矢均分析:矢量之间运算的定义 (A×B)·(C×D A1B2-A2B1)e3+(A3B1-A1B3)e2+(A2B3-A3B2)e1 C1D2-C2D1)3+(C3D1-C1D3)e2+(C2D3-C3D2)1 19/384 A1B2-A2B1)(C1D2-C2D1)+(A3B1-A1B3)(C3D1-C1D3) +(A2B3-A3B2)(C2D3-C3D2) A,C B2D2+ A2 C2B,D1+ A3C3B,D1+AICIB3D3 +A2 C2B3D3+ A3C3B2D2-AlD,B2 C2-A2D2B1C A3D3B1CI-AID,B3 C3- A2D2B3C3-A3D3B2C =(A1C1+A2C2+A3C3)B1D1+B2D2+B3D3) (A1D1+A2D2+A3D3)(B1C1+B2C2+B3C3) A.C(B·D)-(A·D)(B·C (A×B)·(AxB)=(AA)(B.B)-(A.B)(B·A) 空间反射 A轴矢均:A 空间反射 极矢均:A 极矢量x极矢量=轴矢量 ose
19/384 JJ II J I Back Close ➙þ➞Û: ➙þ❷♠✩➂✛➼➶ (A~ × B~ ) · (C~ × D~ ) = [(A1B2 − A2B1)~e3 + (A3B1 − A1B3)~e2 + (A2B3 − A3B2)~e1] ·[(C1D2 − C2D1)~e3 + (C3D1 − C1D3)~e2 + (C2D3 − C3D2)~e1] = (A1B2−A2B1)(C1D2−C2D1) + (A3B1−A1B3)(C3D1−C1D3) +(A2B3 − A3B2)(C2D3 − C3D2) = A1C1B2D2 + A2C2B1D1 + A3C3B1D1 + A1C1B3D3 +A2C2B3D3 + A3C3B2D2 − A1D1B2C2 − A2D2B1C1 −A3D3B1C1 − A1D1B3C3 − A2D2B3C3 − A3D3B2C2 = (A1C1 + A2C2 + A3C3)(B1D1 + B2D2 + B3D3) −(A1D1 + A2D2 + A3D3)(B1C1 + B2C2 + B3C3) = (A~ · C~ )(B~ · D~ ) − (A~ · D~ )(B~ · C~ ) (A~ × B~ ) · (A~ × B~ ) = (A~ · A~ )(B~ · B~ ) − (A~ · B~ )(B~ · A~ ) ✹➙þ➭A~ ➌♠❻✓ − − − → −A~ ➯➙þ➭A~ ➌♠❻✓ − − − → A~ ✹➙þ×✹➙þ=➯➙þ
矢量分析:矢量之间运算的定义 放反角位于坐标原点的三角形,其从原点出发的两条边构成矢量 cos 6- Xi r=团=√x2+x2+x3 20/384 r=vxi 第三条边构成的矢量 R=r-r=∑(x1-x)e R·e;x1 COS R ose
20/384 JJ II J I Back Close ➙þ➞Û: ➙þ❷♠✩➂✛➼➶ ➌❻✍➔✉❿■✝✿✛♥✍✴➜Ù❧✝✿Ñ✉✛ü❫❃✟↕➙þ ~r = X 3 i=1 xi~ei cos θi = xi r r ≡ |~r| = p x 2 1 + x 2 2 + x 2 3 ~r 0 = X 3 i=1 x 0 i ~ei cos θ 0 i = x 0 i r 0 r 0 ≡ |r~0 | = q x 0 1 2 + x 0 2 2 + x 0 3 2 ✶♥❫❃✟↕✛➙þ R~ = ~r −~r 0 = X 3 i=1 (xi − x 0 i )~ei ✁ ✁ ✁ ✁ ✁✕ ✏✏✏✏✏✏✏✏ P✏✶ PP PP P✐ ~r ~r 0 R~ cos φi = R~ · ~ei R = xi − x 0 i R R = vuutX 3 i=1 (xi − x 0 i )(xi − x 0 i )
