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데코수학/ 벡터미적분학/ 원통좌표계 , 구면좌표계

(유튜브 동영상인데 현재는 삭제되어서 내용만 남김)

개념

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원통좌표계 (R,θ,z)(R, \theta, z)
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RR : xy 평면상에서 원점부터의 거리
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θ\theta: x축에서 y축으로 돌아간 각도 (0≤θ<2π)(0 \leq \theta < 2 \pi)
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zz : 높이
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직교 좌표계의 단위 벡터 x^,y^,z^\hat{x}, \hat{y}, \hat{z}를 원통좌표계의 단위벡터 R^,θ^,z^\hat{R}, \hat{\theta}, \hat{z}로 고치기
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R^=cos⁡θx^+sin⁡θy^\hat{R} = \cos \theta \hat{x} + \sin \theta \hat{y}
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θ^=−sin⁡θx^+cos⁡θy^\hat{\theta} = - \sin \theta \hat{x} + \cos \theta \hat{y}
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z^=z^\hat{z} = \hat{z}$latex &s=2$
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x^=cos⁡θR^−sin⁡θθ^\hat{x} = \cos \theta \hat{R} - \sin \theta \hat{\theta}
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y^=sin⁡θR^+cos⁡θθ^\hat{y} = \sin \theta \hat{R} + \cos \theta \hat{\theta}
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z^=z^\hat{z} = \hat{z}
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미소량 dx,dy,dz,dx∧dy,dy∧dz...dx, dy, dz, dx \wedge dy, dy \wedge dz... 등을 dR,dθ,dzdR, d\theta, dz로 고치기
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dx=cos⁡θdR−Rsin⁡θdθdx = \cos \theta dR - R \sin \theta d\theta
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dy=sin⁡θdR+Rcos⁡θdθdy = \sin \theta dR + R \cos \theta d\theta
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dz=dzdz = dz
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dx∧dy=RdR∧dθdx \wedge dy = R dR \wedge d\theta
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dx∧dy∧dz=RdR∧dθ∧dzdx \wedge dy \wedge dz = R dR \wedge d\theta \wedge dz
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dl⃗=dRR^+Rdθθ^+dzz^d \vec{l} = dR \hat{R} + R d\theta \hat{\theta} + dz \hat{z}
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직교좌표계의 편미분 ∂f∂x,∂f∂z{\partial f \over \partial x}, {\partial f \over \partial z}를 ∂∂R,∂∂θ{\partial \over \partial R}, {\partial \over \partial \theta}로 고치기
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∂f∂x=cos⁡θ∂f∂R−sin⁡θR∂f∂θ{\partial f \over \partial x} = \cos \theta {\partial f \over \partial R} - {\sin \theta \over R} {\partial f \over \partial \theta}
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∂f∂y=sin⁡θ∂f∂R+cos⁡θR∂f∂θ{\partial f \over \partial y} = \sin \theta {\partial f \over \partial R} + {\cos \theta \over R} {\partial f \over \partial \theta}
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∂f∂z=∂f∂z{\partial f \over \partial z} = {\partial f \over \partial z}
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∇f,∇⋅F⃗,∇×F⃗,∇2f\nabla f, \nabla \cdot \vec{F}, \nabla \times \vec{F}, \nabla^{2} f를 원통좌표계 표현법으로 고치기
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∇f=∂f∂RR^+1R∂f∂θθ^+∂f∂zz^\nabla f = {\partial f \over \partial R} \hat{R} + {1 \over R} {\partial f \over \partial \theta} \hat{\theta} + {\partial f \over \partial z} \hat{z}
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∇⋅F⃗=div(FRR^+Fθθ^+Fzz^)\nabla \cdot \vec{F} = div(F_{R}\hat{R} + F_{\theta}\hat{\theta} + F_{z}\hat{z})
