곡선좌표계에서의 그래디언트, 다이벌전스, 컬, 라플라시안
📂수리물리 곡선좌표계에서의 그래디언트, 다이벌전스, 컬, 라플라시안 설명 물리학에서 델 연산자 ∇ \nabla ∇ 그래디언트(기울기) , 다이벌전스(발산) , 컬(회전) , 라플라시안 은 매우 중요하다. 따라서 3가지 좌표계에 대한 위 연산 을 반드시 알아야한다. 물론 이 말은 외워야한다는 뜻이 아니다. 물리학 공부는 이런 식을 외우는 것이 아니기 때문이다. 공부하다 보면 자연스레 외워질 것이므로 일부러 외우려하지 말고 공식을 프린트해서 가지고 다니거나, 이 페이지를 즐겨찾기 해놓고 필요할 때 바로 꺼내보자.
공식 f f f A \mathbf A A A = A 1 e ^ 1 + A 2 e ^ 2 + A 3 e ^ 3 \mathbf A= A_{1}\mathbf{\hat e_{1}}+A_2\mathbf{\hat e_2}+A_{3}\mathbf{\hat e_{3}} A = A 1 e ^ 1 + A 2 e ^ 2 + A 3 e ^ 3
그래디언트:
∇ f = e ^ 1 1 h 1 ∂ f ∂ e 1 + e ^ 2 1 h 2 ∂ f ∂ e 2 + e ^ 3 1 h 3 ∂ f ∂ e 3 = ∑ i = 1 3 e ^ i 1 h i ∂ f ∂ e i
\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*}
∇ f = e ^ 1 h 1 1 ∂ e 1 ∂ f + e ^ 2 h 2 1 ∂ e 2 ∂ f + e ^ 3 h 3 1 ∂ e 3 ∂ f = i = 1 ∑ 3 e ^ i h i 1 ∂ e i ∂ f
다이벌전스:
∇ ⋅ A = 1 h 1 h 2 h 3 [ ∂ ∂ e 1 ( h 2 h 3 A 1 ) + ∂ ∂ e 2 ( h 1 h 3 A 2 ) + ∂ ∂ e 3 ( h 1 h 2 A 3 ) ]
\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]
∇ ⋅ A = h 1 h 2 h 3 1 [ ∂ e 1 ∂ ( h 2 h 3 A 1 ) + ∂ e 2 ∂ ( h 1 h 3 A 2 ) + ∂ e 3 ∂ ( h 1 h 2 A 3 ) ]
컬:
∇ × A = 1 h 1 h 2 h 3 ∣ h 1 e ^ 1 h 2 e ^ 2 h 3 e ^ 3 ∂ ∂ e 1 ∂ ∂ e 2 ∂ ∂ e 3 h 1 A 1 h 2 A 2 h 3 A 3 ∣
\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}
∇ × A = h 1 h 2 h 3 1 h 1 e ^ 1 ∂ e 1 ∂ h 1 A 1 h 2 e ^ 2 ∂ e 2 ∂ h 2 A 2 h 3 e ^ 3 ∂ e 3 ∂ h 3 A 3
라플라시안:
∇ ⋅ ( ∇ f ) = ∇ 2 f = 1 h 1 h 2 h 3 [ ∂ ∂ e 1 ( h 2 h 3 h 1 ∂ f ∂ e 1 ) + ∂ ∂ e 2 ( h 1 h 3 h 2 ∂ f ∂ e 2 ) + ∂ ∂ e 3 ( h 1 h 2 h 3 ∂ f ∂ e 3 ) ]
\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*}
= = ∇ ⋅ ( ∇ f ) ∇ 2 f h 1 h 2 h 3 1 [ ∂ e 1 ∂ ( h 1 h 2 h 3 ∂ e 1 ∂ f ) + ∂ e 2 ∂ ( h 2 h 1 h 3 ∂ e 2 ∂ f ) + ∂ e 3 ∂ ( h 3 h 1 h 2 ∂ e 3 ∂ f ) ]
이 때 각 좌표계별 단위벡터, 스케일 팩터 는 다음과 같다.
