Operador nabla en coordenades cilíndriques i esfèriques

En càlcul vectorial, l'operador nabla és un operador diferencial vectorial representat amb el símbol nabla ∇. En coordenades cartesianes tridimensionals R³ amb coordenades (x, y, z), l'operador nabla es pot definir com:

$\mathrm{\nabla }=(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z})$

En els sistemes de coordenades cilíndriques i esfèriques les expressions esdevenen més complexes i es detallen en la següent llista de fórmules de càlcul vectorial.

Notes

• Aquest article utilitza la notació estàndard ISO 80000-2, que reemplaça la ISO 31-11, pel sistema de coordenades esfèriques (altres fonts poden haver revertit la definició dels angles θ i φ):
  • L'angle polar es denota amb la lletra grega θ: es tracta de l'angle entre l'eix positiu z i el radial del vector que connecta l'origen amb el punt en qüestió.
  • L'angle azimutal es denota amb la lletra grega φ i és l'angle entre l'eix x positiu i la projecció del vector radial en el pla xy.
• La funció atan2(x,y) es pot utilitzar en comptes de la funció matemàtica arctan (y/x), atesos el seu domini i imatge. Mentre la clàssica funció arctan té una imatge de (−π/2, +π/2), atan2 es defineix amb una imatge de (−π, π].

Conversions de sistemes de coordenades

Conversions entre sistemes de coordenades cartesianes, cilíndriques i esfèriques

		De
		Cartesià	Cilíndric	Esfèric
A	Cartesià	Plantilla:N/a	$\begin{array}{cc}x& =\rho \cos \phi \\ y& =\rho \sin \phi \\ z& =z\end{array}$	$\begin{array}{cc}x& =r\sin \theta \cos \phi \\ y& =r\sin \theta \sin \phi \\ z& =r\cos \theta \end{array}$
	Cilíndric	$\begin{array}{cc}\rho & =\sqrt{x^2+y^2}\\ \phi & =\arctan \left(\frac{y}{x}\right)\\ z& =z\end{array}$	Plantilla:N/a	$\begin{array}{cc}\rho & =r\sin \theta \\ \phi & =\phi \\ z& =r\cos \theta \end{array}$
	Esfèric	$\begin{array}{cc}r& =\sqrt{x^2+y^2+z^2}\\ \theta & =\arccos \left(\frac{z}{r}\right)\\ \phi & =\arctan \left(\frac{y}{x}\right)\end{array}$	$\begin{array}{cc}r& =\sqrt{{\rho }^2+z^2}\\ \theta & =\arctan \left(\frac{\rho }{z}\right)\\ \phi & =\phi \end{array}$	Plantilla:N/a

Conversions de vectors unitaris

Conversió entre vectors unitaris en sistemes de coordenades cartesianes, cilíndriques i esfèriques en termes de coordenades de destinació

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

Conversió entre vectors unitaris en sistemes de coordenades cartesianes, cilíndriques i esfèriques en termes de coordenades de d'origen

