Cateian Cylindial ˆ= xˆ o + yˆin ˆ = ˆ = xˆin+ yˆ o + = Cylindial to Cateian: ˆ ˆ x o in 0 y in o 0 = 0 0 Cateian to Cylindial: + = + xˆ yˆ o in 0 x = in o 0 y 0 0 Cylindial Spheial ˆ = ˆ + ˆ = ˆ = xˆ ˆ = ˆ = xˆin+ yˆo = ˆ Spheial to Cylindial: ˆ ˆ ˆ 0 = 0 0 o 0 Cylindial to Spheial: ˆ ˆ 0 0 = 0 0 Cateian Spheial ˆ = xˆo+ yˆin + ˆ = ˆ = xˆo+ yˆin + ˆ = ˆ = xˆin+ yˆo = + Spheial to Cateian: ˆ x o o in y = in in o 0 Cateian to Spheial: xˆ ˆ yˆ ˆ o in x o in = y in o 0 Definition and Theoem ( on a path) ( fom a point) Diegene ds di = 0 whee i the olume enloed by loed ufae S. Cul ˆ dl ul = 0 whee i the ufae enloed by loed ontou C and ˆ i the nomal eto of. Diegene theoem : d= ds Stoke ' theoem : ds = dl ( /7 )
正交曲線座標系 (Othogonal Cuilinea Coodinate) The ale fato in diffeent oodinate ytem: Cateian : h = h = h = x y Cylindial : h = h = h = Spheial : h = h = h = in Gadient: ( 梯度 ) dl = uˆdl + uˆ dl + uˆ dl dl = h du 3 3 = uˆ h du + uˆ h du + uˆ h du 3 3 3 ( ) ( ) ( ) ( ) 3 dl ( ) h du ( ) h du ( ) h du [ ] ψ = uˆ ψ + uˆ ψ + uˆ ψ 3 k k k ψi = ψ + ψ + ψ 3 3 3 ψ ψ ψ ψidl = dψ = du + du + du [ B] u u u 3 3 ψ ψ ψ = = = = k = 3 k u h u h u [ ] [ B] ( ψ) h ( ψ) ( ψ) P P ψ ψ ψ ψ = uˆ + uˆ + uˆ 3 h u h u h3 u3 dl = PP dψ = ψ ( P) ψ ( P) = ψψ P P P ψidl = dψ = ψ = ψ ψ P k k ( ) ( ) ( ) ψ = uˆ ψ + uˆ ψ + uˆ ψ 3 3 ψ ψ ψ = uˆ + uˆ + uˆ 3 h u h u h3 u3 ψ ψ ψ ψ = xˆ + yˆ + x y ψ ˆ ψ ψ ψ = ˆ + + ˆ ψ ψ ˆ ψ ψ = + + ˆ whee uˆ ˆ ˆ + u + u3 u u u 3. ψ: ) 指向 ψ 變化最大的方向 ) 由小指向大 3) 垂直於等位面 4) 方向導數 ( dietional deiatie) dψ ψ uˆ = du ( /7 )
Diegene: ( 散度 ) ds di = 0 從一封閉點區域 向量場 所散發出的總通量 向外為正 沿 u 方向的淨流出通量為 : = ( h h ) ( hh) ( h h ) ψ ψ 3 flux = ψ+ ψ+ du = du = dududu3 u u u whee ψ = dl dl = h h du du = dl dl dl = h h h du du du. 3 3 3 3 3 3 flux hh h u 由對稱性知 : 3 3 flux flux hh h u 3 3 = = 3 ( hh ) 3 3 3 ( h h ) ( hh ) ( hh ) flux + flux + flux hh h3 u u u3 ( h h3) ( hh 3 ) ( hh 3 ) = + + 3 u u u 3 3 3 3 3 = = + + Diegene in diffeent oodinate ytem: x y x y = + + ( ) ( ) ( ) = + + ( ) ( ) ( ) = + + u ( 3/7 )
Cul: ( 旋度 ) ˆ dl ul = 0 圍繞一小片區域邊緣 C 向量場 所產生的總環流量 並以該小片區域的法向量為其方向 法向量的產生規則依右手螺旋決定 方向在 uˆ 的淨環流量為 : ( ) 3 ( h ) ( h) ( h ) ( h) f f du du du du dudu u u u u = ˆ = ˆ 3 hh dudu h 3 = uˆ 3 3 u u whee ˆ= uˆ f = dl f = dl = dl dl = h h du du. 3 = huˆ huˆ huˆ 3 3 u u u3 3 h h h 3 3 Cul in diffeent oodinate ytem: = xˆ yˆ x y x y ˆ ˆ = = in ˆ ˆ ( 4/7 )
重要公式 diegene ul gadient dg = 0 通過封閉表面的總通量 d = dl 共有邊界兩兩互相抵消 C ψ = 0 ψ環繞小片區域的總環流量 ψ dl = dψ ψ dl = dψ = ψ ( ) ψ ( ) = 0 C 起點起點 C ψ = ψ Laplae ' Equation ψ ψ ψ ψ = uˆ ˆ ˆ + u + u3 h u h u h3 u3 = + + u u u ( hh) ( hh) ( hh) 3 3 3 3 3 hh 3ψ hh 3 ψ hh ψ ψ = + + 3 u h u u h u u3 h3 u3 ψ ψ ψ x y ψ = + + ψ ψ ψ = + + ψ = + + ψ ψ ψ ψ ψ ψ ψ = + + = ψ ψ ψ + + in ( 5/7 )
* 常用向量恆等式 :( 在正交座標系統中均成立 ) f = f ( u u u3) g = ( u u u3) ala ( u u u3) ( x y ) ( ) ( ) = ( u u u ) B = B( u u u ) eto 3 3 () ( f + g) = f + g + B = + B () ( ) + B = + B (3) ( ) (4) ( f g) = g f + f g f = f + f (5) ( ) f = f + f (6) ( ) (7) ( B ) = ( ) B+ ( B ) + ( B) + B ( ) B = B B (8) ( ) B B B B B (9) ( ) = ( ) ( ) + ( ) ( ) whee ( V ) = V + V + V3 V = B. u u u (0) = 0 () ( ) = ( ) 3 ; f = 0 ( 以 dg 記憶 ) = uˆ + uˆ ˆ + u3 3 = uˆ + + + uˆ + + + uˆ + + () B C = B C B C = C B B C 3 3 u u u3 u u u3 u u u3 (3) ( ) ( ) ( ) ( 6/7 )
Diegene theoem : Stoke ' theoem : d = d d = d 利用散度定理 上述 (5) 式 及 的定義 可得下列的結果 : Let = f g d = d ( f g) d = ( f g) d ( ) f g = f g+ f g = f g+ f g ( f g) d = f g+ f gd Geen ' fit identity 利用對稱性 有 = + ( g f ) d g f g f d 把上式減去此式 ( f gg f ) d = f gg f d Geen ' eond identity Fom Stoke ' theoem : d = d let =< P x y Q x y > d = dxdy ( ) ( ) 0 ˆ Q P = ; d = Pdx+ Qdy x y Q P dxdy = ( P dx + Q dy ) G een ' theoem x y Fom Geen theoem aboe let P = y Q = x ( ) ( ) dxdy = y dx + x dy = = ( xdy ydx) ( 利用線積分求所環繞平面內部區域的面積 ) ( 7/7 )