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	$\begin{array}{ccc}\rho & =& \sqrt{x^2+y^2}\\ \phi & =& \arctan (y/x)\\ z& =& z\end{array}$	$\begin{array}{ccc}x& =& \rho \cos \phi \\ y& =& \rho \sin \phi \\ z& =& z\end{array}$	$\begin{array}{ccc}x& =& r\sin \vartheta \cos \phi \\ y& =& r\sin \vartheta \sin \phi \\ z& =& r\cos \vartheta \end{array}$	$\begin{array}{ccc}x& =& \sigma \tau \\ y& =& \frac{1}{2}({\tau }^2-{\sigma }^2)\\ z& =& z\end{array}$
	$\begin{array}{ccc}r& =& \sqrt{x^2+y^2+z^2}\\ \vartheta & =& \arccos (z/r)\\ \phi & =& \arctan (y/x)\end{array}$	$\begin{array}{ccc}r& =& \sqrt{{\rho }^2+z^2}\\ \vartheta & =& \arctan (\rho /z)\\ \phi & =& \phi \end{array}$	$\begin{array}{ccc}\rho & =& r\sin (\vartheta )\\ \phi & =& \phi \\ z& =& r\cos (\vartheta )\end{array}$	$\begin{array}{ccc}\rho \cos \phi & =& \sigma \tau \\ \rho \sin \phi & =& \frac{1}{2}({\tau }^2-{\sigma }^2)\\ z& =& z\end{array}$
Definicija jediničnih vektora	$\begin{array}{ccc}\hat{\mathit{\rho }}& =& \frac{x}{\rho }\widehat{𝐱}+\frac{y}{\rho }\widehat{𝐲}\\ \hat{\mathit{\phi }}& =& -\frac{y}{\rho }\widehat{𝐱}+\frac{x}{\rho }\widehat{𝐲}\\ \widehat{𝐳}& =& \widehat{𝐳}\end{array}$	$\begin{array}{ccc}\widehat{𝐱}& =& \cos \phi \hat{\mathit{\rho }}-\sin \phi \hat{\mathit{\phi }}\\ \widehat{𝐲}& =& \sin \phi \hat{\mathit{\rho }}+\cos \phi \hat{\mathit{\phi }}\\ \widehat{𝐳}& =& \widehat{𝐳}\end{array}$	$\begin{array}{ccc}\widehat{𝐱}& =& \sin \vartheta \cos \phi \widehat{𝒓}+\cos \vartheta \cos \phi \hat{\mathit{\vartheta }}-\sin \phi \hat{\mathit{\phi }}\\ \widehat{𝐲}& =& \sin \vartheta \sin \phi \widehat{𝒓}+\cos \vartheta \sin \phi \hat{\mathit{\vartheta }}+\cos \phi \hat{\mathit{\phi }}\\ \widehat{𝐳}& =& \cos \vartheta \widehat{𝒓}-\sin \vartheta \hat{\mathit{\vartheta }}\end{array}$	$\begin{array}{ccc}\hat{\mathit{\sigma }}& =& \frac{\tau }{\sqrt{{\tau }^2+{\sigma }^2}}\widehat{𝐱}-\frac{\sigma }{\sqrt{{\tau }^2+{\sigma }^2}}\widehat{𝐲}\\ \hat{\mathit{\tau }}& =& \frac{\sigma }{\sqrt{{\tau }^2+{\sigma }^2}}\widehat{𝐱}+\frac{\tau }{\sqrt{{\tau }^2+{\sigma }^2}}\widehat{𝐲}\\ \widehat{𝐳}& =& \widehat{𝐳}\end{array}$
	$\begin{array}{ccc}\widehat{𝐫}& =& \frac{x\widehat{𝐱}+y\widehat{𝐲}+z\widehat{𝐳}}{r}\\ \hat{\mathit{\vartheta }}& =& \frac{xz\widehat{𝐱}+yz\widehat{𝐲}-{\rho }^2\widehat{𝐳}}{r\rho }\\ \hat{\mathit{\phi }}& =& \frac{-y\widehat{𝐱}+x\widehat{𝐲}}{\rho }\end{array}$	$\begin{array}{ccc}\widehat{𝐫}& =& \frac{\rho }{r}\hat{\mathit{\rho }}+\frac{z}{r}\widehat{𝐳}\\ \hat{\mathit{\vartheta }}& =& \frac{z}{r}\hat{\mathit{\rho }}-\frac{\rho }{r}\widehat{𝐳}\\ \hat{\mathit{\phi }}& =& \hat{\mathit{\phi }}\end{array}$	$\begin{array}{ccc}\hat{\mathit{\rho }}& =& \sin \vartheta \widehat{𝐫}+\cos \vartheta \hat{\mathit{\vartheta }}\\ \hat{\mathit{\phi }}& =& \hat{\mathit{\phi }}\\ \widehat{𝐳}& =& \cos \vartheta \widehat{𝐫}-\sin \vartheta \hat{\mathit{\vartheta }}\end{array}$
