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데코수학/ 벡터미적분학/ 원통좌표계 , 구면좌표계

(유튜브 동영상인데 현재는 삭제되어서 내용만 남김)

개념

원통좌표계 (R,θ,z)(R, \theta, z)
RR : xy 평면상에서 원점부터의 거리
θ\theta: x축에서 y축으로 돌아간 각도 (0θ<2π)(0 \leq \theta < 2 \pi)
zz : 높이
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
R^=cosθx^+sinθy^\hat{R} = \cos \theta \hat{x} + \sin \theta \hat{y}
θ^=sinθx^+cosθy^\hat{\theta} = - \sin \theta \hat{x} + \cos \theta \hat{y}
z^=z^\hat{z} = \hat{z}$latex &s=2$
x^=cosθR^sinθθ^\hat{x} = \cos \theta \hat{R} - \sin \theta \hat{\theta}
y^=sinθR^+cosθθ^\hat{y} = \sin \theta \hat{R} + \cos \theta \hat{\theta}
z^=z^\hat{z} = \hat{z}
미소량 dx,dy,dz,dxdy,dydz...dx, dy, dz, dx \wedge dy, dy \wedge dz... 등을 dR,dθ,dzdR, d\theta, dz로 고치기
dx=cosθdRRsinθdθdx = \cos \theta dR - R \sin \theta d\theta
dy=sinθdR+Rcosθdθdy = \sin \theta dR + R \cos \theta d\theta
dz=dzdz = dz
dxdy=RdRdθdx \wedge dy = R dR \wedge d\theta
dxdydz=RdRdθdzdx \wedge dy \wedge dz = R dR \wedge d\theta \wedge dz
dl=dRR^+Rdθθ^+dzz^d \vec{l} = dR \hat{R} + R d\theta \hat{\theta} + dz \hat{z}
직교좌표계의 편미분 fx,fz{\partial f \over \partial x}, {\partial f \over \partial z}R,θ{\partial \over \partial R}, {\partial \over \partial \theta}로 고치기
fx=cosθfRsinθRfθ{\partial f \over \partial x} = \cos \theta {\partial f \over \partial R} - {\sin \theta \over R} {\partial f \over \partial \theta}
fy=sinθfR+cosθRfθ{\partial f \over \partial y} = \sin \theta {\partial f \over \partial R} + {\cos \theta \over R} {\partial f \over \partial \theta}
fz=fz{\partial f \over \partial z} = {\partial f \over \partial z}
f,F,×F,2f\nabla f, \nabla \cdot \vec{F}, \nabla \times \vec{F}, \nabla^{2} f를 원통좌표계 표현법으로 고치기
f=fRR^+1Rfθθ^+fzz^\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}
F=div(FRR^+Fθθ^+Fzz^)\nabla \cdot \vec{F} = div(F_{R}\hat{R} + F_{\theta}\hat{\theta} + F_{z}\hat{z})
=1RR(RFR)+1RFθθ+Fzz= {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}
×F=curl(FRR^+Fθθ^+Fzz^)\nabla \times \vec{F} = curl(F_{R}\hat{R} + F_{\theta}\hat{\theta} + F_{z}\hat{z})
=(1RFzθFθz)R^+(FRzFRR)θ^+1R(R(RFθ)FRθ)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}
2f=div(f)\nabla^{2} f = div(\nabla f)
=1RR(RfR)+1R22fθ2+2fz2)= {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}})
구면좌표계 (r,θ,ϕ)(r, \theta, \phi)
rr : 원점부터의 거리
θ\theta: xy 평면상에서 x축에서 y축으로 돌아간 각도 (0θ<2π)(0 \leq \theta < 2 \pi)
ϕ\phi: z축과 r사이의 각도 (z축에서 xy평면으로 내려오는 각도 (0ϕ<π)(0 \leq \phi < \pi)
좌표 변환
x=rsinϕcosθx = r \sin \phi \cos \theta