矢量分析:矢量之间运算的定义 三个内角分别记为a(,r);B(r,R);7(r,R) 21/384 sIn a (x-B) Rr R siny sin3sin7|7×r R r R2=(F-r).r-r-rr+rr-2rrr2+r4-2rr'cos a +门)、+=R.R++2R,产=B2-2r2c‖4 (r-R).(F-R)=rT+RR-2rR=r2+R2-2rR 4 ose
21/384 JJ II J I Back Close ➙þ➞Û: ➙þ❷♠✩➂✛➼➶ ♥❻❙✍➞❖P➃α(~r,~r 0 ); β(~r 0 , R~ ); γ(~r, R~ ) ✁ ✁ ✁ ✁ ✁✕ ✏✏✏✏✏✏✏✏ P✏✶ PP PP P✐ ~r ~r 0 R~ α γ β sin α = |~r ×~r 0 | rr0 sin(π − β) = |R~ ×~r 0 | Rr0 = |~r ×~r 0 | Rr0 sin γ = |R~ ×~r| Rr = |~r ×~r 0 | Rr sin α R = sin β r = sin γ r 0 = |~r × ~r0 | rr0R R2 = (~r −~r 0 ) · (~r −~r 0 ) = ~r · ~r +~r 0 · ~r 0 − 2~r · ~r 0= r 2 + r 02 − 2rr0 cos α r 2 = (R~ +~r 0 ) · (R~ +~r 0 ) = R~ · R~ +~r 0 · ~r 0+2R~ · ~r 0= R2+r 02−2rr0 cos β r 02 = (~r − R~ ) · (~r − R~ ) = ~r · ~r + R~ · R~ − 2~r · R~ = r 2 + R2 − 2rR cos γ
矢册分析:张量推广 理维空开中的射阶张册:T=∑T 22/384 die. e ei 1 IA=A. I 阶张册:∑T1… ose
22/384 JJ II J I Back Close ➙þ➞Û: Üþí✷ ♥➅➌♠➙✛✓✣Üþ: ** T ≡ X 3 i,j=1 Tij~ei~ej ** I ≡ X 3 i,j=1 δij~ei~ej = X 3 i=1 ~ei~ei ** I ·A~ = A~ · ** I = A~ n✣Üþ: X 3 i1,i2,...,in=1 Ti1,i2,...,in ~ei1 ~ei2 · · ·~ein
S±T S;ee;±)Tiee Si±Ti)e;e; 张量推广 ∑A)=∑ 23/384 A.T=∑A)·∑T)=∑T;A=∑AT ij, k=1 A Tie;e) Aker TiAl k Eikleiel i、j,k,l=1 T=∑A)x①∑T)=∑TAkx6 ∑ Ak TiCkle ose
23/384 JJ II J I Back Close ** Üþí✷ S ± ** T = X 3 i,j=1 Sij~ei~ej ± X 3 i,j=1 Tij~ei~ej = X 3 i,j=1 (Sij ± Tij)~ei~ej ** T ·A~ = (X 3 i,j=1 Tij~ei~ej) · ( X 3 k=1 Ak~ek) = X 3 i,j,k=1 TijAk~ei(~ej · ~ek) = X 3 i,j=1 TijAj~ei A~ · ** T = (X 3 k=1 Ak~ek) · ( X 3 i,j=1 Tij~ei~ej) = X 3 i,j,k=1 TijAk(~ek · ~ei)~ej = X 3 i,j=1 AiTij~ej ** T ×A~ = (X 3 i,j=1 Tij~ei~ej) × ( X 3 k=1 Ak~ek) = X 3 i,j,k=1 TijAk~ei(~ej × ~ek) = X 3 i,j,k,l=1 TijAkjkl~ei~el A~ × ** T = (X 3 k=1 Ak~ek) × ( X 3 i,j=1 Tij~ei~ej) = X 3 i,j,k=1 TijAk(~ek × ~ei)~ej = X 3 i,j,k,l=1 AkTijkil~el~ej