A vektorsko polje $𝐀$	$A_x\widehat{𝐱}+A_y\widehat{𝐲}+A_z\widehat{𝐳}$	$A_{\rho }\hat{\mathit{\rho }}+A_{\phi }\hat{\mathit{\phi }}+A_z\widehat{𝒛}$	$A_r\widehat{𝒓}+A_{\vartheta }\hat{\mathit{\vartheta }}+A_{\phi }\hat{\mathit{\phi }}$	$A_{\sigma }\hat{\mathit{\sigma }}+A_{\tau }\hat{\mathit{\tau }}+A_{\phi }\widehat{𝒛}$
Gradijent $\mathrm{\nabla }f$	$\frac{\partial f}{\partial x}\widehat{𝐱}+\frac{\partial f}{\partial y}\widehat{𝐲}+\frac{\partial f}{\partial z}\widehat{𝐳}$	$\frac{\partial f}{\partial \rho }\hat{\mathit{\rho }}+\frac{1}{\rho }\frac{\partial f}{\partial \phi }\hat{\mathit{\phi }}+\frac{\partial f}{\partial z}\widehat{𝒛}$	$\frac{\partial f}{\partial r}\widehat{𝒓}+\frac{1}{r}\frac{\partial f}{\partial \vartheta }\hat{\mathit{\vartheta }}+\frac{1}{r\sin \vartheta }\frac{\partial f}{\partial \phi }\hat{\mathit{\phi }}$	$\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial f}{\partial \sigma }\hat{\mathit{\sigma }}+\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial f}{\partial \tau }\hat{\mathit{\tau }}+\frac{\partial f}{\partial z}\widehat{𝒛}$
Divergencija $\mathrm{\nabla }\cdot 𝐀$	$\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}$	$\frac{1}{\rho }\frac{\partial \left(\rho A_{\rho }\right)}{\partial \rho }+\frac{1}{\rho }\frac{\partial A_{\phi }}{\partial \phi }+\frac{\partial A_z}{\partial z}$	$\frac{1}{r^2}\frac{\partial \left(r^2{A}_r\right)}{\partial r}+\frac{1}{r\sin \vartheta }\frac{\partial }{\partial \vartheta }(A_{\vartheta }\sin \vartheta )+\frac{1}{r\sin \vartheta }\frac{\partial A_{\phi }}{\partial \phi }$	$\frac{1}{{\sigma }^2+{\tau }^2}\frac{\partial A_{\sigma }}{\partial \sigma }+\frac{1}{{\sigma }^2+{\tau }^2}\frac{\partial A_{\tau }}{\partial \tau }+\frac{\partial A_z}{\partial z}$
Rotacija $\mathrm{\nabla }\times 𝐀$	$\begin{array}{cc}(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})\widehat{𝐱}& +\\ (\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})\widehat{𝐲}& +\\ (\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})\widehat{𝐳}& \end{array}$	$\begin{array}{cc}(\frac{1}{\rho }\frac{\partial A_z}{\partial \phi }-\frac{\partial A_{\phi }}{\partial z})\hat{\mathit{\rho }}& +\\ (\frac{\partial A_{\rho }}{\partial z}-\frac{\partial