Vektorske operacije u zakrivljenim koordinatama

Ovo je popis najbitnijih relacija i operacija vektorskog računa u sfernom, cilindričnom, paraboličnom i kartezijevu sustavu.

Operacija	Kartezijeve koordinate (x,y,z)	Cilindrične koordinate (ρ,φ,z)	Sferne koordinate (r,ϑ,φ)	Parabolične koordinate (σ,τ,z)
Definicija koordinata	$\begin{matrix} \rho & = & \sqrt{x^2+y^2} \\ \varphi & = & \arctan(y/x) \\ z & = & z \end{matrix}$	$\begin{matrix} x & = & \rho\cos\varphi \\ y & = & \rho\sin\varphi \\ z & = & z \end{matrix}$	$\begin{matrix} x & = & r\sin\vartheta\cos\varphi \\ y & = & r\sin\vartheta\sin\varphi \\ z & = & r\cos\vartheta \end{matrix}$	$\begin{matrix} x & = & \sigma \tau\\ y & = & \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right) \\ z & = & z \end{matrix}$
	$\begin{matrix} r & = & \sqrt{x^2+y^2+z^2} \\ \vartheta & = & \arccos(z/r)\\ \varphi & = & \arctan(y/x) \\ \end{matrix}$	$\begin{matrix} r & = & \sqrt{\rho^2 + z^2} \\ \vartheta & = & \arctan{(\rho/z)}\\ \varphi & = & \varphi \end{matrix}$	$\begin{matrix} \rho & = & r\sin(\vartheta) \\ \varphi & = & \varphi\\ z & = & r\cos(\vartheta) \end{matrix}$	$\begin{matrix} \rho\cos\varphi & = & \sigma \tau\\ \rho\sin\varphi & = & \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right) \\ z & = & z \end{matrix}$
Definicija jediničnih vektora	$\begin{matrix} \boldsymbol{\hat \rho} & = & \frac{x}{\rho}\mathbf{\hat x}+\frac{y}{\rho}\mathbf{\hat y} \\ \boldsymbol{\hat\varphi} & = & -\frac{y}{\rho}\mathbf{\hat x}+\frac{x}{\rho}\mathbf{\hat y} \\ \mathbf{\hat z} & = & \mathbf{\hat z} \end{matrix}$	$\begin{matrix} \mathbf{\hat x} & = & \cos\varphi\boldsymbol{\hat \rho}-\sin\varphi\boldsymbol{\hat\varphi} \\ \mathbf{\hat y} & = & \sin\varphi\boldsymbol{\hat \rho}+\cos\varphi\boldsymbol{\hat\varphi} \\ \mathbf{\hat z} & = & \mathbf{\hat z} \end{matrix}$	$\begin{matrix} \mathbf{\hat x} & = & \sin\vartheta\cos\varphi\boldsymbol{\hat r}+\cos\vartheta\cos\varphi\boldsymbol{\hat\vartheta}-\sin\varphi\boldsymbol{\hat\varphi} \\ \mathbf{\hat y} & = & \sin\vartheta\sin\varphi\boldsymbol{\hat r}+\cos\vartheta\sin\varphi\boldsymbol{\hat\vartheta}+\cos\varphi\boldsymbol{\hat\varphi} \\ \mathbf{\hat z} & = & \cos\vartheta \boldsymbol{\hat r}-\sin\vartheta \boldsymbol{\hat\vartheta} \\ \end{matrix}$	$\begin{matrix} \boldsymbol{\hat \sigma} & = & \frac{\tau}{\sqrt{\tau^2+\sigma^2}}\mathbf{\hat x}-\frac{\sigma}{\sqrt{\tau^2+\sigma^2}}\mathbf{\hat y} \\ \boldsymbol{\hat\tau} & = & \frac{\sigma}{\sqrt{\tau^2+\sigma^2}}\mathbf{\hat x}+\frac{\tau}{\sqrt{\tau^2+\sigma^2}}\mathbf{\hat y} \\ \mathbf{\hat z} & = & \mathbf{\hat z} \end{matrix}$
