebf\u65b9\u7a0b\u4e3a
(x-1)/2=(y-1)/4=(z-3)/(-1)
\u6c42\u504f\u5bfc
zx=2x
zy=6y
\u6240\u4ee5,(1,1,3)\u5904\u7684\u6cd5\u5411\u91cf\u4e3a
(zx,zy,-1)=(2,4,-1)
\u5207\u5e73\u9762\u65b9\u7a0b\u4e3a
2(x-1)+4(x-1)-(x-3)=0
\u5373\u4e3a
2x+4y-z-3=0
\u6cd5\u7ebf\u65b9\u7a0b\u4e3a
(x-1)/2=(y-1)/4=(z-3)/(-1)
(水平面与其交线是椭圆,竖平面若与其相交则交线是抛物线);
令 F=x²/4+y²/9 -z,该曲面的法线方向数可表示为 {∂F/∂x,∂F/∂y,∂F/∂z} 即 {x/2,2y/9,-1};
点 (-2,-3,2) 处的切平面法线方向数:{-1,-2/3,-1} (或{1,2/3,1});
指定点处切平面方程:(x+2)+2(y+3)/3+(z-2)=0,化简后:3x+2y+3z+6=0;
对应法线方程:(x+2)=3(y+3)/2=(z-2);
绛旓細绌洪棿鏇查潰鐨鍒囧钩闈㈠拰娉曠嚎.璁剧┖闂存洸闈㈢殑鏂圭▼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,z0).鍒囧钩闈㈡柟绋嬶細Fx(x0,y0,...
绛旓細闅愬嚱鏁扮殑鍑犱綍搴旂敤锛氬垏绾裤佹硶骞抽潰銆娉曠嚎涓庡垏骞抽潰 1. 鍒囩嚎锛氬湪鏇茬嚎涓婏紝涓ょ偣闂寸殑鍓茬嚎鏂圭▼鐪嬩技绠鍗曪紝瀹為檯涓婅暣鍚潃娣卞埢鍚箟銆傛棤闄愭帴杩戝湴瑙傚療鏇茬嚎锛屾垜浠細鎵惧埌閭f潯涓庢洸绾夸翰瀵嗘帴瑙︾殑鍒囩嚎銆傚垏绾跨殑鏂滅巼锛屽嵆鍒囧悜閲忥紝鏄嚱鏁板湪璇ョ偣鐨勫彉鍖栫巼銆2. 娉曞钩闈細浠讳綍绌胯繃娉曞钩闈㈢殑鐩寸嚎閮戒笌鍒囩嚎姝d氦銆傛硶骞抽潰鏂圭▼涓 锛屽叾涓 鏄...
绛旓細娉曠嚎涓庡垏闈㈢殑瀹氫箟</娉曠嚎锛屽氨鍍忕毊鐞冧笂鐨勭粡绾嚎锛屾槸鍨傜洿浜庢洸闈㈠湪鏌愮偣P鐨勭洿绾匡紝瀹冪殑鏂瑰悜鎸囧悜鏇查潰鐨勬渶闄″抄澶勩傚湪鏁板涓婏紝娉曞悜閲N鍙互閫氳繃鍑芥暟鐨勬搴︽潵璁$畻锛屽嵆锛歂 = ∇F(P)鑰鍒囧钩闈鍒欐槸杩欎釜娉曠嚎鍦ㄧ偣P澶勭殑鎶曞奖锛屽畠鏄笌鏇查潰鍦ㄧ偣P澶勬渶鎺ヨ繎鐨勫钩闈傚垏骞抽潰鐨勬柟绋嬪彲浠ュ啓涓猴細F(x, y, z) - F(...
绛旓細棣栧厛锛屸滃垏绾库濅竴璇嶇浉瀵瑰鏄撶悊瑙c傝嫳鏂囦腑鐨勨渢angent鈥濇剰鍛崇潃鎭板ソ鎺ヨ繎銆傝嚦浜庘娉曠嚎鈥濓紝鍦ㄨ嫳鏂囦腑瀹冩槸鈥渘ormal鈥濄傚師鏈紝鈥渘ormal鈥濊〃绀鸿鑼冦佹瑙勩佹甯告垨鏍囧噯锛屼緥濡傚笀鑼冨ぇ瀛︾殑鑻辨枃鍚嶁渘ormal university鈥濆嵆浣跨敤浜嗏渘ormal鈥濈殑鏈剰銆傜劧鑰岋紝鈥渘ormal鈥濆悗鏉ュ彂灞曞嚭浜嗏滄浜も鍜鈥滃瀭鐩粹濈殑鍚箟銆傛垜鎺ㄦ祴杩欎釜鍚嶇О鐨...
