Differential operators in three dimensions
出典: フリー百科事典『ウィキペディア（Wikipedia）』 (2022/04/02 01:23 UTC 版)
「直交曲線座標」の記事における「Differential operators in three dimensions」の解説
詳細は「del」を参照 これらの演算は応用上共通なので、本節ではすべてのベクトル成分を正規化基底を用いて以下のように示す。 F i = F ⋅ e ^ i {\displaystyle F_{i}=\mathbf {F} \cdot {\hat {\mathbf {e} }}_{i}} . OperatorExpressionGradient of a scalar field ∇ ϕ = e ^ 1 h 1 ∂ ϕ ∂ q 1 + e ^ 2 h 2 ∂ ϕ ∂ q 2 + e ^ 3 h 3 ∂ ϕ ∂ q 3 {\displaystyle \nabla \phi ={\frac {{\hat {\mathbf {e} }}_{1}}{h_{1}}}{\frac {\partial \phi }{\partial q^{1}}}+{\frac {{\hat {\mathbf {e} }}_{2}}{h_{2}}}{\frac {\partial \phi }{\partial q^{2}}}+{\frac {{\hat {\mathbf {e} }}_{3}}{h_{3}}}{\frac {\partial \phi }{\partial q^{3}}}} Divergence of a vector field ∇ ⋅ F = 1 h 1 h 2 h 3 [ ∂ ∂ q 1 ( F 1 h 2 h 3 ) + ∂ ∂ q 2 ( F 2 h 3 h 1 ) + ∂ ∂ q 3 ( F 3 h 1 h 2 ) ] {\displaystyle \nabla \cdot \mathbf {F} ={\frac {1}{h_{1}h_{2}h_{3}}}\left[{\frac {\partial }{\partial q^{1}}}\left(F_{1}h_{2}h_{3}\right)+{\frac {\partial }{\partial q^{2}}}\left(F_{2}h_{3}h_{1}\right)+{\frac {\partial }{\partial q^{3}}}\left(F_{3}h_{1}h_{2}\right)\right]} Curl of a vector field ∇ × F = e ^ 1 h 2 h 3 [ ∂ ∂ q 2 ( h 3 F 3 ) − ∂ ∂ q 3 ( h 2 F 2 ) ] + e ^ 2 h 3 h 1 [ ∂ ∂ q 3 ( h 1 F 1 ) − ∂ ∂ q 1 ( h 3 F 3 ) ] + e ^ 3 h 1 h 2 [ ∂ ∂ q 1 ( h 2 F 2 ) − ∂ ∂ q 2 ( h 1 F 1 ) ] = 1 h 1 h 2 h 3 | h 1 e ^ 1 h 2 e ^ 2 h 3 e ^ 3 ∂ ∂ q 1 ∂ ∂ q 2 ∂ ∂ q 3 h 1 F 1 h 2 F 2 h 3 F 3 | {\displaystyle {\begin{aligned}\nabla \times \mathbf {F} &={\frac {{\hat {\mathbf {e} }}_{1}}{h_{2}h_{3}}}\left[{\frac {\partial }{\partial q^{2}}}\left(h_{3}F_{3}\right)-{\frac {\partial }{\partial q^{3}}}\left(h_{2}F_{2}\right)\right]+{\frac {{\hat {\mathbf {e} }}_{2}}{h_{3}h_{1}}}\left[{\frac {\partial }{\partial q^{3}}}\left(h_{1}F_{1}\right)-{\frac {\partial }{\partial q^{1}}}\left(h_{3}F_{3}\right)\right]\\[10pt]&+{\frac {{\hat {\mathbf {e} }}_{3}}{h_{1}h_{2}}}\left[{\frac {\partial }{\partial q^{1}}}\left(h_{2}F_{2}\right)-{\frac {\partial }{\partial q^{2}}}\left(h_{1}F_{1}\right)\right]={\frac {1}{h_{1}h_{2}h_{3}}}{\begin{vmatrix}h_{1}{\hat {\mathbf {e} }}_{1}&h_{2}{\hat {\mathbf {e} }}_{2}&h_{3}{\hat {\mathbf {e} }}_{3}\\{\dfrac {\partial }{\partial q^{1}}}&{\dfrac {\partial }{\partial q^{2}}}&{\dfrac {\partial }{\partial q^{3}}}\\h_{1}F_{1}&h_{2}F_{2}&h_{3}F_{3}\end{vmatrix}}\end{aligned}}} Laplacian of a scalar field ∇ 2 ϕ = 1 h 1 h 2 h 3 [ ∂ ∂ q 1 ( h 2 h 3 h 1 ∂ ϕ ∂ q 1 ) + ∂ ∂ q 2 ( h 3 h 1 h 2 ∂ ϕ ∂ q 2 ) + ∂ ∂ q 3 ( h 1 h 2 h 3 ∂ ϕ ∂ q 3 ) ] {\displaystyle \nabla ^{2}\phi ={\frac {1}{h_{1}h_{2}h_{3}}}\left[{\frac {\partial }{\partial q^{1}}}\left({\frac {h_{2}h_{3}}{h_{1}}}{\frac {\partial \phi }{\partial q^{1}}}\right)+{\frac {\partial }{\partial q^{2}}}\left({\frac {h_{3}h_{1}}{h_{2}}}{\frac {\partial \phi }{\partial q^{2}}}\right)+{\frac {\partial }{\partial q^{3}}}\left({\frac {h_{1}h_{2}}{h_{3}}}{\frac {\partial \phi }{\partial q^{3}}}\right)\right]} 上記の式は、レヴィ＝チヴィタ記号を用いてより簡潔に書くことができる。 ϵ i j k {\displaystyle \epsilon _{ijk}} とヤコビ行列式 J = h 1 h 2 h 3 {\displaystyle J=h_{1}h_{2}h_{3}} で、繰り返し添字に対する和を考える。 OperatorExpressionGradient of a scalar field ∇ ϕ = e ^ k h k ∂ ϕ ∂ q k {\displaystyle \nabla \phi ={\frac {{\hat {\mathbf {e} }}_{k}}{h_{k}}}{\frac {\partial \phi }{\partial q^{k}}}} Divergence of a vector field ∇ ⋅ F = 1 J ∂ ∂ q k ( J h k F k ) {\displaystyle \nabla \cdot \mathbf {F} ={\frac {1}{J}}{\frac {\partial }{\partial q^{k}}}\left({\frac {J}{h_{k}}}F_{k}\right)} Curl of a vector field (3D only) ∇ × F = h k e ^ k J ϵ i j k ∂ ∂ q i ( h j F j ) {\displaystyle \nabla \times \mathbf {F} ={\frac {h_{k}{\hat {\mathbf {e} }}_{k}}{J}}\epsilon _{ijk}{\frac {\partial }{\partial q^{i}}}\left(h_{j}F_{j}\right)} Laplacian of a scalar field ∇ 2 ϕ = 1 J ∂ ∂ q k ( J h k 2 ∂ ϕ ∂ q k ) {\displaystyle \nabla ^{2}\phi ={\frac {1}{J}}{\frac {\partial }{\partial q^{k}}}\left({\frac {J}{h_{k}^{2}}}{\frac {\partial \phi }{\partial q^{k}}}\right)} また、スカラー場の勾配は正準偏導関数を含むヤコビ行列式 J で表現できることに注意。 J = [ ∂ ϕ ∂ q 1 , ∂ ϕ ∂ q 2 , ∂ ϕ ∂ q 3 ] {\displaystyle \mathbf {J} =\left[{\frac {\partial \phi }{\partial q^{1}}},{\frac {\partial \phi }{\partial q^{2}}},{\frac {\partial \phi }{\partial q^{3}}}\right]} upon a change of basis: ∇ ϕ = S R T J T {\displaystyle \nabla \phi =\mathbf {S} \mathbf {R} ^{T}\mathbf {J} ^{T}} where the rotation and scaling matrices are: R = [ e 1 , e 2 , e 3 ] {\displaystyle \mathbf {R} =[\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}]} S = d i a g ( [ h 1 − 1 , h 2 − 1 , h 3 − 1 ] ) . {\displaystyle \mathbf {S} =\mathrm {diag} ([h_{1}^{-1},h_{2}^{-1},h_{3}^{-1}]).}
※この「Differential operators in three dimensions」の解説は、「直交曲線座標」の解説の一部です。
「Differential operators in three dimensions」を含む「直交曲線座標」の記事については、「直交曲線座標」の概要を参照ください。
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