직교 좌표계:
e 1 ^ = x ^ , e 2 ^ = y ^ , e 3 ^ = z ^ , h 1 = 1 , h 2 = 1 , h 3 = 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
e 1 ^ = x ^ , e 2 ^ = y ^ , e 3 ^ = z ^ , h 1 = 1 , h 2 = 1 , h 3 = 1
원통 좌표계:
e 1 ^ = ρ ^ , e 2 ^ = ϕ ^ , e 3 ^ = z ^ , h 1 = 1 , h 2 = ρ , h 3 = 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
e 1 ^ = ρ ^ , e 2 ^ = ϕ ^ , e 3 ^ = z ^ , h 1 = 1 , h 2 = ρ , h 3 = 1
구면 좌표계
e 1 ^ = r ^ , e 2 ^ = θ ^ , e 3 ^ = ϕ ^ , h 1 = 1 , h 2 = r , h 3 = r sin 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
e 1 ^ = r ^ , e 2 ^ = θ ^ , e 3 ^ = ϕ ^ , h 1 = 1 , h 2 = r , h 3 = r sin θ
직교 좌표계 ∇ f = ∂ f ∂ x x ^ + ∂ f ∂ y y ^ + ∂ f ∂ z z ^
\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}}}
∇ f = ∂ x ∂ f x ^ + ∂ y ∂ f y ^ + ∂ z ∂ f z ^
∇ ⋅ A = ∂ A x ∂ x + ∂ A y ∂ y + ∂ A z ∂ z
\nabla \cdot \mathbf A=\frac{\partial A_{x}}{\partial x}+\frac{\partial A_{y}}{\partial y}+\frac{\partial A_{z}}{\partial z}
∇ ⋅ A = ∂ x ∂ A x + ∂ y ∂ A y + ∂ z ∂ A z
∇ × A = ( ∂ A z ∂ y − ∂ A y ∂ z ) x ^ + ( ∂ A x ∂ z − ∂ A z ∂ x ) y ^ + ( ∂ A y ∂ x − ∂ A x ∂ y ) z ^ = ∣ x ^ y ^ z ^ ∂ ∂ x ∂ ∂ y ∂ ∂ z A x A y A z ∣
\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*}
∇ × A = ( ∂ y ∂ A z − ∂ z ∂ A y ) x ^ + ( ∂ z ∂ A x − ∂ x ∂ A z ) y ^ + ( ∂ x ∂ A y − ∂ y ∂ A x ) z ^ = x ^ ∂ x ∂ A x y ^ ∂ y ∂ A y z ^ ∂ z ∂ A z
∇ ⋅ ( ∇ f ) = ∇ 2 f = ( ∂ ∂ x x ^ + ∂ ∂ y y ^ + ∂ ∂ z z ^ ) ⋅ ( ∂ f ∂ x x ^ + ∂ f ∂ y y ^ + ∂ f ∂ z z ^ ) = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2
\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*}
∇ ⋅ ( ∇ f ) = ∇ 2 f = ( ∂ x ∂ x ^ + ∂ y ∂ y ^ + ∂ z ∂ z ^ ) ⋅ ( ∂ x ∂ f x ^ + ∂ y ∂ f y ^ + ∂ z ∂ f z ^ ) = ∂ x 2 ∂ 2 f + ∂ y 2 ∂ 2 f + ∂ z 2 ∂ 2 f
원통 좌표계 ∇ f = ∂ f ∂ ρ ρ ^ + 1 ρ ∂ f ∂ ϕ ϕ ^ + ∂ f ∂ z z ^
\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}}}
∇ f = ∂ ρ ∂ f ρ ^ + ρ 1 ∂ ϕ ∂ f ϕ ^ + ∂ z ∂ f z ^
∇ ⋅ A = 1 ρ ∂ ( ρ A ρ ) ∂ ρ + 1 ρ ∂ A ϕ ∂ ϕ + ∂ A z ∂ 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}