	$\begin{matrix} \mathbf{\hat r} & = & \frac{x\mathbf{\hat x}+y\mathbf{\hat y}+z\mathbf{\hat z}}{r} \\ \boldsymbol{\hat\vartheta} & = & \frac{xz\mathbf{\hat x}+yz\mathbf{\hat y}-\rho^2\mathbf{\hat z}}{r \rho} \\ \boldsymbol{\hat\varphi} & = & \frac{-y\mathbf{\hat x}+x\mathbf{\hat y}}{\rho} \end{matrix}$	$\begin{matrix} \mathbf{\hat r} & = & \frac{\rho}{r}\boldsymbol{\hat \rho}+\frac{ z}{r}\mathbf{\hat z} \\ \boldsymbol{\hat\vartheta} & = & \frac{z }{r}\boldsymbol{\hat \rho}-\frac{\rho}{r}\mathbf{\hat z} \\ \boldsymbol{\hat\varphi} & = & \boldsymbol{\hat\varphi} \end{matrix}$	$\begin{matrix} \boldsymbol{\hat \rho} & = & \sin\vartheta\mathbf{\hat r}+\cos\vartheta\boldsymbol{\hat\vartheta} \\ \boldsymbol{\hat\varphi} & = & \boldsymbol{\hat\varphi} \\ \mathbf{\hat z} & = & \cos\vartheta\mathbf{\hat r}-\sin\vartheta\boldsymbol{\hat\vartheta} \\ \end{matrix}$	$\begin{matrix} \end{matrix}$
A vektorsko polje $\mathbf{A}$	$A_x\mathbf{\hat x} + A_y\mathbf{\hat y} + A_z\mathbf{\hat z}$	$A_\rho\boldsymbol{\hat \rho} + A_\varphi\boldsymbol{\hat \varphi} + A_z\boldsymbol{\hat z}$	$A_r\boldsymbol{\hat r} + A_\vartheta\boldsymbol{\hat \vartheta} + A_\varphi\boldsymbol{\hat \varphi}$	$A_\sigma\boldsymbol{\hat \sigma} + A_\tau\boldsymbol{\hat \tau} + A_\varphi\boldsymbol{\hat z}$
Gradijent $\nabla f$	${\partial f \over \partial x}\mathbf{\hat x} + {\partial f \over \partial y}\mathbf{\hat y} + {\partial f \over \partial z}\mathbf{\hat z}$	${\partial f \over \partial \rho}\boldsymbol{\hat \rho} + {1 \over \rho}{\partial f \over \partial \varphi}\boldsymbol{\hat \varphi} + {\partial f \over \partial z}\boldsymbol{\hat z}$	${\partial f \over \partial r}\boldsymbol{\hat r} + {1 \over r}{\partial f \over \partial \vartheta}\boldsymbol{\hat \vartheta} + {1 \over r\sin\vartheta}{\partial f \over \partial \varphi}\boldsymbol{\hat \varphi}$	$\frac{1}{\sqrt{\sigma^{2} + \tau^{2}}} {\partial f \over \partial \sigma}\boldsymbol{\hat \sigma} + \frac{1}{\sqrt{\sigma^{2} + \tau^{2}}} {\partial f \over \partial \tau}\boldsymbol{\hat \tau} + {\partial f \over \partial z}\boldsymbol{\hat z}$
Divergencija $\nabla \cdot \mathbf{A}$	${\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}$	${1 \over \rho}{\partial \left( \rho A_\rho \right) \over \partial \rho} + {1 \over \rho}{\partial A_\varphi \over \partial \varphi} + {\partial A_z \over \partial z}$	${1 \over r^2}{\partial \left( r^2 A_r \right) \over \partial r} + {1 \over r\sin\vartheta}{\partial \over \partial \vartheta} \left( A_\vartheta\sin\vartheta \right) + {1 \over r\sin\vartheta}{\partial A_\varphi \over \partial \varphi}$	$\frac{1}{\sigma^{2} + \tau^{2}}{\partial A_\sigma \over \partial \sigma} + \frac{1}{\sigma^{2} + \tau^{2}}{\partial A_\tau \over \partial \tau} + {\partial A_z \over \partial z}$
