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曲線座標系における勾配、発散、回転、ラプラシアン 📂数理物理学

曲線座標系における勾配、発散、回転、ラプラシアン

解説

物理学で、デル演算子∇\nablaを含む4つの演算、勾配、発散、回転、ラプラシアンは非常に重要である。そのため、3つの座標系における上記の演算を必ず知っておく必要がある。もちろん、これが暗記しなければならないという意味ではない。物理学の勉強は公式を暗記するのではなく、勉強しているうちに自然と覚えることになるので、わざわざ暗記しようとせず、公式を印刷して持ち歩くか、このページをブックマークして必要な時にすぐに取り出せるようにしよう。

公式

ffをスカラー関数、ベクトル関数A\mathbf AをA=A1e^1+A2e^2+A3e^3\mathbf A= A_{1}\mathbf{\hat e_{1}}+A_2\mathbf{\hat e_2}+A_{3}\mathbf{\hat e_{3}}とする。

  • 勾配:

    ∇f=e^11h1∂f∂e1+e^21h2∂f∂e2+e^31h3∂f∂e3=∑i=13e^i1hi∂f∂ei \begin{align*} \nabla f &= \mathbf{\hat e_{1}}\frac{1}{h_{1}}\frac{\partial f}{\partial e_{1}}+ \mathbf{\hat e_2}\frac{1}{h_2}\frac{\partial f}{\partial e_2}+\mathbf{\hat e_{3}}\frac{1}{h_{3}}\frac{\partial f}{\partial e_{3}} \\ &= \sum \limits_{i=1}^3 \mathbf{\hat e_{i}}\frac{1}{h_{i}}\frac{\partial f}{\partial e_{i}} \end{align*}

  • 発散:

    ∇⋅A=1h1h2h3[∂∂e1(h2h3A1)+∂∂e2(h1h3A2)+∂∂e3(h1h2A3)] \nabla \cdot \mathbf A=\frac{1}{h_{1}h_2h_{3}} \left[ \frac{\partial}{\partial e_{1}} (h_2h_{3}A_{1}) + \frac{\partial}{\partial e_2} (h_{1}h_{3}A_2) + \frac{\partial}{\partial e_{3}} (h_{1}h_2A_{3}) \right]

  • 回転:

    ∇×A=1h1h2h3∣h1e^1h2e^2h3e^3∂∂e1∂∂e2∂∂e3h1A1h2A2h3A3∣ \nabla \times \mathbf A =\frac{1}{h_{1}h_2h_{3}} \begin{vmatrix} h_{1} \mathbf{\hat e_{1}} & h_2 \mathbf{\hat e_2} & h_{3} \mathbf{\hat e_{3}} \\[0.5em] \dfrac{\partial}{\partial e_{1}} & \dfrac{\partial }{\partial e_2} & \dfrac{\partial}{\partial e_{3}} \\[1em] h_{1}A_{1} & h_2A_2 & h_{3}A_{3} \end{vmatrix}

  • ラプラシアン:

    ∇⋅(∇f)= ∇2f= 1h1h2h3[∂∂e1(h2h3h1∂f∂e1)+∂∂e2(h1h3h2∂f∂e2)+∂∂e3(h1h2h3∂f∂e3)] \begin{align*} & \nabla \cdot (\nabla f) \\ =&\ \nabla ^2 f \\ =&\ \frac{1}{h_{1}h_2h_{3}} \left[ \frac{\partial }{\partial e_{1}} \left( \frac{h_2h_{3}}{h_{1}} \frac{\partial f}{\partial e_{1}} \right) +\frac{\partial }{\partial e_2} \left( \frac{h_{1}h_{3}}{h_2} \frac{\partial f}{\partial e_2} \right) + \frac{\partial }{\partial e_{3}} \left( \frac{h_{1}h_2}{h_{3}} \frac{\partial f}{\partial e_{3}} \right) \right] \end{align*}

このとき、各座標系ごとの単位ベクトル、スケールファクターは以下の通りである。

  • 直交座標系:

    e1^=x^,e2^=y^,e3^=z^,h1=1,h2=1,h3=1 \mathbf{\hat{e_{1}}}=\mathbf{\hat{\mathbf{x}}},\quad\mathbf{\hat{e_{2}}}=\mathbf{\hat{\mathbf{y}}},\quad\mathbf{\hat{e_{3}}}=\mathbf{\hat{\mathbf{z}}},\quad h_{1}=1,\quad h_{2}=1,\quad h_{3}=1

  • 円筒座標系:

    e1^=ρ^,e2^=ϕ^,e3^=z^,h1=1,h2=ρ,h3=1 \mathbf{\hat{e_{1}}}=\boldsymbol{\hat \rho},\quad\mathbf{\hat{e_{2}}}=\boldsymbol{\hat \phi},\quad\mathbf{\hat{e_{3}}}=\mathbf{\hat{\mathbf{z}}},\quad h_{1}=1,\quad h_{2}=\rho,\quad h_{3}=1

