Microsoft Word - CoordinatesTransform.doc

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1 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 )

2 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 dl ( ) h du ( ) h du ( ) h du [ ] ψ = uˆ ψ + uˆ ψ + uˆ ψ 3 k k k ψi = ψ + ψ + ψ 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 )

3 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 flux hh h u 由對稱性知 : 3 3 flux flux hh h u 3 3 = = 3 ( hh ) ( h h ) ( hh ) ( hh ) flux + flux + flux hh h3 u u u3 ( h h3) ( hh 3 ) ( hh 3 ) = u u u = = + + Diegene in diffeent oodinate ytem: x y x y = + + ( ) ( ) ( ) = + + ( ) ( ) ( ) = + + u ( 3/7 )

4 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 )

5 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) hh 3ψ hh 3 ψ hh ψ ψ = 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 )

6 * 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 )

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 )

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