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

Fórmules amb l'operador nabla

Taula amb l'operador nabla en coordenades cartesianes, cilíndriques i esfèriques

Operació	Coordenades cartesianes Plantilla:Math	Coordenades cilíndriques Plantilla:Math	Coordenades esfèriques Plantilla:Math, on θ és l'angle polar i Plantilla:Math és l'angle azimutalPlantilla:Ref
Un camp vectorial Plantilla:Math	$A_x\widehat{𝐱}+A_y\widehat{𝐲}+A_z\widehat{𝐳}$	$A_{\rho }\hat{\mathit{\rho }}+A_{\phi }\hat{\mathit{\phi }}+A_z\widehat{𝐳}$	$A_r\widehat{𝐫}+A_{\theta }\hat{\mathit{\theta }}+A_{\phi }\hat{\mathit{\phi }}$
Gradient Plantilla:Math	$\frac{\partial f}{\partial x}\widehat{𝐱}+\frac{\partial f}{\partial y}\widehat{𝐲}+\frac{\partial f}{\partial z}\widehat{𝐳}$	$\frac{\partial f}{\partial \rho }\hat{\mathit{\rho }}+\frac{1}{\rho }\frac{\partial f}{\partial \phi }\hat{\mathit{\phi }}+\frac{\partial f}{\partial z}\widehat{𝐳}$	$\frac{\partial f}{\partial r}\widehat{𝐫}+\frac{1}{r}\frac{\partial f}{\partial \theta }\hat{\mathit{\theta }}+\frac{1}{r\sin \theta }\frac{\partial f}{\partial \phi }\hat{\mathit{\phi }}$
Divergència Plantilla:Math	$\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}$	$\frac{1}{\rho }\frac{\partial \left(\rho A_{\rho }\right)}{\partial \rho }+\frac{1}{\rho }\frac{\partial A_{\phi }}{\partial \phi }+\frac{\partial A_z}{\partial z}$	$\frac{1}{r^2}\frac{\partial \left(r^2{A}_r\right)}{\partial r}+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }(A_{\theta }\sin \theta )+\frac{1}{r\sin \theta }\frac{\partial A_{\phi }}{\partial \phi }$
Rotacional Plantilla:Math	$\begin{array}{cc}(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})& \widehat{𝐱}\\ +(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})& \widehat{𝐲}\\ +(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})& \widehat{𝐳}\end{array}$	$\begin{array}{cc}(\frac{1}{\rho }\frac{\partial A_z}{\partial \phi }-\frac{\partial A_{\phi }}{\partial z})& \hat{\mathit{\rho }}\\ +(\frac{\partial A_{\rho }}{\partial z}-\frac{\partial A_z}{\partial \rho })& \hat{\mathit{\phi }}\\ +\frac{1}{\rho }(\frac{\partial \left(\rho A_{\phi }\right)}{\partial \rho }-\frac{\partial A_{\rho }}{\partial \phi })& \widehat{𝐳}\end{array}$	$\begin{array}{cc}\frac{1}{r\sin \theta }(\frac{\partial }{\partial \theta }(A_{\phi }\sin \theta )-\frac{\partial A_{\theta }}{\partial \phi })& \widehat{𝐫}\\ +\frac{1}{r}(\frac{1}{\sin \theta }\frac{\partial A_r}{\partial \phi }-\frac{\partial }{\partial r}\left(r{A}_{\phi }\right))& \hat{\mathit{\theta }}\\ +\frac{1}{r}(\frac{\partial }{\partial r}\left(r{A}_{\theta }\right)-\frac{\partial A_r}{\partial \theta })& \hat{\mathit{\phi }}\end{array}$
Operador laplacià Plantilla:Math	$\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{\rho }\frac{\partial }{\partial \rho }\left(\rho \frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2}\frac{{\partial }^{2}f}{\partial {\phi }^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial f}{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial f}{\partial \theta })+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^{2}f}{\partial {\phi }^2}$
Vector laplacià Plantilla:Math	${\mathrm{\nabla }}^2{A}_x\widehat{𝐱}+{\mathrm{\nabla }}^2{A}_y\widehat{𝐲}+{\mathrm{\nabla }}^2{A}_z\widehat{𝐳}$	$\begin{array}{cc}({\mathrm{\nabla }}^2{A}_{\rho }-\frac{A_{\rho }}{{\rho }^2}-\frac{2}{{\rho }^2}\frac{\partial A_{\phi }}{\partial \phi })& \hat{\mathit{\rho }}\\ +({\mathrm{\nabla }}^2{A}_{\phi }-\frac{A_{\phi }}{{\rho }^2}+\frac{2}{{\rho }^2}\frac{\partial A_{\rho }}{\partial \phi })& \hat{\mathit{\phi }}\\ +{\mathrm{\nabla }}^2{A}_z& \widehat{𝐳}\end{array}$	$\begin{array}{cc}({\mathrm{\nabla }}^2{A}_r-\frac{2A_r}{r^2}-\frac{2}{r^2\sin \theta }\frac{\partial (A_{\theta }\sin \theta )}{\partial \theta }-\frac{2}{r^2\sin \theta }\frac{\partial A_{\phi }}{\partial \phi })& \widehat{𝐫}\\ +({\mathrm{\nabla }}^2{A}_{\theta }-\frac{A_{\theta }}{r^2{\sin }^2\theta }+\frac{2}{r^2}\frac{\partial A_r}{\partial \theta }-\frac{2\cos \theta }{r^2{\sin }^2\theta }\frac{\partial A_{\phi }}{\partial \phi })& \hat{\mathit{\theta }}\\ +({\mathrm{\nabla }}^2{A}_{\phi }-\frac{A_{\phi }}{r^2{\sin }^2\theta }+\frac{2}{r^2\sin \theta }\frac{\partial A_r}{\partial \phi }+\frac{2\cos \theta }{r^2{\sin }^2\theta }\frac{\partial A_{\theta }}{\partial \phi })& \hat{\mathit{\phi }}\end{array}$