Rotacija $\nabla \times \mathbf{A}$	$\begin{matrix} \displaystyle\left({\partial A_z \over \partial y} - {\partial A_y \over \partial z}\right) \mathbf{\hat x} & + \\ \displaystyle\left({\partial A_x \over \partial z} - {\partial A_z \over \partial x}\right) \mathbf{\hat y} & + \\ \displaystyle\left({\partial A_y \over \partial x} - {\partial A_x \over \partial y}\right) \mathbf{\hat z} & \ \end{matrix}$	$\begin{matrix} \displaystyle\left({1 \over \rho}{\partial A_z \over \partial \varphi} - {\partial A_\varphi \over \partial z}\right) \boldsymbol{\hat \rho} & + \\ \displaystyle\left({\partial A_\rho \over \partial z} - {\partial A_z \over \partial \rho}\right) \boldsymbol{\hat \varphi} & + \\ \displaystyle{1 \over \rho}\left({\partial \left( \rho A_\varphi \right) \over \partial \rho} - {\partial A_\rho \over \partial \varphi}\right) \boldsymbol{\hat z} & \ \end{matrix}$	$\begin{matrix} \displaystyle{1 \over r\sin\vartheta}\left({\partial \over \partial \vartheta} \left( A_\varphi\sin\vartheta \right) - {\partial A_\vartheta \over \partial \varphi}\right) \boldsymbol{\hat r} & + \\ \displaystyle{1 \over r}\left({1 \over \sin\vartheta}{\partial A_r \over \partial \varphi} - {\partial \over \partial r} \left( r A_\varphi \right) \right) \boldsymbol{\hat \vartheta} & + \\ \displaystyle{1 \over r}\left({\partial \over \partial r} \left( r A_\vartheta \right) - {\partial A_r \over \partial \vartheta}\right) \boldsymbol{\hat \varphi} & \ \end{matrix}$	$\begin{matrix} \displaystyle\left(\frac{1}{\sqrt{\sigma^{2} + \tau^{2}}}{\partial A_z \over \partial \tau} - {\partial A_\tau \over \partial z}\right) \boldsymbol{\hat \sigma} & - \\ \displaystyle\left(\frac{1}{\sqrt{\sigma^{2} + \tau^{2}}}{\partial A_z \over \partial \sigma}- {\partial A_\sigma \over \partial z}\right) \boldsymbol{\hat \tau} & + \\ \displaystyle\frac{1}{\sqrt{\sigma^{2} + \tau^{2}}}\left({\partial \left( \rho A_\varphi \right) \over \partial \rho} - {\partial A_\rho \over \partial \varphi}\right) \boldsymbol{\hat z} & \ \end{matrix}$
Laplasijan $\Delta f = \nabla^2 f$	${\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2}$	${1 \over \rho}{\partial \over \partial \rho}\left(\rho {\partial f \over \partial \rho}\right) + {1 \over \rho^2}{\partial^2 f \over \partial \varphi^2} + {\partial^2 f \over \partial z^2}$	${1 \over r^2}{\partial \over \partial r}\!\left(r^2 {\partial f \over \partial r}\right) \!+\!{1 \over r^2\!\sin\vartheta}{\partial \over \partial \vartheta}\!\left(\sin\vartheta {\partial f \over \partial \vartheta}\right) \!+\!{1 \over r^2\!\sin^2\vartheta}{\partial^2 f \over \partial \varphi^2}$	$\frac{1}{\sigma^{2} + \tau^{2}} \left( \frac{\partial^{2} f}{\partial \sigma^{2}} + \frac{\partial^{2} f}{\partial \tau^{2}} \right) + \frac{\partial^{2} f}{\partial z^{2}}$
Vekotrski laplasijan $\Delta \mathbf{A} = \nabla^2 \mathbf{A}$	$\Delta A_x \mathbf{\hat x} + \Delta A_y \mathbf{\hat y} + \Delta A_z \mathbf{\hat z}$	$\begin{matrix} \displaystyle\left(\Delta A_\rho - {A_\rho \over \rho^2} - {2 \over \rho^2}{\partial A_\varphi \over \partial \varphi}\right) \boldsymbol{\hat \rho} & + \\ \displaystyle\left(\Delta A_\varphi - {A_\varphi \over \rho^2} + {2 \over \rho^2}{\partial A_\rho \over \partial \varphi}\right) \boldsymbol{\hat\varphi} & + \\ \displaystyle\left(\Delta A_z \right) \boldsymbol{\hat