  • 球座標系

    e1^=r^,e2^=θ^,e3^=ϕ^,h1=1,h2=r,h3=rsin⁡θ \mathbf{\hat{e_{1}}}=\mathbf{\hat r},\quad\mathbf{\hat{e_{2}}}=\boldsymbol{\hat \theta},\quad\mathbf{\hat{e_{3}}}=\boldsymbol{\hat \phi},\quad h_{1}=1,\quad h_{2}=r,\quad h_{3}=r\sin\theta

直交座標系

  • 勾配

∇f=∂f∂xx^+∂f∂yy^+∂f∂zz^ \nabla f = \frac{\partial f}{\partial x}\mathbf{\hat{\mathbf{x}} }+ \frac{\partial f}{\partial y}\mathbf{\hat{\mathbf{y}}} + \frac{\partial f}{\partial z}\mathbf{\hat{\mathbf{z}}}

  • 発散

∇⋅A=∂Ax∂x+∂Ay∂y+∂Az∂z \nabla \cdot \mathbf A=\frac{\partial A_{x}}{\partial x}+\frac{\partial A_{y}}{\partial y}+\frac{\partial A_{z}}{\partial z}

  • 回転

∇×A=(∂Az∂y−∂Ay∂z)x^+(∂Ax∂z−∂Az∂x)y^+(∂Ay∂x−∂Ax∂y)z^=∣x^y^z^∂∂x∂∂y∂∂zAxAyAz∣ \begin{align*} \nabla \times \mathbf A&=\left(\frac{\partial A_{z}}{\partial y}-\frac{\partial A_{y}}{\partial z} \right) \mathbf{\hat{\mathbf{x}}}+\left(\frac{\partial A_{x}}{\partial z}-\frac{\partial A_{z}}{\partial x} \right) \mathbf{\hat{\mathbf{y}}}+\left(\frac{\partial A_{y}}{\partial x}-\frac{\partial A_{x}}{\partial y} \right) \mathbf{\hat{\mathbf{z}}} \\ &= \begin{vmatrix} \mathbf{\hat{\mathbf{x}}} & \mathbf{\hat{\mathbf{y}}} & \mathbf{\hat{\mathbf{z}}} \\ \dfrac{\partial}{\partial x} & \dfrac{\partial }{\partial y} & \dfrac{\partial}{\partial z} \\ A_{x} & A_{y} & A_{z} \end{vmatrix} \end{align*}

  • ラプラシアン

∇⋅(∇f)=∇2f=(∂∂xx^+∂∂yy^+∂∂zz^)⋅(∂f∂xx^+∂f∂yy^+∂f∂zz^)=∂2f∂x2+∂2f∂y2+∂2f∂z2 \begin{align*} \nabla \cdot (\nabla f) = \nabla ^2 f &= \left( \frac{\partial}{\partial x}\mathbf{\hat{\mathbf{x}}}+\frac{\partial}{\partial y}\mathbf{\hat{\mathbf{y}}}+\frac{\partial}{\partial z}\mathbf{\hat{\mathbf{z}}} \right) \cdot \left( \frac{\partial f}{\partial x}\mathbf{\hat{\mathbf{x}}}+\frac{\partial f}{\partial y}\mathbf{\hat{\mathbf{y}}}+\frac{\partial f}{\partial z}\mathbf{\hat{\mathbf{z}}} \right) \\ &= \frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+\frac{\partial^2 f}{\partial z^2} \end{align*}

円筒座標系

  • 勾配

∇f=∂f∂ρρ^+1ρ∂f∂ϕϕ^+∂f∂zz^ \nabla f = \frac{\partial f}{\partial \rho}\boldsymbol{\hat \rho} + \frac{1}{\rho}\frac{\partial f}{\partial \phi}\boldsymbol{\hat \phi} + \frac{\partial f}{\partial z}\mathbf{\hat{\mathbf{z}}}

  • 発散

∇⋅A=1ρ∂(ρAρ)∂ρ+1ρ∂Aϕ∂ϕ+∂Az∂z \nabla \cdot \mathbf A=\frac{1}{\rho}\frac{\partial (\rho A_\rho)}{\partial \rho}+\frac{1}{\rho}\frac{\partial A_\phi}{\partial \phi}+\frac{\partial A_{z}}{\partial z}

