En càlcul vectorial , l'operador nabla és un operador diferencial vectorial representat amb el símbol nabla ∇. En coordenades cartesianes tridimensionals R 3 amb coordenades (x , y , z ), l'operador nabla es pot definir com:
∇
=
(
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
{\displaystyle \nabla =\left({\cfrac {\partial }{\partial x}},{\cfrac {\partial }{\partial y}},{\cfrac {\partial }{\partial z}}\right)}
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 .
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 [ modifica ]
Conversions entre sistemes de coordenades cartesianes, cilíndriques i esfèriques
De
Cartesià
Cilíndric
Esfèric
A
Cartesià
N/A
x
=
ρ
cos
φ
y
=
ρ
sin
φ
z
=
z
{\displaystyle {\begin{aligned}x&=\rho \cos \varphi \\y&=\rho \sin \varphi \\z&=z\end{aligned}}}
x
=
r
sin
θ
cos
φ
y
=
r
sin
θ
sin
φ
z
=
r
cos
θ
{\displaystyle {\begin{aligned}x&=r\sin \theta \cos \varphi \\y&=r\sin \theta \sin \varphi \\z&=r\cos \theta \end{aligned}}}
Cilíndric
ρ
=
x
2
+
y
2
φ
=
arctan
(
y
x
)
z
=
z
{\displaystyle {\begin{aligned}\rho &={\sqrt {x^{2}+y^{2}}}\\\varphi &=\arctan \left({\frac {y}{x}}\right)\\z&=z\end{aligned}}}
N/A
ρ
=
r
sin
θ
φ
=
φ
z
=
r
cos
θ
{\displaystyle {\begin{aligned}\rho &=r\sin \theta \\\varphi &=\varphi \\z&=r\cos \theta \end{aligned}}}
Esfèric
r
=
x
2
+
y
2
+
z
2
θ
=
arccos
(
z
r
)
φ
=
arctan
(
y
x
)
{\displaystyle {\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arccos \left({\frac {z}{r}}\right)\\\varphi &=\arctan \left({\frac {y}{x}}\right)\end{aligned}}}
r
=
ρ
2
+
z
2
θ
=
arctan
(
ρ
z
)
φ
=
φ
{\displaystyle {\begin{aligned}r&={\sqrt {\rho ^{2}+z^{2}}}\\\theta &=\arctan {\left({\frac {\rho }{z}}\right)}\\\varphi &=\varphi \end{aligned}}}
N/A
Conversions de vectors unitaris [ modifica ]
Conversió entre vectors unitaris en sistemes de coordenades cartesianes, cilíndriques i esfèriques en termes de coordenades de destinació
Cartesià
Cilíndric
Esfèric
Cartesià
N/A
x
^
=
cos
φ
ρ
^
−
sin
φ
φ
^
y
^
=
sin
φ
ρ
^
+
cos
φ
φ
^
z
^
=
z
^
{\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}&=\cos \varphi {\hat {\boldsymbol {\rho }}}-\sin \varphi {\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {y} }}&=\sin \varphi {\hat {\boldsymbol {\rho }}}+\cos \varphi {\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}}
x
^
=
sin
θ
cos
φ
r
^
+
cos
θ
cos
φ
θ
^
−
sin
φ
φ
^
y
^
=
sin
θ
sin
φ
r
^
+
cos
θ
sin
φ
θ
^
+
cos
φ
φ
^
z
^
=
cos
θ
r
^
−
sin
θ
θ
^
{\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}&=\sin \theta \cos \varphi {\hat {\mathbf {r} }}+\cos \theta \cos \varphi {\hat {\boldsymbol {\theta }}}-\sin \varphi {\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {y} }}&=\sin \theta \sin \varphi {\hat {\mathbf {r} }}+\cos \theta \sin \varphi {\hat {\boldsymbol {\theta }}}+\cos \varphi {\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {z} }}&=\cos \theta {\hat {\mathbf {r} }}-\sin \theta {\hat {\boldsymbol {\theta }}}\end{aligned}}}