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=1R∂∂R(RFR)+1R∂Fθ∂θ+∂Fz∂z= {1 \over R} {\partial \over \partial R} (R F_{R}) + {1 \over R} {\partial F_{\theta} \over \partial \theta} + {\partial F_{z} \over \partial z}
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∇×F⃗=curl(FRR^+Fθθ^+Fzz^)\nabla \times \vec{F} = curl(F_{R}\hat{R} + F_{\theta}\hat{\theta} + F_{z}\hat{z})
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=(1R∂Fz∂θ−∂Fθ∂z)R^+(∂FR∂z−∂FR∂R)θ^+1R(∂∂R(RFθ)−∂FR∂θ)z^= ({1 \over R} {\partial F_{z} \over \partial \theta} - {\partial F_{\theta} \over \partial z}) \hat{R} + ({\partial F_{R} \over \partial z} - {\partial F_{R} \over \partial R}) \hat{\theta} + {1 \over R} ({\partial \over \partial R} (R F_{\theta}) - {\partial F_{R} \over \partial \theta}) \hat{z}
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∇2f=div(∇f)\nabla^{2} f = div(\nabla f)
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=1R∂∂R(R∂f∂R)+1R2∂2f∂θ2+∂2f∂z2)= {1 \over R} {\partial \over \partial R} (R {\partial f \over \partial R}) + {1 \over R^{2}} {\partial^{2} f \over \partial \theta^{2}} + {\partial^{2} f \over \partial z^{2}})
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구면좌표계 (r,θ,ϕ)(r, \theta, \phi)
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rr : 원점부터의 거리
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θ\theta: xy 평면상에서 x축에서 y축으로 돌아간 각도 (0≤θ<2π)(0 \leq \theta < 2 \pi)
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ϕ\phi: z축과 r사이의 각도 (z축에서 xy평면으로 내려오는 각도 (0≤ϕ<π)(0 \leq \phi < \pi)
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좌표 변환
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x=rsin⁡ϕcos⁡θx = r \sin \phi \cos \theta
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y=rsin⁡ϕsin⁡θy = r \sin \phi \sin \theta
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z=rcos⁡ϕz = r \cos \phi
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r=x2+y2+z2r = \sqrt{x^{2} + y^{2} + z^{2}}
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θ=arctan⁡yx\theta = \arctan {y \over x}
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ϕ=arctan⁡x2+y2z\phi = \arctan {\sqrt{x^{2} + y^{2}} \over z}
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단위벡터 변환
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r^=sin⁡ϕcos⁡θx^+sin⁡ϕsin⁡θy^+cos⁡ϕz^\hat{r} = \sin \phi \cos \theta \hat{x} + \sin \phi \sin \theta \hat{y} + \cos \phi \hat{z}
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θ^=−sin⁡θx^+cos⁡θy^\hat{\theta} = - \sin \theta \hat{x} + \cos \theta \hat{y}
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ϕ=θ^×r^=cos⁡θcos⁡ϕx^+sin⁡θcos⁡ϕy^+sin⁡ϕz^\phi = \hat{\theta} \times \hat{r} = \cos \theta \cos \phi \hat{x} + \sin \theta \cos \phi \hat{y} + \sin \phi \hat{z}
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x^=cos⁡θsin⁡ϕr^−sin⁡θθ^+cos⁡θcos⁡ϕϕ^\hat{x} = \cos \theta \sin \phi \hat{r} - \sin \theta \hat{\theta} + \cos \theta \cos \phi \hat{\phi}
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y^=sin⁡θsin⁡ϕr^+cos⁡θθ^+sin⁡θcos⁡ϕϕ^\hat{y} = \sin \theta \sin \phi \hat{r} + \cos \theta \hat{\theta} + \sin \theta \cos \phi \hat{\phi}
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z^=cos⁡ϕr^−sin⁡ϕϕ^\hat{z} = \cos \phi \hat{r} - \sin \phi \hat{\phi}
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미소량 표현
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dx,dy,dz↔dr,dθ,dϕdx, dy, dz \leftrightarrow dr, d\theta, d\phi
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dx∧dy∧dz=r2sin⁡ϕdr∧dθ∧dϕ=dvdx \wedge dy \wedge dz = r^{2} \sin \phi dr \wedge d\theta \wedge d\phi = dv
