《数学分析》课程教学课件（讲稿，打印版）第9章 多变量函数的微分学（8/8）

‡©/ªm ‡©/ª È ‡© Poincar´e Ún §9.8 ‡©/ª  V ⊂ Rn ´‡«, f : V → R ´‡Œ‡¼ê, K f 3: x = (x1, x2, · · · , xn) ‡©´ df(x) = ∂f ∂x1 (x)dx1 + ∂f ∂x2 (x)dx2 · · · + ∂f ∂xn (x)dxn, §´ dx1, dx2, · · · , dxn ‚5|Ü, X괃AuˆCþ ê. éu‰½ V þ¼ê A1(x), A2(x), · · · , An(x), ‚5|Ü ω = A1(x)dx1 + A2(x)dx2 + · · · + An(x)dxn (9.1) ؽ´,¼ê‡©.  ïļꇩ, ·‚‡ïÄ/X (9.1) ‡© /ª. ·‚r¤k/X (9.1) ‡©/ªŠ‡N(m)5ïÄ, ‡ww 3ù‡m¥´ÄŒ±½Â$Ž. 1/13 kJ Ik J I ˆ£ ¶ '4 òÑ

‡©/ªm ‡©/ª È ‡© Poincar´e Ún aqu‚5m, ÄkŒ±½Âê¦Ú\{$Ž.  λ(x) ´ V þ¼ ê, ω1 = A1(x)dx1 + A2(x)dx2 + · · · + An(x)dxn ω2 = B1(x)dx1 + B2(x)dx2 + · · · + Bn(x)dxn, ½Âê¦$Ž: λ(x)ω1 = λ(x)A1(x)dx1 + λ(x)A2(x)dx2 + · · · + λ(x)An(x)dxn, Ú\{$Ž: ω1+ω2 = (A1(x)+B1(x))dx1+(A2(x)+B2(x))dx1+· · ·+(An(x)+Bn(x))dx1. w, Uìê¦Ú\{$Ž/X (9.1) ‡©/ªN´‡‚5m. UØU½Â‡©/ª¦{Q? ù‡w dx1, dx2, · · · , dxn ƒmUĽ ¦È. 2/13 kJ Ik J I ˆ£ ¶ '4 òÑ

‡©/ªm ‡©/ª È ‡© Poincar´e Ún 9.8.1 ‡©/ªm  {üå„, ?Øn‘mœ¹, = V ⊂ R3 .  f : V → R ´ ‡nŒ‡¼ê, K f ‡©´ df = ∂f ∂xdx + ∂f ∂ydy + ∂f ∂zdz.  ½Â‡©/ª¦È, ½Â dx ∧ dx = dy ∧ dy = dz ∧ dz = 0, dx ∧ dy = −dy ∧ dx, dy ∧ dz = −dz ∧ dy, dz ∧ dx = −dx ∧ dz ¡ dx ∧ dy  dx † dy  È, ¿5½ È÷v(ÜÇÚ©Ç. lþ¡ ½Âw ȏ÷v‡é¡5. dx ∧ dy ∧ dz ½ÂÐ . Kk (dx ∧ dy ∧ dz) ∧ dx = 0. 3/13 kJ Ik J I ˆ£ ¶ '4 òÑ

‡©/ªm ‡©/ª È ‡© Poincar´e Ún ć©/ª$Žƒ￾, ‡©/ªŒ±©Aa: "g‡©/ª: V þŒ‡¼êN. g‡©/ª: /X Adx + Bdy + Cdz, g‡©/ª: /X Ady ∧ dz + Bdz ∧ dx + Cdx ∧ dy, ng‡©/ª: /X Ddx ∧ dy ∧ dz, Ù¥ A, B, C, D ´ V þŒ‡¼ê. ˆa‡©/ªÑ/¤‚5m. ØÓa‡©/ªØ½Â\{. 4/13 kJ Ik J I ˆ£ ¶ '4 òÑ

‡©/ªm ‡©/ª È ‡© Poincar´e Ún 9.8.2 ‡©/ª È 1 ◦ ü‡g‡©/ª È.  λ1 = A1dx + B1dy + C1dz, λ2 = A2dx + B2dy + C2dz, Kk λ1 ∧ λ2 = (A1dx + B1dy + C1dz) ∧ (A2dx + B2dy + C2dz) = A1B2dx ∧ dy + A1C2dx ∧ dz + B1A2dy ∧ dx + B1C2dy ∧ dz + C1A2dz ∧ dx + C1B2dz ∧ dy = (B1C2 − C1B2)dy ∧ dz + (C1A2 − A1C2)dz ∧ dx + (A1B2 − B1A2)dx ∧ dy =         dy ∧ dz dz ∧ dx dx ∧ dy A1 B1 C1 A2 B2 C2         . 5/13 kJ Ik J I ˆ£ ¶ '4 òÑ