Cilíndric
ρ
^
=
x
x
^
+
y
y
^
x
2
+
y
2
φ
^
=
−
y
x
^
+
x
y
^
x
2
+
y
2
z
^
=
z
^
{\displaystyle {\begin{aligned}{\hat {\boldsymbol {\rho }}}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\boldsymbol {\varphi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}}
N/A
ρ
^
=
sin
θ
r
^
+
cos
θ
θ
^
φ
^
=
φ
^
z
^
=
cos
θ
r
^
−
sin
θ
θ
^
{\displaystyle {\begin{aligned}{\hat {\boldsymbol {\rho }}}&=\sin \theta {\hat {\mathbf {r} }}+\cos \theta {\hat {\boldsymbol {\theta }}}\\{\hat {\boldsymbol {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {z} }}&=\cos \theta {\hat {\mathbf {r} }}-\sin \theta {\hat {\boldsymbol {\theta }}}\end{aligned}}}
Esfèric
r
^
=
x
x
^
+
y
y
^
+
z
z
^
x
2
+
y
2
+
z
2
θ
^
=
(
x
x
^
+
y
y
^
)
z
−
(
x
2
+
y
2
)
z
^
x
2
+
y
2
+
z
2
x
2
+
y
2
φ
^
=
−
y
x
^
+
x
y
^
x
2
+
y
2
{\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}+z{\hat {\mathbf {z} }}}{\sqrt {x^{2}+y^{2}+z^{2}}}}\\{\hat {\boldsymbol {\theta }}}&={\frac {\left(x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}\right)z-\left(x^{2}+y^{2}\right){\hat {\mathbf {z} }}}{{\sqrt {x^{2}+y^{2}+z^{2}}}{\sqrt {x^{2}+y^{2}}}}}\\{\hat {\boldsymbol {\varphi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\end{aligned}}}
r
^
=
ρ
ρ
^
+
z
z
^
ρ
2
+
z
2
θ
^
=
z
ρ
^
−
ρ
z
^
ρ
2
+
z
2
φ
^
=
φ
^
{\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&={\frac {\rho {\hat {\boldsymbol {\rho }}}+z{\hat {\mathbf {z} }}}{\sqrt {\rho ^{2}+z^{2}}}}\\{\hat {\boldsymbol {\theta }}}&={\frac {z{\hat {\boldsymbol {\rho }}}-\rho {\hat {\mathbf {z} }}}{\sqrt {\rho ^{2}+z^{2}}}}\\{\hat {\boldsymbol {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\end{aligned}}}
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
N/A
x
^
=
x
ρ
^
−
y
φ
^
x
2
+
y
2
y
^
=
y
ρ
^
+
x
φ
^
x
2
+
y
2
z
^
=
z
^
{\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}&={\frac {x{\hat {\boldsymbol {\rho }}}-y{\hat {\boldsymbol {\varphi }}}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\mathbf {y} }}&={\frac {y{\hat {\boldsymbol {\rho }}}+x{\hat {\boldsymbol {\varphi }}}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}}
x
^
=
x
(
x
2
+
y
2
r
^
+
z
θ
^
)
−
y
x
2
+
y
2
+
z
2
φ
^
x
2
+
y
2
x
2
+
y
2
+
z
2
y
^
=
y
(
x
2
+
y
2
r
^
+
z
θ
^
)
+
x
x
2
+
y
2
+
z
2
φ
^
x
2
+
y
2
x
2
+
y
2
+
z
2
z
^
=
z
r
^
−
x
2
+
y
2
θ
^
x
2
+
y
2
+
z
2
{\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}&={\frac {x\left({\sqrt {x^{2}+y^{2}}}{\hat {\mathbf {r} }}+z{\hat {\boldsymbol {\theta }}}\right)-y{\sqrt {x^{2}+y^{2}+z^{2}}}{\hat {\boldsymbol {\varphi }}}}{{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}}\\{\hat {\mathbf {y} }}&={\frac {y\left({\sqrt {x^{2}+y^{2}}}{\hat {\mathbf {r} }}+z{\hat {\boldsymbol {\theta }}}\right)+x{\sqrt {x^{2}+y^{2}+z^{2}}}{\hat {\boldsymbol {\varphi }}}}{{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}}\\{\hat {\mathbf {z} }}&={\frac {z{\hat {\mathbf {r} }}-{\sqrt {x^{2}+y^{2}}}{\hat {\boldsymbol {\theta }}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}\end{aligned}}}