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dl⃗=drr^+rsin⁡ϕdθθ^+rdϕϕ^d\vec{l} = dr\hat{r} + r \sin \phi d\theta \hat{\theta} + r d\phi \hat{\phi}
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∂f∂x=cos⁡θsin⁡ϕ∂f∂r−sin⁡θrsin⁡ϕ∂f∂θ+cos⁡θcos⁡ϕr∂f∂ϕ{\partial f \over \partial x} = \cos \theta \sin \phi {\partial f \over \partial r} - {\sin \theta \over r \sin \phi} {\partial f \over \partial \theta} + {\cos \theta \cos \phi \over r} {\partial f \over \partial \phi}
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∂f∂y=sin⁡θsin⁡ϕ∂f∂r−cos⁡θrsin⁡ϕ∂f∂θ+sin⁡θcos⁡ϕr∂f∂ϕ{\partial f \over \partial y} = \sin \theta \sin \phi {\partial f \over \partial r} - {\cos \theta \over r \sin \phi} {\partial f \over \partial \theta} + {\sin \theta \cos \phi \over r} {\partial f \over \partial \phi}
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∂f∂z=cos⁡ϕ∂f∂r−sin⁡ϕr∂f∂ϕ{\partial f \over \partial z} = \cos \phi {\partial f \over \partial r} - {\sin \phi \over r} {\partial f \over \partial \phi}
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∇f,∇⋅F⃗,∇×F⃗,∇2f\nabla f, \nabla \cdot \vec{F}, \nabla \times \vec{F}, \nabla^{2} f 를 구면좌표계 표현법으로 고치기
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∇f=∂f∂rr^+1rsin⁡ϕ∂f∂θθ^+1r∂f∂ϕϕ^\nabla f = {\partial f \over \partial r} \hat{r} + {1 \over r \sin \phi} {\partial f \over \partial \theta} \hat{\theta} + {1 \over r} {\partial f \over \partial \phi} \hat{\phi}
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∇⋅F⃗=div(Frr^+Fθθ^+Fϕϕ^)\nabla \cdot \vec{F} = div(F_{r}\hat{r} + F_{\theta}\hat{\theta} + F_{\phi}\hat{\phi})
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=1r2∂∂r(r2Fr)+1rsin⁡ϕ∂Fθ∂θ+1rsin⁡ϕ∂∂ϕ(sin⁡ϕFϕ)= {1 \over r^{2}} {\partial \over \partial r} (r^{2} F_{r}) + {1 \over r \sin \phi} {\partial F_{\theta} \over \partial \theta} + {1 \over r \sin \phi} {\partial \over \partial \phi} (\sin \phi F_{\phi})
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∇×F⃗=curl(Frr^+Fθθ^+Fϕϕ^)\nabla \times \vec{F} = curl(F_{r}\hat{r} + F_{\theta}\hat{\theta} + F_{\phi}\hat{\phi})
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=1rsin⁡ϕ(∂∂ϕ(sin⁡ϕFθ)−∂Fθ∂θ)r^+1r(∂∂r(rFϕ)−∂Fr∂ϕ)θ^+1r(1sin⁡ϕ∂Fr∂θ−∂∂r(rFθ))ϕ^= {1 \over r \sin \phi} ({\partial \over \partial \phi} (\sin \phi F_{\theta}) - {\partial F_{\theta} \over \partial \theta}) \hat{r} + {1 \over r} ({\partial \over \partial r} (r F_{\phi}) - {\partial F_{r} \over \partial \phi}) \hat{\theta} + {1 \over r} ({1 \over \sin \phi} {\partial F_{r} \over \partial \theta} - {\partial \over \partial r} (r F_{\theta})) \hat{\phi}
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∇2f=div(∇f)\nabla^{2} f = div(\nabla f)
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=1r2∂∂r(r2∂f∂r)+1r2sin⁡2ϕ∂2f∂θ2+1r2sin⁡ϕ∂∂ϕ(sin⁡ϕ∂f∂ϕ)= {1 \over r^{2}} {\partial \over \partial r} (r^{2} {\partial f \over \partial r}) + {1 \over r^{2} \sin^{2} \phi} {\partial^{2} f \over \partial \theta^{2}} + {1 \over r^{2} \sin \phi} {\partial \over \partial \phi}(\sin \phi {\partial f \over \partial \phi})
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좌표계와 무관한 div, curl의 정의
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divF⃗=lim⁡v→01v∫∂vF⃗⋅dA⃗div \vec{F} = \lim_{v \to 0} {1 \over v} \int_{\partial v} \vec{F} \cdot d\vec{A}
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curlF⃗=lim⁡v→01v∫∂vF⃗×dA⃗curl \vec{F} = \lim_{v \to 0} {1 \over v} \int_{\partial v} \vec{F} \times d\vec{A}
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