‡©/ªm ‡©/ª È ‡© Poincar´e Ún 2 ◦ n‡g‡©/ª È.  λ1 = A1dx + B1dy + C1dz, λ2 = A2dx + B2dy + C2dz, λ3 = A3dx + B3dy + C3dz. λ1 ∧ λ2 = (B1C2 − C1B2)dy ∧ dz + (C1A2 − A1C2)dz ∧ dx + (A1B2 − B1A2)dx ∧ dy, λ1 ∧ λ2 ∧ λ3 =  A3(B1C2 − C1B2) + B3(C1A2 − A1C2) + C3(A1B2 − B1A2)  dx ∧ dy ∧ dz =  (A1, B1, C1) × (A2, B2, C2) · (A3, B3, C3)  dx ∧ dy ∧ dz =         A1 B1 C1 A2 B2 C2 A3 B3 C3         dx ∧ dy ∧ dz. 6/13 kJ Ik J I ˆ£ ¶ '4 òÑ

‡©/ªm ‡©/ª È ‡© Poincar´e Ún AO,  u, v, w ∈ C1 (V ) ž, k du = ∂u ∂xdx + ∂u ∂ydy + ∂u ∂zdz dv = ∂v ∂xdx + ∂v ∂ydy + ∂v ∂zdz dw = ∂w ∂x dx + ∂w ∂y dy + ∂w ∂z dz, Ïd du ∧ dv ∧ dw = ∂(u,v,w) ∂(x,y,z) dx ∧ dy ∧ dz. 3 ◦ g‡©/ª†g‡©/ª È.  λ = Adx + Bdy + Cdz, ω = P dy ∧ dz + Qdz ∧ dx + Rdx ∧ dy, Kk λ ∧ ω = (AP + BQ + CR)dx ∧ dy ∧ dz 7/13 kJ Ik J I ˆ£ ¶ '4 òÑ

‡©/ªm ‡©/ª È ‡© Poincar´e Ún 9.8.3 ‡©/ª ‡© ‡©/ª ω  ‡©´l ω ' ω pg‡©/ªN: d : ω → dω. äN½ÂXe: 1 ◦  f ´"g‡©/ª, =, f ∈ C1 (V ), K df = ∂f ∂xdx + ∂f ∂ydy + ∂f ∂zdz. 2 ◦  ω ´g‡©/ª, ω = P dx + Qdy + Rdz, P, Q, R ∈ C 1 (V ), 8/13 kJ Ik J I ˆ£ ¶ '4 òÑ

‡©/ªm ‡©/ª È ‡© Poincar´e Ún K dω = dP ∧ dx + dQ ∧ dy + dR ∧ dz =  ∂P ∂x dx + ∂P ∂y dy + ∂P ∂z dz ∧ dx +  ∂Q ∂x dx + ∂Q ∂y dy + ∂Q ∂z dz ∧ dy +  ∂R ∂x dx + ∂R ∂y dy + ∂R ∂z dz ∧ dz =  ∂R ∂y − ∂Q ∂z  dy ∧ dz + ￾ ∂P ∂z − ∂R ∂x
dz ∧ dx +  ∂Q ∂x − ∂P ∂y  dx ∧ dy =         dy ∧ dz dz ∧ dx dx ∧ dy ∂ ∂x ∂ ∂y ∂ ∂z P Q R         . 9/13 kJ Ik J I ˆ£ ¶ '4 òÑ

‡©/ªm ‡©/ª È ‡© Poincar´e Ún 3 ◦  ω ´g‡©/ª, ω = P dy ∧ dz + Qdz ∧ dx + Rdx ∧ dy, P, Q, R ∈ C 1 (V ), K dω = dP ∧ dy ∧ dz + dQ ∧ dz ∧ dx + dR ∧ dx ∧ dy =  ∂P ∂x dx + ∂P ∂y dy + ∂P ∂z dz ∧ dy ∧ dz +  ∂Q ∂x dx + ∂Q ∂y dy + ∂Q ∂z dz ∧ dz ∧ dx +  ∂R ∂x dx + ∂R ∂y dy + ∂R ∂z dz ∧ dx ∧ dy =  ∂P ∂x + ∂Q ∂y + ∂R ∂z  dx ∧ dy ∧ dz. 4 ◦  ω ´ng‡©/ª, ω = P dx ∧ dy ∧ dz, P ∈ C 1 (V ), K dω = dP ∧ dx ∧ dy ∧ dz = 0. 10/13 kJ Ik J I ˆ£ ¶ '4 òÑ