Cilíndric
ρ
^
=
cos
φ
x
^
+
sin
φ
y
^
φ
^
=
−
sin
φ
x
^
+
cos
φ
y
^
z
^
=
z
^
{\displaystyle {\begin{aligned}{\hat {\boldsymbol {\rho }}}&=\cos \varphi {\hat {\mathbf {x} }}+\sin \varphi {\hat {\mathbf {y} }}\\{\hat {\boldsymbol {\varphi }}}&=-\sin \varphi {\hat {\mathbf {x} }}+\cos \varphi {\hat {\mathbf {y} }}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}}
N/A
ρ
^
=
ρ
r
^
+
z
θ
^
ρ
2
+
z
2
φ
^
=
φ
^
z
^
=
z
r
^
−
ρ
θ
^
ρ
2
+
z
2
{\displaystyle {\begin{aligned}{\hat {\boldsymbol {\rho }}}&={\frac {\rho {\hat {\mathbf {r} }}+z{\hat {\boldsymbol {\theta }}}}{\sqrt {\rho ^{2}+z^{2}}}}\\{\hat {\boldsymbol {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {z} }}&={\frac {z{\hat {\mathbf {r} }}-\rho {\hat {\boldsymbol {\theta }}}}{\sqrt {\rho ^{2}+z^{2}}}}\end{aligned}}}
Esfèric
r
^
=
sin
θ
(
cos
φ
x
^
+
sin
φ
y
^
)
+
cos
θ
z
^
θ
^
=
cos
θ
(
cos
φ
x
^
+
sin
φ
y
^
)
−
sin
θ
z
^
φ
^
=
−
sin
φ
x
^
+
cos
φ
y
^
{\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&=\sin \theta \left(\cos \varphi {\hat {\mathbf {x} }}+\sin \varphi {\hat {\mathbf {y} }}\right)+\cos \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\theta }}}&=\cos \theta \left(\cos \varphi {\hat {\mathbf {x} }}+\sin \varphi {\hat {\mathbf {y} }}\right)-\sin \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\varphi }}}&=-\sin \varphi {\hat {\mathbf {x} }}+\cos \varphi {\hat {\mathbf {y} }}\end{aligned}}}
r
^
=
sin
θ
ρ
^
+
cos
θ
z
^
θ
^
=
cos
θ
ρ
^
−
sin
θ
z
^
φ
^
=
φ
^
{\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&=\sin \theta {\hat {\boldsymbol {\rho }}}+\cos \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\theta }}}&=\cos \theta {\hat {\boldsymbol {\rho }}}-\sin \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\end{aligned}}}
N/A
Fórmules amb l'operador nabla [ modifica ]
Taula amb l'operador nabla en coordenades cartesianes, cilíndriques i esfèriques
Operació
Coordenades cartesianes (x , y , z )
Coordenades cilíndriques (ρ , φ , z )
Coordenades esfèriques (r , θ , φ ) , on θ és l'angle polar i φ α
Un camp vectorial A
A
x
x
^
+
A
y
y
^
+
A
z
z
^
{\displaystyle A_{x}{\hat {\mathbf {x} }}+A_{y}{\hat {\mathbf {y} }}+A_{z}{\hat {\mathbf {z} }}}
A
ρ
ρ
^
+
A
φ
φ
^
+
A
z
z
^
{\displaystyle A_{\rho }{\hat {\boldsymbol {\rho }}}+A_{\varphi }{\hat {\boldsymbol {\varphi }}}+A_{z}{\hat {\mathbf {z} }}}
A
r
r
^
+
A
θ
θ
^
+
A
φ
φ
^
{\displaystyle A_{r}{\hat {\mathbf {r} }}+A_{\theta }{\hat {\boldsymbol {\theta }}}+A_{\varphi }{\hat {\boldsymbol {\varphi }}}}
Gradient ∇f
∂
f
∂
x
x
^
+
∂