Derivada materialPlantilla:Ref Plantilla:Math	$𝐀\cdot \mathrm{\nabla }{B}_x\widehat{𝐱}+𝐀\cdot \mathrm{\nabla }{B}_y\widehat{𝐲}+𝐀\cdot \mathrm{\nabla }{B}_z\widehat{𝐳}$	$\begin{array}{cc}(A_{\rho }\frac{\partial B_{\rho }}{\partial \rho }+\frac{A_{\phi }}{\rho }\frac{\partial B_{\rho }}{\partial \phi }+A_z\frac{\partial B_{\rho }}{\partial z}-\frac{A_{\phi }{B}_{\phi }}{\rho })& \hat{\mathit{\rho }}\\ +(A_{\rho }\frac{\partial B_{\phi }}{\partial \rho }+\frac{A_{\phi }}{\rho }\frac{\partial B_{\phi }}{\partial \phi }+A_z\frac{\partial B_{\phi }}{\partial z}+\frac{A_{\phi }{B}_{\rho }}{\rho })& \hat{\mathit{\phi }}\\ +(A_{\rho }\frac{\partial B_z}{\partial \rho }+\frac{A_{\phi }}{\rho }\frac{\partial B_z}{\partial \phi }+A_z\frac{\partial B_z}{\partial z})& \widehat{𝐳}\end{array}$	$\begin{array}{cc}(A_r\frac{\partial B_r}{\partial r}+\frac{A_{\theta }}{r}\frac{\partial B_r}{\partial \theta }+\frac{A_{\phi }}{r\sin \theta }\frac{\partial B_r}{\partial \phi }-\frac{A_{\theta }{B}_{\theta }+A_{\phi }{B}_{\phi }}{r})& \widehat{𝐫}\\ +(A_r\frac{\partial B_{\theta }}{\partial r}+\frac{A_{\theta }}{r}\frac{\partial B_{\theta }}{\partial \theta }+\frac{A_{\phi }}{r\sin \theta }\frac{\partial B_{\theta }}{\partial \phi }+\frac{A_{\theta }{B}_r}{r}-\frac{A_{\phi }{B}_{\phi }\cot \theta }{r})& \hat{\mathit{\theta }}\\ +(A_r\frac{\partial B_{\phi }}{\partial r}+\frac{A_{\theta }}{r}\frac{\partial B_{\phi }}{\partial \theta }+\frac{A_{\phi }}{r\sin \theta }\frac{\partial B_{\phi }}{\partial \phi }+\frac{A_{\phi }{B}_r}{r}+\frac{A_{\phi }{B}_{\theta }\cot \theta }{r})& \hat{\mathit{\phi }}\end{array}$
Tensor de divergència Plantilla:Math	$\begin{array}{cc}(\frac{\partial T_{xx}}{\partial x}+\frac{\partial T_{yx}}{\partial y}+\frac{\partial T_{zx}}{\partial z})& \widehat{𝐱}\\ +(\frac{\partial T_{xy}}{\partial x}+\frac{\partial T_{yy}}{\partial y}+\frac{\partial T_{zy}}{\partial z})& \widehat{𝐲}\\ +(\frac{\partial T_{xz}}{\partial x}+\frac{\partial T_{yz}}{\partial y}+\frac{\partial T_{zz}}{\partial z})& \widehat{𝐳}\end{array}$	$\begin{array}{cc}[\frac{\partial T_{\rho \rho }}{\partial \rho }+\frac{1}{\rho }\frac{\partial T_{\phi \rho }}{\partial \phi }+\frac{\partial T_{z\rho }}{\partial z}+\frac{1}{\rho }(T_{\rho \rho }-T_{\phi \phi })]& \hat{\mathit{\rho }}\\ +[\frac{\partial T_{\rho \phi }}{\partial \rho }+\frac{1}{\rho }\frac{\partial T_{\phi \phi }}{\partial \phi }+\frac{\partial T_{z\phi }}{\partial z}+\frac{1}{\rho }(T_{\rho \phi }+T_{\phi \rho })]& \hat{\mathit{\phi }}\\ +[\frac{\partial T_{\rho z}}{\partial \rho }+\frac{1}{\rho }\frac{\partial T_{\phi z}}{\partial \phi }+\frac{\partial T_{zz}}{\partial z}+\frac{T_{\rho z}}{\rho }]& \widehat{𝐳}\end{array}$	$\begin{array}{cc}[\frac{\partial T_{rr}}{\partial r}+2\frac{T_{rr}}{r}+\frac{1}{r}\frac{\partial T_{\theta r}}{\partial \theta }+\frac{\cot \theta }{r}{T}_{\theta r}+\frac{1}{r\sin \theta }\frac{\partial T_{\phi r}}{\partial \phi }-\frac{1}{r}(T_{\theta \theta }+T_{\phi \phi })]& \widehat{𝐫}\\ +[\frac{\partial T_{r\theta }}{\partial r}+2\frac{T_{r\theta }}{r}+\frac{1}{r}\frac{\partial T_{\theta \theta }}{\partial \theta }+\frac{\cot \theta }{r}{T}_{\theta \theta }+\frac{1}{r\sin \theta }\frac{\partial T_{\phi \theta }}{\partial \phi }+\frac{T_{\theta r}}{r}-\frac{\cot \theta }{r}{T}_{\phi \phi }]& \hat{\mathit{\theta }}\\ +[\frac{\partial T_{r\phi }}{\partial r}+2\frac{T_{r\phi }}{r}+\frac{1}{r}\frac{\partial T_{\theta \phi }}{\partial \theta }+\frac{1}{r\sin \theta }\frac{\partial T_{\phi \phi }}{\partial \phi }+\frac{T_{\phi r}}{r}+\frac{\cot \theta }{r}(T_{\theta \phi }+T_{\phi \theta })]& \hat{\mathit{\phi }}\end{array}$
Desplaçament diferencial Plantilla:Math	$dx\widehat{𝐱}+dy\widehat{𝐲}+dz\widehat{𝐳}$	$d\rho \hat{\mathit{\rho }}+\rho d\phi \hat{\mathit{\phi }}+dz\widehat{𝐳}$	$dr\widehat{𝐫}+rd\theta \hat{\mathit{\theta }}+r\sin \theta d\phi \hat{\mathit{\phi }}$
Normal d'àrea diferencial Plantilla:Math	$\begin{array}{cc}dydz& \widehat{𝐱}\\ +dxdz& \widehat{𝐲}\\ +dxdy& \widehat{𝐳}\end{array}$	$\begin{array}{cc}\rho d\phi dz& \hat{\mathit{\rho }}\\ +d\rho dz& \hat{\mathit{\phi }}\\ +\rho d\rho d\phi & \widehat{𝐳}\end{array}$	$\begin{array}{cc}{r}^2\sin \theta d\theta d\phi & \widehat{𝐫}\\ +r\sin \theta drd\phi & \hat{\mathit{\theta }}\\ +rdrd\theta & \hat{\mathit{\phi }}\end{array}$
Volum diferencial Plantilla:Math	$dxdydz$	$\rho d\rho d\phi dz$	$r^2\sin \theta drd\theta d\phi$