z} & \ \end{matrix}$	$\begin{matrix} \left(\Delta A_r - {2 A_r \over r^2} - {2 \over r^2\sin\vartheta}{\partial \left(A_\vartheta \sin\vartheta\right) \over \partial\vartheta} - {2 \over r^2\sin\vartheta}{\partial A_\varphi \over \partial \varphi}\right) \boldsymbol{\hat r} & + \\ \left(\Delta A_\vartheta - {A_\vartheta \over r^2\sin^2\vartheta} + {2 \over r^2}{\partial A_r \over \partial \vartheta} - {2 \cos\vartheta \over r^2\sin^2\vartheta}{\partial A_\varphi \over \partial \varphi}\right) \boldsymbol{\hat\vartheta} & + \\ \left(\Delta A_\varphi - {A_\varphi \over r^2\sin^2\vartheta} + {2 \over r^2\sin\vartheta}{\partial A_r \over \partial \varphi} + {2 \cos\vartheta \over r^2\sin^2\vartheta}{\partial A_\vartheta \over \partial \varphi}\right) \boldsymbol{\hat\varphi} & \end{matrix}$
Element duljine	$d\mathbf{l} = dx\mathbf{\hat x} + dy\mathbf{\hat y} + dz\mathbf{\hat z}$	$d\mathbf{l} = d\rho\boldsymbol{\hat \rho} + \rho d\varphi\boldsymbol{\hat \varphi} + dz\boldsymbol{\hat z}$	$d\mathbf{l} = dr\mathbf{\hat r} + rd\vartheta\boldsymbol{\hat \vartheta} + r\sin\vartheta d\varphi\boldsymbol{\hat \varphi}$	$d\mathbf{l} = \sqrt{\sigma^{2} + \tau^{2}} d\sigma\boldsymbol{\hat \sigma} + \sqrt{\sigma^{2} + \tau^{2}} d\tau\boldsymbol{\hat \tau} + dz\boldsymbol{\hat z}$
Element površine	$\begin{matrix}d\mathbf{S} = &dy\,dz\,\mathbf{\hat x} + \\ &dx\,dz\,\mathbf{\hat y} + \\ &dx\,dy\,\mathbf{\hat z}\end{matrix}$	$\begin{matrix} d\mathbf{S} = & \rho\, d\varphi\, dz\,\boldsymbol{\hat \rho} + \\ & d\rho \,dz\,\boldsymbol{\hat \varphi} + \\ & \rho \,d\rho d\varphi \,\mathbf{\hat z} \end{matrix}$	$\begin{matrix} d\mathbf{S} = & r^2 \sin\vartheta \,d\vartheta \,d\varphi \,\mathbf{\hat r} + \\ & r\sin\vartheta \,dr\,d\varphi \,\boldsymbol{\hat \vartheta} + \\ & r\,dr\,d\vartheta\,\boldsymbol{\hat \varphi} \end{matrix}$	$\begin{matrix} d\mathbf{S} = & \sqrt{\sigma^{2} + \tau^{2}}, d\tau\, dz\,\boldsymbol{\hat \sigma} + \\ & \sqrt{\sigma^{2} + \tau^{2}} d\sigma\,dz\,\boldsymbol{\hat \tau} + \\ & \sigma^{2} + \tau^{2} d\sigma, d\tau \,\mathbf{\hat z} \end{matrix}$
Element obujma	$dV = dx\,dy\,dz \,$	$dV = \rho\, d\rho\, d\varphi\, dz\,$	$dV = r^2\sin\vartheta \,dr\,d\vartheta\, d\varphi\,$	$dV = \left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau dz,$
Netrivijalne kombinacije vektorskih operacija: $\operatorname{div\ grad\ } f = \nabla \cdot (\nabla f) = \nabla^2 f = \Delta f$ (laplasijan) $\operatorname{rot\ grad\ } f = \nabla \times (\nabla f) = \mathbf{0}$ $\operatorname{div\ rot\ } \mathbf{A} = \nabla \cdot (\nabla \times \mathbf{A}) = 0$ $\operatorname{rot\ rot\ } \mathbf{A} = \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$ $\Delta f g = f \Delta g + 2 \nabla f \cdot \nabla g + g\Delta f$

Vezani pojmovi

• Koordinatni sustav
• Sferni koodrinatni sustav
• Cilindrični koordinatni sustav
• Vektorsko polje
• Divergencija
• Gradijent
• Rotacija
• Laplasijan