y=rsinϕsinθy = r \sin \phi \sin \theta
z=rcosϕz = r \cos \phi
r=x2+y2+z2r = \sqrt{x^{2} + y^{2} + z^{2}}
θ=arctanyx\theta = \arctan {y \over x}
ϕ=arctanx2+y2z\phi = \arctan {\sqrt{x^{2} + y^{2}} \over z}
단위벡터 변환
r^=sinϕcosθx^+sinϕsinθy^+cosϕz^\hat{r} = \sin \phi \cos \theta \hat{x} + \sin \phi \sin \theta \hat{y} + \cos \phi \hat{z}
θ^=sinθx^+cosθy^\hat{\theta} = - \sin \theta \hat{x} + \cos \theta \hat{y}
ϕ=θ^×r^=cosθcosϕx^+sinθcosϕy^+sinϕz^\phi = \hat{\theta} \times \hat{r} = \cos \theta \cos \phi \hat{x} + \sin \theta \cos \phi \hat{y} + \sin \phi \hat{z}
x^=cosθsinϕr^sinθθ^+cosθcosϕϕ^\hat{x} = \cos \theta \sin \phi \hat{r} - \sin \theta \hat{\theta} + \cos \theta \cos \phi \hat{\phi}
y^=sinθsinϕr^+cosθθ^+sinθcosϕϕ^\hat{y} = \sin \theta \sin \phi \hat{r} + \cos \theta \hat{\theta} + \sin \theta \cos \phi \hat{\phi}
z^=cosϕr^sinϕϕ^\hat{z} = \cos \phi \hat{r} - \sin \phi \hat{\phi}
미소량 표현
dx,dy,dzdr,dθ,dϕdx, dy, dz \leftrightarrow dr, d\theta, d\phi
dxdydz=r2sinϕdrdθdϕ=dvdx \wedge dy \wedge dz = r^{2} \sin \phi dr \wedge d\theta \wedge d\phi = dv
dl=drr^+rsinϕdθθ^+rdϕϕ^d\vec{l} = dr\hat{r} + r \sin \phi d\theta \hat{\theta} + r d\phi \hat{\phi}
fx=cosθsinϕfrsinθrsinϕfθ+cosθcosϕrfϕ{\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}
fy=sinθsinϕfrcosθrsinϕfθ+sinθcosϕrfϕ{\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}
fz=cosϕfrsinϕrfϕ{\partial f \over \partial z} = \cos \phi {\partial f \over \partial r} - {\sin \phi \over r} {\partial f \over \partial \phi}
f,F,×F,2f\nabla f, \nabla \cdot \vec{F}, \nabla \times \vec{F}, \nabla^{2} f 를 구면좌표계 표현법으로 고치기
f=frr^+1rsinϕfθθ^+1rfϕϕ^\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}
F=div(Frr^+Fθθ^+Fϕϕ^)\nabla \cdot \vec{F} = div(F_{r}\hat{r} + F_{\theta}\hat{\theta} + F_{\phi}\hat{\phi})
=1r2r(r2Fr)+1rsinϕFθθ+1rsinϕϕ(sinϕFϕ)= {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})
×F=curl(Frr^+Fθθ^+Fϕϕ^)\nabla \times \vec{F} = curl(F_{r}\hat{r} + F_{\theta}\hat{\theta} + F_{\phi}\hat{\phi})
=1rsinϕ(ϕ(sinϕFθ)Fθθ)r^+1r(r(rFϕ)Frϕ)θ^+1r(1sinϕFrθr(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}
2f=div(f)\nabla^{2} f = div(\nabla f)
=1r2r(r2fr)+1r2sin2ϕ2fθ2+1r2sinϕϕ(sinϕfϕ)= {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})
좌표계와 무관한 div, curl의 정의
divF=limv01vvFdAdiv \vec{F} = \lim_{v \to 0} {1 \over v} \int_{\partial v} \vec{F} \cdot d\vec{A}
curlF=limv01vvF×dAcurl \vec{F} = \lim_{v \to 0} {1 \over v} \int_{\partial v} \vec{F} \times d\vec{A}