∇ ⋅ A = ρ 1 ∂ ρ ∂ ( ρ A ρ ) + ρ 1 ∂ ϕ ∂ A ϕ + ∂ z ∂ A z
∇ × A = [ 1 ρ ∂ A z ∂ ϕ − ∂ A ϕ ∂ z ] ρ ^ + [ ∂ A ρ ∂ z − ∂ A z ∂ ρ ] ϕ ^ + 1 ρ [ ∂ ( ρ A ϕ ) ∂ ρ − ∂ A ρ ∂ ϕ ] z ^ = 1 ρ ∣ ρ ^ ρ ϕ ^ z ^ ∂ ∂ ρ ∂ ∂ ϕ ∂ ∂ z A ρ ρ A ϕ A z ∣
\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*}
∇ × A = [ ρ 1 ∂ ϕ ∂ A z − ∂ z ∂ A ϕ ] ρ ^ + [ ∂ z ∂ A ρ − ∂ ρ ∂ A z ] ϕ ^ + ρ 1 [ ∂ ρ ∂ ( ρ A ϕ ) − ∂ ϕ ∂ A ρ ] z ^ = ρ 1 ρ ^ ∂ ρ ∂ A ρ ρ ϕ ^ ∂ ϕ ∂ ρ A ϕ z ^ ∂ z ∂ A z
∇ ⋅ ( ∇ f ) = ∇ 2 f = 1 ρ ∂ ∂ ρ ( ρ ∂ f ∂ ρ ) + 1 ρ 2 ∂ 2 f ∂ ϕ 2 + ∂ 2 f ∂ z 2
\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}
∇ ⋅ ( ∇ f ) = ∇ 2 f = ρ 1 ∂ ρ ∂ ( ρ ∂ ρ ∂ f ) + ρ 2 1 ∂ ϕ 2 ∂ 2 f + ∂ z 2 ∂ 2 f
구 좌표계 ∇ f = ∂ f ∂ r r ^ + 1 r ∂ f ∂ θ θ ^ + 1 r sin θ ∂ f ∂ ϕ ϕ ^
\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}
∇ f = ∂ r ∂ f r ^ + r 1 ∂ θ ∂ f θ ^ + r sin θ 1 ∂ ϕ ∂ f ϕ ^
∇ ⋅ A = 1 r 2 ∂ ( r 2 A r ) ∂ r + 1 r sin θ ∂ ( sin θ A θ ) ∂ θ + 1 r sin θ ∂ A ϕ ∂ 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}
∇ ⋅ A = r 2 1 ∂ r ∂ ( r 2 A r ) + r sin θ 1 ∂ θ ∂ ( sin θ A θ ) + r sin θ 1 ∂ ϕ ∂ A ϕ
∇ × A = 1 r sin θ [ ∂ ( sin θ A ϕ ) ∂ θ − ∂ A θ ∂ ϕ ] r ^ + 1 r [ 1 sin θ ∂ A r ∂ ϕ − ∂ ( r A ϕ ) ∂ r ] θ ^ + 1 r [ ∂ ( r A θ ) ∂ r − ∂ A r ∂ θ ] ϕ ^ = 1 r 2 sin θ ∣ r ^ r θ ^ r sin θ ϕ ^ ∂ ∂ r ∂ ∂ θ ∂ ∂ ϕ A r r A θ r sin θ A ϕ ∣
\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*}
∇ × A = r sin θ 1 [ ∂ θ ∂ ( sin θ A ϕ ) − ∂ ϕ ∂ A θ ] r ^ + r 1 [ sin θ 1 ∂ ϕ ∂ A r − ∂ r ∂ ( r A ϕ ) ] θ ^ + r 1 [ ∂ r ∂ ( r A θ ) − ∂ θ ∂ A r ] ϕ ^ = r 2 sin θ 1 r ^ ∂ r ∂ A r r θ ^ ∂ θ ∂ r A θ r sin θ ϕ ^ ∂ ϕ ∂ r sin θ A ϕ
∇ ⋅ ( ∇ f ) = ∇ 2 f = 1 r 2 ∂ ∂ r ( r 2 ∂ f ∂ r ) + 1 r 2 sin θ ∂ ∂ θ ( sin θ ∂ f ∂ θ ) + 1 r 2 sin 2 θ ∂ 2 f ∂ 2 ϕ
\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}
∇ ⋅ ( ∇ f ) = ∇ 2 f = r 2 1 ∂ r ∂ ( r 2 ∂ r ∂ f ) + r 2 sin θ 1 ∂ θ ∂ ( sin θ ∂ θ ∂ f ) + r 2 sin 2 θ 1 ∂ 2 ϕ ∂ 2 f