f
∂
y
y
^
+
∂
f
∂
z
z
^
{\displaystyle {\partial f \over \partial x}{\hat {\mathbf {x} }}+{\partial f \over \partial y}{\hat {\mathbf {y} }}+{\partial f \over \partial z}{\hat {\mathbf {z} }}}
∂
f
∂
ρ
ρ
^
+
1
ρ
∂
f
∂
φ
φ
^
+
∂
f
∂
z
z
^
{\displaystyle {\partial f \over \partial \rho }{\hat {\boldsymbol {\rho }}}+{1 \over \rho }{\partial f \over \partial \varphi }{\hat {\boldsymbol {\varphi }}}+{\partial f \over \partial z}{\hat {\mathbf {z} }}}
∂
f
∂
r
r
^
+
1
r
∂
f
∂
θ
θ
^
+
1
r
sin
θ
∂
f
∂
φ
φ
^
{\displaystyle {\partial f \over \partial r}{\hat {\mathbf {r} }}+{1 \over r}{\partial f \over \partial \theta }{\hat {\boldsymbol {\theta }}}+{1 \over r\sin \theta }{\partial f \over \partial \varphi }{\hat {\boldsymbol {\varphi }}}}
Divergència ∇ ⋅ A
∂
A
x
∂
x
+
∂
A
y
∂
y
+
∂
A
z
∂
z
{\displaystyle {\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}}
1
ρ
∂
(
ρ
A
ρ
)
∂
ρ
+
1
ρ
∂
A
φ
∂
φ
+
∂
A
z
∂
z
{\displaystyle {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
r
2
∂
(
r
2
A
r
)
∂
r
+
1
r
sin
θ
∂
∂
θ
(
A
θ
sin
θ
)
+
1
r
sin
θ
∂
A
φ
∂
φ
{\displaystyle {1 \over r^{2}}{\partial \left(r^{2}A_{r}\right) \over \partial r}+{1 \over r\sin \theta }{\partial \over \partial \theta }\left(A_{\theta }\sin \theta \right)+{1 \over r\sin \theta }{\partial A_{\varphi } \over \partial \varphi }}
Rotacional ∇ × A
(
∂
A
z
∂
y
−
∂
A
y
∂
z
)
x
^
+
(
∂
A
x
∂
z
−
∂
A
z
∂
x
)
y
^
+
(
∂
A
y
∂
x
−
∂
A
x
∂
y
)
z
^
{\displaystyle {\begin{aligned}\left({\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\right)&{\hat {\mathbf {x} }}\\+\left({\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}}\right)&{\hat {\mathbf {y} }}\\+\left({\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}\right)&{\hat {\mathbf {z} }}\end{aligned}}}
(
1
ρ
∂
A
z
∂
φ
−
∂
A
φ
∂
z
)
ρ
^
+
(
∂
A
ρ
∂
z
−
∂
A
z
∂
ρ
)
φ
^
+
1
ρ
(
∂
(
ρ
A
φ
)
∂
ρ
−
∂
A
ρ
∂
φ
)
z
^
{\displaystyle {\begin{aligned}\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}-{\frac {\partial A_{\varphi }}{\partial z}}\right)&{\hat {\boldsymbol {\rho }}}\\+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right)&{\hat {\boldsymbol {\varphi }}}\\{}+{\frac {1}{\rho }}\left({\frac {\partial \left(\rho A_{\varphi }\right)}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \varphi }}\right)&{\hat {\mathbf {z} }}\end{aligned}}}
1
r
sin
θ
(
∂
∂
θ
(
A
φ
sin
θ
)
−
∂
A
θ
∂
φ
)
r
^
+
1
r
(
1
sin
θ
∂
A
r
∂
φ
−
∂
∂
r
(
r
A
φ
)
)
θ
^
+
1
r
(
∂
∂
r
(
r
A
θ
)
−
∂
A
r
∂
θ
)
φ
^
{\displaystyle {\begin{aligned}{\frac {1}{r\sin \theta }}\left({\frac {\partial }{\partial \theta }}\left(A_{\varphi }\sin \theta \right)-{\frac {\partial A_{\theta }}{\partial \varphi }}\right)&{\hat {\mathbf {r} }}\\{}+{\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \varphi }}-{\frac {\partial }{\partial r}}\left(rA_{\varphi }\right)\right)&{\hat {\boldsymbol {\theta }}}\\{}+{\frac {1}{r}}\left({\frac {\partial }{\partial r}}\left(rA_{\theta }\right)-{\frac {\partial A_{r}}{\partial \theta }}\right)&{\hat {\boldsymbol {\varphi }}}\end{aligned}}}