A_z}{\partial \rho })\hat{\mathit{\phi }}& +\\ \frac{1}{\rho }(\frac{\partial \left(\rho A_{\phi }\right)}{\partial \rho }-\frac{\partial A_{\rho }}{\partial \phi })\widehat{𝒛}& \end{array}$	$\begin{array}{cc}\frac{1}{r\sin \vartheta }(\frac{\partial }{\partial \vartheta }(A_{\phi }\sin \vartheta )-\frac{\partial A_{\vartheta }}{\partial \phi })\widehat{𝒓}& +\\ \frac{1}{r}(\frac{1}{\sin \vartheta }\frac{\partial A_r}{\partial \phi }-\frac{\partial }{\partial r}\left(r{A}_{\phi }\right))\hat{\mathit{\vartheta }}& +\\ \frac{1}{r}(\frac{\partial }{\partial r}\left(r{A}_{\vartheta }\right)-\frac{\partial A_r}{\partial \vartheta })\hat{\mathit{\phi }}& \end{array}$	$\begin{array}{cc}(\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial A_z}{\partial \tau }-\frac{\partial A_{\tau }}{\partial z})\hat{\mathit{\sigma }}& -\\ (\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial A_z}{\partial \sigma }-\frac{\partial A_{\sigma }}{\partial z})\hat{\mathit{\tau }}& +\\ \frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}(\frac{\partial \left(\rho A_{\phi }\right)}{\partial \rho }-\frac{\partial A_{\rho }}{\partial \phi })\widehat{𝒛}& \end{array}$
Laplasijan $\mathrm{\Delta }f={\mathrm{\nabla }}^{2}f$	$\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{\rho }\frac{\partial }{\partial \rho }\left(\rho \frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2}\frac{{\partial }^{2}f}{\partial {\phi }^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial f}{\partial r}\right)+\frac{1}{r^2\sin \vartheta }\frac{\partial }{\partial \vartheta }(\sin \vartheta \frac{\partial f}{\partial \vartheta })+\frac{1}{r^2{\sin }^2\vartheta }\frac{{\partial }^{2}f}{\partial {\phi }^2}$	$\frac{1}{{\sigma }^2+{\tau }^2}(\frac{{\partial }^{2}f}{\partial {\sigma }^2}+\frac{{\partial }^{2}f}{\partial {\tau }^2})+\frac{{\partial }^{2}f}{\partial z^2}$
Vekotrski laplasijan $\mathrm{\Delta }𝐀={\mathrm{\nabla }}^2𝐀$	$\mathrm{\Delta }{A}_x\widehat{𝐱}+\mathrm{\Delta }{A}_y\widehat{𝐲}+\mathrm{\Delta }{A}_z\widehat{𝐳}$	$\begin{array}{cc}(\mathrm{\Delta }{A}_{\rho }-\frac{A_{\rho }}{{\rho }^2}-\frac{2}{{\rho }^2}\frac{\partial A_{\phi }}{\partial \phi })\hat{\mathit{\rho }}& +\\ (\mathrm{\Delta }{A}_{\phi }-\frac{A_{\phi }}{{\rho }^2}+\frac{2}{{\rho }^2}\frac{\partial A_{\rho }}{\partial \phi })\hat{\mathit{\phi }}& +\\ \left(\mathrm{\Delta }{A}_z\right)\widehat{𝒛}& \end{array}$	$\begin{array}{cc}(\mathrm{\Delta }{A}_r-\frac{2A_r}{r^2}-\frac{2}{r^2\sin \vartheta }\frac{\partial (A_{\vartheta }\sin \vartheta )}{\partial \vartheta }-\frac{2}{r^2\sin \vartheta }\frac{\partial A_{\phi }}{\partial \phi })\widehat{𝒓}& +\\ (\mathrm{\Delta }{A}_{\vartheta }-\frac{A_{\vartheta }}{r^2{\sin }^2\vartheta }+\frac{2}{r^2}\frac{\partial A_r}{\partial \vartheta }-\frac{2\cos \vartheta }{r^2{\sin }^2\vartheta }\frac{\partial A_{\phi }}{\partial \phi })\hat{\mathit{\vartheta }}& +\\ (\mathrm{\Delta }{A}_{\phi }-\frac{A_{\phi }}{r^2{\sin }^2\vartheta }+\frac{2}{r^2\sin \vartheta }\frac{\partial A_r}{\partial \phi }+\frac{2\cos \vartheta }{r^2{\sin }^2\vartheta }\frac{\partial A_{\vartheta }}{\partial \phi })\hat{\mathit{\phi }}& \end{array}$
Element duljine	$d𝐥=dx\widehat{𝐱}+dy\widehat{𝐲}+dz\widehat{𝐳}$	$d𝐥=d\rho \hat{\mathit{\rho }}+\rho d\phi \hat{\mathit{\phi }}+dz\widehat{𝒛}$	$d𝐥=dr\widehat{𝐫}+rd\vartheta \hat{\mathit{\vartheta }}+r\sin \vartheta d\phi \hat{\mathit{\phi }}$	$d𝐥=\sqrt{{\sigma }^2+{\tau }^2}d\sigma \hat{\mathit{\sigma }}+\sqrt{{\sigma }^2+{\tau }^2}d\tau \hat{\mathit{\tau }}+dz\widehat{𝒛}$
Element površine	$\begin{array}{cc}d𝐒=& dydz\widehat{𝐱}+\\ & dxdz\widehat{𝐲}+\\ & dxdy\widehat{𝐳}\end{array}$	$\begin{array}{cc}d𝐒=& \rho d\phi dz\hat{\mathit{\rho }}+\\ & d\rho dz\hat{\mathit{\phi }}+\\ & \rho d\rho d\phi \widehat{𝐳}\end{array}$	$\begin{array}{cc}d𝐒=& r^2\sin \vartheta d\vartheta d\phi \widehat{𝐫}+\\ & r\sin \vartheta drd\phi \hat{\mathit{\vartheta }}+\\ & rdrd\vartheta \hat{\mathit{\phi }}\end{array}$	$\begin{array}{cc}d𝐒=& \sqrt{{\sigma }^2+{\tau }^2},d\tau dz\hat{\mathit{\sigma }}+\\ & \sqrt{{\sigma }^2+{\tau }^2}d\sigma dz\hat{\mathit{\tau }}+\\ & {\sigma }^2+{\tau }^{2}d\sigma ,d\tau \widehat{𝐳}\end{array}$
Element obujma	$dV=dxdydz$	$dV=\rho d\rho d\phi dz$	$dV=r^2\sin \vartheta drd\vartheta d\phi$	$dV=({\sigma }^2+{\tau }^2)d\sigma d\tau dz,$
Netrivijalne kombinacije vektorskih operacija: $\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f=\mathrm{\nabla }\cdot (\mathrm{\nabla }f)={\mathrm{\nabla }}^{2}f=\mathrm{\Delta }f$ (laplasijan) $\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f=\mathrm{\nabla }\times (\mathrm{\nabla }f)=\mathrm{𝟎}$ $\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{r}\mathrm{o}\mathrm{t}𝐀=\mathrm{\nabla }\cdot (\mathrm{\nabla }\times 𝐀)=0$ $\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{t}𝐀=\mathrm{\nabla }\times (\mathrm{\nabla }\times 𝐀)=\mathrm{\nabla }(\mathrm{\nabla }\cdot 𝐀)-{\mathrm{\nabla }}^2𝐀$ $\mathrm{\Delta }fg=f\mathrm{\Delta }g+2\mathrm{\nabla }f\cdot \mathrm{\nabla }g+g\mathrm{\Delta }f$

Vezani pojmovi

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