Nabla in verschillende assenstelsels

Dit is een lijst van enkele formules uit vectoranalyse voor het werken met veelvoorkomende kromlijnige coördinatenstelsels: cartesische coördinaten, cilindercoördinaten, bolcoördinaten.

Conversies tussen stelsels

conversies tussen cartesische, cilinder- en bolcoördinaten

		van
		cartesisch	cilindrisch	bol
naar	cartesisch	$\begin{array}{cc}x& =x\\ y& =y\\ z& =z\end{array}$	$\begin{array}{cc}x& =\rho \cos \phi \\ y& =\rho \sin \phi \\ z& =z\end{array}$	$\begin{array}{cc}x& =r\sin \theta \cos \phi \\ y& =r\sin \theta \sin \phi \\ z& =r\cos \theta \end{array}$
	cilindrisch	$\begin{array}{cc}\rho & =\sqrt{x^2+y^2}\\ \phi & =\arctan \left(\frac{y}{x}\right)\\ z& =z\end{array}$	$\begin{array}{cc}\rho & =\rho \\ \phi & =\phi \\ z& =z\end{array}$	$\begin{array}{cc}\rho & =r\sin \theta \\ \phi & =\phi \\ z& =r\cos \theta \end{array}$
	bol	$\begin{array}{cc}r& =\sqrt{x^2+y^2+z^2}\\ \theta & =\arctan \left(\frac{\sqrt{x^2+y^2}}{z}\right)\\ \phi & =\arctan \left(\frac{y}{x}\right)\end{array}$	$\begin{array}{cc}r& =\sqrt{{\rho }^2+z^2}\\ \theta & =\arctan \left(\frac{\rho }{z}\right)\\ \phi & =\phi \end{array}$	$\begin{array}{cc}r& =r\\ \phi & =\phi \\ \theta & =\theta \\ \end{array}$

Conversies tussen eenheidsvectoren

conversies tussen eenheidsvectoren in cartesische, cilindrische en bolcoördinaten in termen van bestemmingscoördinaten

	cartesische	cilindrische	bol
cartesische	Sjabloon:N/a	$\begin{array}{cc}\widehat{𝐱}& =\cos \phi \hat{\mathit{\rho }}-\sin \phi \hat{\mathit{\phi }}\\ \widehat{𝐲}& =\sin \phi \hat{\mathit{\rho }}+\cos \phi \hat{\mathit{\phi }}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$	$\begin{array}{cc}\widehat{𝐱}& =\sin \theta \cos \phi \widehat{𝐫}+\cos \theta \cos \phi \hat{\mathit{\theta }}-\sin \phi \hat{\mathit{\phi }}\\ \widehat{𝐲}& =\sin \theta \sin \phi \widehat{𝐫}+\cos \theta \sin \phi \hat{\mathit{\theta }}+\cos \phi \hat{\mathit{\phi }}\\ \widehat{𝐳}& =\cos \theta \widehat{𝐫}-\sin \theta \hat{\mathit{\theta }}\end{array}$
cilindrische	$\begin{array}{cc}\hat{\mathit{\rho }}& =\frac{x\widehat{𝐱}+y\widehat{𝐲}}{\sqrt{x^2+y^2}}\\ \hat{\mathit{\phi }}& =\frac{-y\widehat{𝐱}+x\widehat{𝐲}}{\sqrt{x^2+y^2}}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$	Sjabloon:N/a	$\begin{array}{cc}\hat{\mathit{\rho }}& =\sin \theta \widehat{𝐫}+\cos \theta \hat{\mathit{\theta }}\\ \hat{\mathit{\phi }}& =\hat{\mathit{\phi }}\\ \widehat{𝐳}& =\cos \theta \widehat{𝐫}-\sin \theta \hat{\mathit{\theta }}\end{array}$
bol	$\begin{array}{cc}\widehat{𝐫}& =\frac{x\widehat{𝐱}+y\widehat{𝐲}+z\widehat{𝐳}}{\sqrt{x^2+y^2+z^2}}\\ \hat{\mathit{\theta }}& =\frac{(x\widehat{𝐱}+y\widehat{𝐲})z-(x^2+y^2)\widehat{𝐳}}{\sqrt{x^2+y^2+z^2}\sqrt{x^2+y^2}}\\ \hat{\mathit{\phi }}& =\frac{-y\widehat{𝐱}+x\widehat{𝐲}}{\sqrt{x^2+y^2}}\end{array}$	$\begin{array}{cc}\widehat{𝐫}& =\frac{\rho \hat{\mathit{\rho }}+z\widehat{𝐳}}{\sqrt{{\rho }^2+z^2}}\\ \hat{\mathit{\theta }}& =\frac{z\hat{\mathit{\rho }}-\rho \widehat{𝐳}}{\sqrt{{\rho }^2+z^2}}\\ \hat{\mathit{\phi }}& =\hat{\mathit{\phi }}\end{array}$	Sjabloon:N/a

conversies tussen eenheidsvectoren in cartesische, cilindrische en bolcoördinaten in termen van oorsprongscoördinaten