Operador laplacià ∇²f ≡ ∆f
∂
2
f
∂
x
2
+
∂
2
f
∂
y
2
+
∂
2
f
∂
z
2
{\displaystyle {\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}}
1
ρ
∂
∂
ρ
(
ρ
∂
f
∂
ρ
)
+
1
ρ
2
∂
2
f
∂
φ
2
+
∂
2
f
∂
z
2
{\displaystyle {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
r
2
∂
∂
r
(
r
2
∂
f
∂
r
)
+
1
r
2
sin
θ
∂
∂
θ
(
sin
θ
∂
f
∂
θ
)
+
1
r
2
sin
2
θ
∂
2
f
∂
φ
2
{\displaystyle {1 \over r^{2}}{\partial \over \partial r}\!\left(r^{2}{\partial f \over \partial r}\right)\!+\!{1 \over r^{2}\!\sin \theta }{\partial \over \partial \theta }\!\left(\sin \theta {\partial f \over \partial \theta }\right)\!+\!{1 \over r^{2}\!\sin ^{2}\theta }{\partial ^{2}f \over \partial \varphi ^{2}}}
Vector laplacià ∇²A ≡ ∆A
∇
2
A
x
x
^
+
∇
2
A
y
y
^
+
∇
2
A
z
z
^
{\displaystyle \nabla ^{2}A_{x}{\hat {\mathbf {x} }}+\nabla ^{2}A_{y}{\hat {\mathbf {y} }}+\nabla ^{2}A_{z}{\hat {\mathbf {z} }}}
(
∇
2
A
ρ
−
A
ρ
ρ
2
−
2
ρ
2
∂
A
φ
∂
φ
)
ρ
^
+
(
∇
2
A
φ
−
A
φ
ρ
2
+
2
ρ
2
∂
A
ρ
∂
φ
)
φ
^
+
∇
2
A
z
z
^
{\displaystyle {\begin{aligned}{\mathopen {}}\left(\nabla ^{2}A_{\rho }-{\frac {A_{\rho }}{\rho ^{2}}}-{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\varphi }}{\partial \varphi }}\right){\mathclose {}}&{\hat {\boldsymbol {\rho }}}\\+{\mathopen {}}\left(\nabla ^{2}A_{\varphi }-{\frac {A_{\varphi }}{\rho ^{2}}}+{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\rho }}{\partial \varphi }}\right){\mathclose {}}&{\hat {\boldsymbol {\varphi }}}\\{}+\nabla ^{2}A_{z}&{\hat {\mathbf {z} }}\end{aligned}}}
(
∇
2
A
r
−
2
A
r
r
2
−
2
r
2
sin
θ
∂
(
A
θ
sin
θ
)
∂
θ
−
2
r
2
sin
θ
∂
A
φ
∂
φ
)
r
^
+
(
∇
2
A
θ
−
A
θ
r
2
sin
2
θ
+
2
r
2
∂
A
r
∂
θ
−
2
cos
θ
r
2
sin
2
θ
∂
A
φ
∂
φ
)
θ
^
+
(
∇
2
A
φ
−
A
φ
r
2
sin
2
θ
+
2
r
2
sin
θ
∂
A
r
∂
φ
+
2
cos
θ
r
2
sin
2
θ
∂
A
θ
∂
φ
)
φ
^
{\displaystyle {\begin{aligned}\left(\nabla ^{2}A_{r}-{\frac {2A_{r}}{r^{2}}}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial \left(A_{\theta }\sin \theta \right)}{\partial \theta }}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{\varphi }}{\partial \varphi }}\right)&{\hat {\mathbf {r} }}\\+\left(\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_{\varphi }}{\partial \varphi }}\right)&{\hat {\boldsymbol {\theta }}}\\+\left(\nabla ^{2}A_{\varphi }-{\frac {A_{\varphi }}{r^{2}\sin ^{2}\theta }}+{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{r}}{\partial \varphi }}+{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial A_{\theta }}{\partial \varphi }}\right)&{\hat {\boldsymbol {\varphi }}}\end{aligned}}}
Derivada materialα [1] (A ⋅ ∇)B
A
⋅
∇
B
x
x
^
+
A
⋅
∇
B
y
y
^
+
A
⋅
∇
B
z
z
^