Plantilla:Note Aquesta pàgina utilitza $\theta$ per l'angle polar i $\phi$ per l'angle azimutal, que és la notació habitual en física. La font que s'utilitza per aquestes fórmules utilitza $\theta$ per l'azimut i $\phi$ per l'angle polar, que és la notació habitual en matemàtiques. Per tal d'obternir les fórmules en notació matemàtica, canviï's $\theta$ i $\phi$ en les fórmules de la taula.

Normes de càlcul no trivials

1. $\mathrm{div}\mathrm{grad}f\equiv \mathrm{\nabla }\cdot \mathrm{\nabla }f\equiv {\mathrm{\nabla }}^{2}f$ (Operador laplacià)
2. $\mathrm{curl}\mathrm{grad}f\equiv \mathrm{\nabla }\times \mathrm{\nabla }f=\mathrm{𝟎}$
3. $\mathrm{div}\mathrm{curl}𝐀\equiv \mathrm{\nabla }\cdot (\mathrm{\nabla }\times 𝐀)=0$
4. $\mathrm{curl}\mathrm{curl}𝐀\equiv \mathrm{\nabla }\times (\mathrm{\nabla }\times 𝐀)=\mathrm{\nabla }(\mathrm{\nabla }\cdot 𝐀)-{\mathrm{\nabla }}^2𝐀$
5. ${\mathrm{\nabla }}^2(fg)=f{\mathrm{\nabla }}^{2}g+2\mathrm{\nabla }f\cdot \mathrm{\nabla }g+g{\mathrm{\nabla }}^{2}f$