	cartesische	cilindrische	bol
cartesische	Sjabloon:N/a	$\begin{array}{cc}\widehat{𝐱}& =\frac{x\hat{\mathit{\rho }}-y\hat{\mathit{\phi }}}{\sqrt{x^2+y^2}}\\ \widehat{𝐲}& =\frac{y\hat{\mathit{\rho }}+x\hat{\mathit{\phi }}}{\sqrt{x^2+y^2}}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$	$\begin{array}{cc}\widehat{𝐱}& =\frac{x(\sqrt{x^2+y^2}\widehat{𝐫}+z\hat{\mathit{\theta }})-y\sqrt{x^2+y^2+z^2}\hat{\mathit{\phi }}}{\sqrt{x^2+y^2}\sqrt{x^2+y^2+z^2}}\\ \widehat{𝐲}& =\frac{y(\sqrt{x^2+y^2}\widehat{𝐫}+z\hat{\mathit{\theta }})+x\sqrt{x^2+y^2+z^2}\hat{\mathit{\phi }}}{\sqrt{x^2+y^2}\sqrt{x^2+y^2+z^2}}\\ \widehat{𝐳}& =\frac{z\widehat{𝐫}-\sqrt{x^2+y^2}\hat{\mathit{\theta }}}{\sqrt{x^2+y^2+z^2}}\end{array}$
cilindrische	$\begin{array}{cc}\hat{\mathit{\rho }}& =\cos \phi \widehat{𝐱}+\sin \phi \widehat{𝐲}\\ \hat{\mathit{\phi }}& =-\sin \phi \widehat{𝐱}+\cos \phi \widehat{𝐲}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$	Sjabloon:N/a	$\begin{array}{cc}\hat{\mathit{\rho }}& =\frac{\rho \widehat{𝐫}+z\hat{\mathit{\theta }}}{\sqrt{{\rho }^2+z^2}}\\ \hat{\mathit{\phi }}& =\hat{\mathit{\phi }}\\ \widehat{𝐳}& =\frac{z\widehat{𝐫}-\rho \hat{\mathit{\theta }}}{\sqrt{{\rho }^2+z^2}}\end{array}$
bol	$\begin{array}{cc}\widehat{𝐫}& =\sin \theta (\cos \phi \widehat{𝐱}+\sin \phi \widehat{𝐲})+\cos \theta \widehat{𝐳}\\ \hat{\mathit{\theta }}& =\cos \theta (\cos \phi \widehat{𝐱}+\sin \phi \widehat{𝐲})-\sin \theta \widehat{𝐳}\\ \hat{\mathit{\phi }}& =-\sin \phi \widehat{𝐱}+\cos \phi \widehat{𝐲}\end{array}$	$\begin{array}{cc}\widehat{𝐫}& =\sin \theta \hat{\mathit{\rho }}+\cos \theta \widehat{𝐳}\\ \hat{\mathit{\theta }}& =\cos \theta \hat{\mathit{\rho }}-\sin \theta \widehat{𝐳}\\ \hat{\mathit{\phi }}& =\hat{\mathit{\phi }}\end{array}$	Sjabloon:N/a