  • 回転

∇×A=[1ρ∂Az∂ϕ−∂Aϕ∂z]ρ^+[∂Aρ∂z−∂Az∂ρ]ϕ^+1ρ[∂(ρAϕ)∂ρ−∂Aρ∂ϕ]z^=1ρ∣ρ^ρϕ^z^∂∂ρ∂∂ϕ∂∂zAρρAϕAz∣ \begin{align*} \nabla \times \mathbf A&=\left[\frac{1}{\rho}\frac{\partial A_{z}}{\partial \phi}-\frac{\partial A_\phi}{\partial z} \right] \boldsymbol{\hat \rho}+\left[\frac{\partial A_\rho}{\partial z}-\frac{\partial A_{z}}{\partial \rho} \right] \boldsymbol{\hat \phi}+\frac{1}{\rho}\left[\frac{\partial (\rho A_\phi)}{\partial \rho}-\frac{\partial A_\rho}{\partial \phi} \right] \mathrm{\hat{\mathbf{z}}} \\ &= \frac{1}{\rho}\begin{vmatrix} \boldsymbol{\hat \rho} & \rho\boldsymbol{ \hat \phi} & \mathbf{\hat{\mathbf{z}}} \\ \dfrac{\partial}{\partial \rho} & \dfrac{\partial }{\partial \phi} & \dfrac{\partial}{\partial z} \\ A_\rho & \rho A_\phi & A_{z} \end{vmatrix} \end{align*}

  • ラプラシアン

∇⋅(∇f)=∇2f=1ρ∂∂ρ(ρ∂f∂ρ)+1ρ2∂2f∂ϕ2+∂2f∂z2 \nabla \cdot (\nabla f) = \nabla ^2 f = \frac{1}{\rho}\frac{\partial}{\partial \rho}\left( \rho \frac{\partial f}{\partial \rho} \right) + \frac{1}{\rho^2}\frac{\partial^2 f}{\partial \phi^2} + \frac{\partial^2 f}{\partial z^2}

球座標系

  • 勾配

∇f=∂f∂rr^+1r∂f∂θθ^+1rsin⁡θ∂f∂ϕϕ^ \nabla f = \frac{\partial f}{\partial r} \mathbf{\hat{\mathbf{r}}} + \frac{1}{r}\frac{\partial f}{\partial \theta} \boldsymbol{\hat{\boldsymbol{\theta}}} + \frac{1}{r\sin\theta}\frac{\partial f}{\partial \phi}\boldsymbol{\hat \phi}

  • 発散

∇⋅A=1r2∂(r2Ar)∂r+1rsin⁡θ∂(sin⁡θAθ)∂θ+1rsin⁡θ∂Aϕ∂ϕ \nabla \cdot \mathbf A=\frac{1}{r^2}\frac{\partial (r^2 A_{r})}{\partial r}+\frac{1}{r\sin\theta}\frac{\partial (\sin\theta A_\theta)}{\partial \theta}+\frac{1}{r\sin\theta}\frac{\partial A_\phi}{\partial \phi}

  • 回転

∇×A=1rsin⁡θ[∂(sin⁡θAϕ)∂θ−∂Aθ∂ϕ]r^+1r[1sin⁡θ∂Ar∂ϕ−∂(rAϕ)∂r]θ^+1r[∂(rAθ)∂r−∂Ar∂θ]ϕ^=1r2sin⁡θ∣r^rθ^rsin⁡θϕ^∂∂r∂∂θ∂∂ϕArrAθrsin⁡θAϕ∣ \begin{align*} \nabla \times \mathbf A &=\frac{1}{r\sin\theta} \left[\frac{\partial (\sin\theta A_\phi)}{\partial \theta}-\frac{\partial A_\theta}{\partial \phi} \right]\mathbf{\hat{\mathbf{r}}}+\frac{1}{r}\left[\frac{1}{\sin\theta} \frac{\partial A_{r}}{\partial \phi}-\frac{\partial (rA_\phi)}{\partial r} \right] \boldsymbol{\hat{\boldsymbol{\theta}}} \\ & \quad+ \frac{1}{r} \left[\frac{\partial (rA_\theta)}{\partial r}-\frac{\partial A_{r}}{\partial \theta} \right]\boldsymbol{\hat \phi} \\ &= \frac{1}{r^2\sin\theta}\begin{vmatrix} \mathbf{\hat{\mathbf{r}}} & r\boldsymbol{\hat{\boldsymbol{\theta}}} & r\sin\theta\boldsymbol{\hat \phi} \\ \dfrac{\partial}{\partial r} & \dfrac{\partial }{\partial \theta} & \dfrac{\partial}{\partial \phi} \\ A_{r} & r A_\theta & r\sin\theta A_\phi \end{vmatrix} \end{align*}

  • ラプラシアン

∇⋅(∇f)=∇2f=1r2∂∂r(r2∂f∂r)+1r2sin⁡θ∂∂θ(sin⁡θ∂f∂θ)+1r2sin⁡2θ∂2f∂2ϕ \nabla \cdot (\nabla f) = \nabla ^2 f = \frac{1}{r^2}\frac{\partial}{\partial r} \left( r^2\frac{\partial f}{\partial r} \right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\left( \sin\theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2 f}{\partial^2 \phi}