Derivació cartesiana

Element infinitesimal en coordenades cartesianes

$\begin{array}{cc}\mathrm{div}𝐀=\underset{V\to 0}{\lim }\frac{\underset{\partial V}{\iint }𝐀\cdot d𝐒}{\underset{V}{\iiint }dV}& =\frac{A_x(x+dx)dydz-A_x(x)dydz+A_y(y+dy)dxdz-A_y(y)dxdz+A_z(z+dz)dxdy-A_z(z)dxdy}{dxdydz}\\ & =\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}\end{array}$

$\begin{array}{cc}(\mathrm{curl}𝐀{)}_x=\underset{S^{\perp \widehat{𝐱}}\to 0}{\lim }\frac{\underset{\partial S}{\int }𝐀\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}& =\frac{A_z(z+dz)dz-A_z(z)dz+A_y(y)dy-A_y(y+dy)dy}{dydz}\\ & =\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}\end{array}$

Les expressions per $(\mathrm{curl}𝐀{)}_y$ i $(\mathrm{curl}𝐀{)}_z$ s'obtenen de la mateixa manera.

Derivació cilíndrica

Element infinitesimal en coordenades cilíndriques

$\begin{array}{cc}\mathrm{div}𝐀=\underset{V\to 0}{\lim }\frac{\underset{\partial V}{\iint }𝐀\cdot d𝐒}{\underset{V}{\iiint }dV}& =\frac{A_{\rho }(\rho +d\rho )(\rho +d\rho )d\varphi dz-A_{\rho }(\rho )\rho d\varphi dz+A_{\varphi }(\varphi +d\varphi )d\rho dz-A_{\varphi }(\varphi )d\rho dz+A_z(z+dz)d\rho (\rho +d\rho /2)d\varphi -A_z(z)d\rho (\rho +d\rho /2)d\varphi }{(\rho +d\rho /2)d\varphi d\rho dz}\\ & =\frac{1}{\rho }\frac{\partial (\rho A_{\rho })}{\partial \rho }+\frac{1}{\rho }\frac{\partial A_{\varphi }}{\partial \varphi }+\frac{\partial A_z}{\partial z}\end{array}$

$\begin{array}{cc}(\mathrm{curl}𝐀{)}_{\rho }=\underset{S^{\perp \hat{\mathit{\rho }}}\to 0}{\lim }\frac{\underset{\partial S}{\int }𝐀\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}& =\frac{A_{\varphi }(z)(\rho +d\rho )d\varphi -A_{\varphi }(z+dz)(\rho +d\rho )d\varphi +A_z(\varphi +d\varphi )dz-A_z(\varphi )dz}{(\rho +d\rho )d\varphi dz}\\ & =-\frac{\partial A_{\varphi }}{\partial z}+\frac{1}{\rho }\frac{\partial A_z}{\partial \varphi }\end{array}$

$\begin{array}{cc}(\mathrm{curl}𝐀{)}_{\varphi }=\underset{S^{\perp \hat{\mathit{\varphi }}}\to 0}{\lim }\frac{\underset{\partial S}{\int }𝐀\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}& =\frac{A_z(\rho )dz-A_z(\rho +d\rho )dz+A_{\rho }(z+dz)d\rho -A_{\rho }(z)d\rho }{d\rho dz}\\ & =-\frac{\partial A_z}{\partial \rho }+\frac{\partial A_{\rho }}{\partial z}\end{array}$

$\begin{array}{cc}(\mathrm{curl}𝐀{)}_z=\underset{S^{\perp \widehat{𝒛}}\to 0}{\lim }\frac{\underset{\partial S}{\int }𝐀\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}& =\frac{A_{\rho }(\varphi )d\rho -A_{\rho }(\varphi +d\varphi )d\rho +A_{\varphi }(\rho +d\rho )(\rho +d\rho )d\varphi -A_{\varphi }(\rho )\rho d\varphi }{(\rho +d\rho /2)d\rho d\varphi }\\ & =-\frac{1}{\rho }\frac{\partial A_{\rho }}{\partial \varphi }+\frac{1}{\rho }\frac{\partial (\rho A_{\varphi })}{\partial \rho }\end{array}$

$\mathrm{curl}𝐀=(\mathrm{curl}𝐀{)}_{\rho }\hat{\mathit{\rho }}+(\mathrm{curl}𝐀{)}_{\varphi }\hat{\mathit{\varphi }}+(\mathrm{curl}𝐀{)}_z\widehat{𝒛}=(\frac{1}{\rho }\frac{\partial A_z}{\partial \varphi }-\frac{\partial A_{\varphi }}{\partial z})\hat{\mathit{\rho }}+(\frac{\partial A_{\rho }}{\partial z}-\frac{\partial A_z}{\partial \rho })\hat{\mathit{\varphi }}+\frac{1}{\rho }(\frac{\partial (\rho A_{\varphi })}{\partial \rho }-\frac{\partial A_{\rho }}{\partial \varphi })\widehat{𝒛}$

Derivació esfèrica

Element infinitesimal en coordenades esfèriques.