Formules met de gradiënt

Tabel met nabla-operator in cartesische, cilindrische en bolcoördinaten

operatie	cartesische coördinaten (x, y, z)	cilindercoördinaten (ρ, φ, z)	bolcoördinaten (r, θ, φ), waar φ de azimutale en θ de polaire hoek is
vectorveld A	$A_x\widehat{𝐱}+A_y\widehat{𝐲}+A_z\widehat{𝐳}$	$A_{\rho }\hat{\mathit{\rho }}+A_{\phi }\hat{\mathit{\phi }}+A_z\widehat{𝐳}$	$A_r\widehat{𝐫}+A_{\theta }\hat{\mathit{\theta }}+A_{\phi }\hat{\mathit{\phi }}$
gradiënt ∇f	$\frac{\partial f}{\partial x}\widehat{𝐱}+\frac{\partial f}{\partial y}\widehat{𝐲}+\frac{\partial f}{\partial z}\widehat{𝐳}$	$\frac{\partial f}{\partial \rho }\hat{\mathit{\rho }}+\frac{1}{\rho }\frac{\partial f}{\partial \phi }\hat{\mathit{\phi }}+\frac{\partial f}{\partial z}\widehat{𝐳}$	$\frac{\partial f}{\partial r}\widehat{𝐫}+\frac{1}{r}\frac{\partial f}{\partial \theta }\hat{\mathit{\theta }}+\frac{1}{r\sin \theta }\frac{\partial f}{\partial \phi }\hat{\mathit{\phi }}$
divergentie ∇ ⋅ A	$\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}$	$\frac{1}{\rho }\frac{\partial \left(\rho A_{\rho }\right)}{\partial \rho }+\frac{1}{\rho }\frac{\partial A_{\phi }}{\partial \phi }+\frac{\partial A_z}{\partial z}$	$\frac{1}{r^2}\frac{\partial \left(r^2{A}_r\right)}{\partial r}+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }(A_{\theta }\sin \theta )+\frac{1}{r\sin \theta }\frac{\partial A_{\phi }}{\partial \phi }$
rotatie ∇ × A	$\begin{array}{cc}(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})& \widehat{𝐱}\\ +(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})& \widehat{𝐲}\\ +(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})& \widehat{𝐳}\end{array}$	$\begin{array}{cc}(\frac{1}{\rho }\frac{\partial A_z}{\partial \phi }-\frac{\partial A_{\phi }}{\partial z})& \hat{\mathit{\rho }}\\ +(\frac{\partial A_{\rho }}{\partial z}-\frac{\partial A_z}{\partial \rho })& \hat{\mathit{\phi }}\\ +\frac{1}{\rho }(\frac{\partial \left(\rho A_{\phi }\right)}{\partial \rho }-\frac{\partial A_{\rho }}{\partial \phi })& \widehat{𝐳}\end{array}$	$\begin{array}{cc}\frac{1}{r\sin \theta }(\frac{\partial }{\partial \theta }(A_{\phi }\sin \theta )-\frac{\partial A_{\theta }}{\partial \phi })& \widehat{𝐫}\\ +\frac{1}{r}(\frac{1}{\sin \theta }\frac{\partial A_r}{\partial \phi }-\frac{\partial }{\partial r}\left(r{A}_{\phi }\right))& \hat{\mathit{\theta }}\\ +\frac{1}{r}(\frac{\partial }{\partial r}\left(r{A}_{\theta }\right)-\frac{\partial A_r}{\partial \theta })& \hat{\mathit{\phi }}\end{array}$
laplace-operator ∇2f ≡ ∆f	$\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{\rho }\frac{\partial }{\partial \rho }\left(\rho \frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2}\frac{{\partial }^{2}f}{\partial {\phi }^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial f}{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial f}{\partial \theta })+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^{2}f}{\partial {\phi }^2}$
vector Laplaciaan ∇2A ≡ ∆A	${\mathrm{\nabla }}^2{A}_x\widehat{𝐱}+{\mathrm{\nabla }}^2{A}_y\widehat{𝐲}+{\mathrm{\nabla }}^2{A}_z\widehat{𝐳}$	$\begin{array}{cc}({\mathrm{\nabla }}^2{A}_{\rho }-\frac{A_{\rho }}{{\rho }^2}-\frac{2}{{\rho }^2}\frac{\partial A_{\phi }}{\partial \phi })& \hat{\mathit{\rho }}\\ +({\mathrm{\nabla }}^2{A}_{\phi }-\frac{A_{\phi }}{{\rho }^2}+\frac{2}{{\rho }^2}\frac{\partial A_{\rho }}{\partial \phi })& \hat{\mathit{\phi }}\\ +{\mathrm{\nabla }}^2{A}_z& \widehat{𝐳}\end{array}$	$\begin{array}{cc}({\mathrm{\nabla }}^2{A}_r-\frac{2A_r}{r^2}-\frac{2}{r^2\sin \theta }\frac{\partial (A_{\theta }\sin \theta )}{\partial \theta }-\frac{2}{r^2\sin \theta }\frac{\partial A_{\phi }}{\partial \phi })& \widehat{𝐫}\\ +({\mathrm{\nabla }}^2{A}_{\theta }-\frac{A_{\theta }}{r^2{\sin }^2\theta }+\frac{2}{r^2}\frac{\partial A_r}{\partial \theta }-\frac{2\cos \theta }{r^2{\sin }^2\theta }\frac{\partial A_{\phi }}{\partial \phi })& \hat{\mathit{\theta }}\\ +({\mathrm{\nabla }}^2{A}_{\phi }-\frac{A_{\phi }}{r^2{\sin }^2\theta }+\frac{2}{r^2\sin \theta }\frac{\partial A_r}{\partial \phi }+\frac{2\cos \theta }{r^2{\sin }^2\theta }\frac{\partial A_{\theta }}{\partial \phi })& \hat{\mathit{\phi }}\end{array}$

Rekenregels

1. $\mathrm{div}\mathrm{grad}f\equiv \mathrm{\nabla }\cdot \mathrm{\nabla }f\equiv {\mathrm{\nabla }}^{2}f$
2. $\mathrm{rot}\mathrm{grad}f\equiv \mathrm{\nabla }\times \mathrm{\nabla }f=\mathrm{𝟎}$
3. $\mathrm{div}\mathrm{rot}𝐀\equiv \mathrm{\nabla }\cdot (\mathrm{\nabla }\times 𝐀)=0$
4. $\mathrm{rot}\mathrm{rot}𝐀\equiv \mathrm{\nabla }\times (\mathrm{\nabla }\times 𝐀)=\mathrm{\nabla }(\mathrm{\nabla }\cdot 𝐀)-{\mathrm{\nabla }}^2𝐀$, Lagrange's formule voor de gradiënt
5. ${\mathrm{\nabla }}^2(fg)=f{\mathrm{\nabla }}^{2}g+2\mathrm{\nabla }f\cdot \mathrm{\nabla }g+g{\mathrm{\nabla }}^{2}f$