绛旓細1銆佷簩娆℃洸闈㈣繃鍦ㄧ偣澶勭殑鍒囧钩闈鍙娉曠嚎鏂圭▼濡備笅锛歠(x锛寉锛寊) = x^2+2y^2+3z^2-36锛屽垯 fx ' = 2x = 2锛宖y ' = 4y = 8锛宖z ' = 6z = 18锛屽垏骞抽潰鏂圭▼涓 2(x-1)+8(y-2)+18(z-3) = 0锛屾硶绾挎柟绋嬩负 (x-1)/2 = (y-2)/8 = (z-3)/18 銆2銆佸垏骞抽潰鍙婃硶绾挎柟绋嬭绠...
绛旓細锛堟按骞抽潰涓鍏朵氦绾挎槸妞渾锛岀珫骞抽潰鑻ヤ笌鍏剁浉浜ゅ垯浜ょ嚎鏄姏鐗╃嚎锛夛紱浠 F=x²/4+y²/9 -z锛岃鏇查潰鐨勬硶绾挎柟鍚戞暟鍙〃绀轰负 {∂F/∂x,∂F/∂y,∂F/∂z} 鍗 {x/2,2y/9,-1}锛涚偣 (-2,-3,2) 澶勭殑鍒囧钩闈㈡硶绾鏂瑰悜鏁帮細{-1,-2/3,-1} 锛...
绛旓細涓ょ鐨勬柟绋嬪叕寮忓涓嬶細1銆鍒囧钩闈鏂圭▼=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銆娉曠嚎鏂圭▼=-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...
绛旓細姹傛洸闈㈢殑鍒囧钩闈㈡柟绋嬪拰娉曠嚎鏂圭▼鏄痭=(x/2锛2y锛2z/9)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垚鐨勮建杩癸紝褰㈡垚鏇查潰鐨勫姩绾跨О涓烘瘝绾匡紝姣嶇嚎鍦ㄦ洸闈腑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洸闈㈡槸鐩寸嚎鎴栨洸绾垮湪涓瀹氱害鏉熸潯浠朵笅鐨勮繍鍔ㄨ建杩癸紝杩欐牴杩愬姩鐨勭洿绾挎垨鏇茬嚎...
绛旓細杩欎釜闂娑夊強鍒扳滄硶鈥鍜鈥滃垏鈥濈殑鏈剰锛屾垜涓嶆槸寰堢‘瀹氬畠浠殑鏉ュ巻锛屼絾鍙互鍋氬嚭鍑犱釜鐚滄祴銆傜煡涔庝笂鏈夌被浼肩殑闂锛 娉曠嚎涓轰粈涔堝彨娉曠嚎锛熼鍏堝叧浜庘滃垏 *鈥濓紝鐩稿濂界悊瑙o紝tangent *锛屽彲浠ョ湅鍋氬垰濂芥帴杩戠殑鎰忔濄傚叧浜庘滄硶*鈥濓紝鑻辨枃鏄 normal *锛屾湰鎰忔槸锛氳鑼冿紝姝h锛屾甯革紝鏍囧噯銆傚儚甯堣寖澶у鐨勮嫳鏂囧彨鍋 normal ...
绛旓細锛堟按骞抽潰涓鍏朵氦绾挎槸妞渾锛岀珫骞抽潰鑻ヤ笌鍏剁浉浜ゅ垯浜ょ嚎鏄姏鐗╃嚎锛夛紱浠 F=x²/4+y²/9 -z锛岃鏇查潰鐨勬硶绾挎柟鍚戞暟鍙〃绀轰负 {∂F/∂x,∂F/∂y,∂F/∂z} 鍗 {x/2,2y/9,-1}锛涚偣 (-2,-3,2) 澶勭殑鍒囧钩闈㈡硶绾鏂瑰悜鏁帮細{-1,-2/3,-1} 锛...