$\begin{array}{cc}\mathrm{div}𝐀& =\underset{V\to 0}{\lim }\frac{\underset{\partial V}{\iint }𝐀\cdot d𝐒}{\underset{V}{\iiint }dV}\\ & =\frac{A_r(r+dr)(r+dr)d\theta (r+dr)\sin \theta d\varphi -A_r(r)rd\theta r\sin \theta d\varphi +A_{\theta }(\theta +d\theta )\sin (\theta +d\theta )rdrd\varphi -A_{\theta }(\theta )\sin (\theta )rdrd\varphi +A_{\varphi }(\varphi +d\varphi )(r+dr/2)drd\theta -A_{\varphi }(\varphi )(r+dr/2)drd\theta }{drrd\theta r\sin \theta d\varphi }\\ & =\frac{1}{r^2}\frac{\partial (r^2{A}_r)}{\partial r}+\frac{1}{r\sin \theta }\frac{\partial (A_{\theta }\sin \theta )}{\partial \theta }+\frac{1}{r\sin \theta }\frac{\partial A_{\varphi }}{\partial \varphi }\end{array}$

$\begin{array}{cc}(\mathrm{curl}𝐀{)}_r=\underset{S^{\perp \widehat{𝒓}}\to 0}{\lim }\frac{\underset{\partial S}{\int }𝐀\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}& =\frac{A_{\theta }(\varphi )rd\theta +A_{\varphi }(\theta +d\theta )r\sin (\theta +d\theta )d\varphi -A_{\theta }(\varphi +d\varphi )rd\theta -A_{\varphi }(\theta )r\sin (\theta )d\varphi }{rd\theta r\sin \theta d\varphi }\\ & =\frac{1}{r\sin \theta }\frac{\partial (A_{\varphi }\sin \theta )}{\partial \theta }-\frac{1}{r\sin \theta }\frac{\partial A_{\theta }}{\partial \varphi }\end{array}$

$\begin{array}{cc}(\mathrm{curl}𝐀{)}_{\theta }=\underset{S^{\perp \hat{\mathit{\theta }}}\to 0}{\lim }\frac{\underset{\partial S}{\int }𝐀\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}& =\frac{A_{\varphi }(r)r\sin \theta d\varphi +A_r(\varphi +d\varphi )dr-A_{\varphi }(r+dr)(r+dr)\sin \theta d\varphi -A_r(\varphi )dr}{drr\sin \theta d\varphi }\\ & =\frac{1}{r\sin \theta }\frac{\partial A_r}{\partial \varphi }-\frac{1}{r}\frac{\partial (r{A}_{\varphi })}{\partial r}\end{array}$

$\begin{array}{cc}(\mathrm{curl}𝐀{)}_{\varphi }=\underset{S^{\perp \hat{\mathit{\varphi }}}\to 0}{\lim }\frac{\underset{\partial S}{\int }𝐀\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}& =\frac{A_r(\theta )dr+A_{\theta }(r+dr)(r+dr)d\theta -A_r(\theta +d\theta )dr-A_{\theta }(r)rd\theta }{(r+dr/2)drd\theta }\\ & =\frac{1}{r}\frac{\partial (r{A}_{\theta })}{\partial r}-\frac{1}{r}\frac{\partial A_r}{\partial \theta }\end{array}$

$\mathrm{curl}𝐀=(\mathrm{curl}𝐀{)}_r\widehat{𝒓}+(\mathrm{curl}𝐀{)}_{\theta }\hat{\mathit{\theta }}+(\mathrm{curl}𝐀{)}_{\varphi }\hat{\mathit{\varphi }}=\frac{1}{r\sin \theta }(\frac{\partial (A_{\varphi }\sin \theta )}{\partial \theta }-\frac{\partial A_{\theta }}{\partial \varphi })\widehat{𝒓}+\frac{1}{r}(\frac{1}{\sin \theta }\frac{\partial A_r}{\partial \varphi }-\frac{\partial (r{A}_{\varphi })}{\partial r})\hat{\mathit{\theta }}+\frac{1}{r}(\frac{\partial (r{A}_{\theta })}{\partial r}-\frac{\partial A_r}{\partial \theta })\hat{\mathit{\varphi }}$

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