{\displaystyle \mathbf {A} \cdot \nabla B_{x}{\hat {\mathbf {x} }}+\mathbf {A} \cdot \nabla B_{y}{\hat {\mathbf {y} }}+\mathbf {A} \cdot \nabla B_{z}{\hat {\mathbf {z} }}}
(
A
ρ
∂
B
ρ
∂
ρ
+
A
φ
ρ
∂
B
ρ
∂
φ
+
A
z
∂
B
ρ
∂
z
−
A
φ
B
φ
ρ
)
ρ
^
+
(
A
ρ
∂
B
φ
∂
ρ
+
A
φ
ρ
∂
B
φ
∂
φ
+
A
z
∂
B
φ
∂
z
+
A
φ
B
ρ
ρ
)
φ
^
+
(
A
ρ
∂
B
z
∂
ρ
+
A
φ
ρ
∂
B
z
∂
φ
+
A
z
∂
B
z
∂
z
)
z
^
{\displaystyle {\begin{aligned}\left(A_{\rho }{\frac {\partial B_{\rho }}{\partial \rho }}+{\frac {A_{\varphi }}{\rho }}{\frac {\partial B_{\rho }}{\partial \varphi }}+A_{z}{\frac {\partial B_{\rho }}{\partial z}}-{\frac {A_{\varphi }B_{\varphi }}{\rho }}\right)&{\hat {\boldsymbol {\rho }}}\\+\left(A_{\rho }{\frac {\partial B_{\varphi }}{\partial \rho }}+{\frac {A_{\varphi }}{\rho }}{\frac {\partial B_{\varphi }}{\partial \varphi }}+A_{z}{\frac {\partial B_{\varphi }}{\partial z}}+{\frac {A_{\varphi }B_{\rho }}{\rho }}\right)&{\hat {\boldsymbol {\varphi }}}\\+\left(A_{\rho }{\frac {\partial B_{z}}{\partial \rho }}+{\frac {A_{\varphi }}{\rho }}{\frac {\partial B_{z}}{\partial \varphi }}+A_{z}{\frac {\partial B_{z}}{\partial z}}\right)&{\hat {\mathbf {z} }}\end{aligned}}}
(
A
r
∂
B
r
∂
r
+
A
θ
r
∂
B
r
∂
θ
+
A
φ
r
sin
θ
∂
B
r
∂
φ
−
A
θ
B
θ
+
A
φ
B
φ
r
)
r
^
+
(
A
r
∂
B
θ
∂
r
+
A
θ
r
∂
B
θ
∂
θ
+
A
φ
r
sin
θ
∂
B
θ
∂
φ
+
A
θ
B
r
r
−
A
φ
B
φ
cot
θ
r
)
θ
^
+
(
A
r
∂
B
φ
∂
r
+
A
θ
r
∂
B
φ
∂
θ
+
A
φ
r
sin
θ
∂
B
φ
∂
φ
+
A
φ
B
r
r
+
A
φ
B
θ
cot
θ
r
)
φ
^
{\displaystyle {\begin{aligned}\left(A_{r}{\frac {\partial B_{r}}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{r}}{\partial \theta }}+{\frac {A_{\varphi }}{r\sin \theta }}{\frac {\partial B_{r}}{\partial \varphi }}-{\frac {A_{\theta }B_{\theta }+A_{\varphi }B_{\varphi }}{r}}\right)&{\hat {\mathbf {r} }}\\+\left(A_{r}{\frac {\partial B_{\theta }}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{\theta }}{\partial \theta }}+{\frac {A_{\varphi }}{r\sin \theta }}{\frac {\partial B_{\theta }}{\partial \varphi }}+{\frac {A_{\theta }B_{r}}{r}}-{\frac {A_{\varphi }B_{\varphi }\cot \theta }{r}}\right)&{\hat {\boldsymbol {\theta }}}\\+\left(A_{r}{\frac {\partial B_{\varphi }}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{\varphi }}{\partial \theta }}+{\frac {A_{\varphi }}{r\sin \theta }}{\frac {\partial B_{\varphi }}{\partial \varphi }}+{\frac {A_{\varphi }B_{r}}{r}}+{\frac {A_{\varphi }B_{\theta }\cot \theta }{r}}\right)&{\hat {\boldsymbol {\varphi }}}\end{aligned}}}
Tensor de divergència ∇ ⋅ T
(
∂
T
x
x
∂
x
+
∂
T
y
x
∂
y
+
∂
T
z
x
∂
z
)
x
^
+
(
∂
T
x
y
∂
x
+
∂
T
y
y
∂
y
+
∂
T
z
y
∂
z
)
y
^
+
(
∂
T
x
z
∂
x
+
∂
T
y
z
∂
y
+
∂
T
z
z
∂
z
)
z
^
{\displaystyle {\begin{aligned}\left({\frac {\partial T_{xx}}{\partial x}}+{\frac {\partial T_{yx}}{\partial y}}+{\frac {\partial T_{zx}}{\partial z}}\right)&{\hat {\mathbf {x} }}\\+\left({\frac {\partial T_{xy}}{\partial x}}+{\frac {\partial T_{yy}}{\partial y}}+{\frac {\partial T_{zy}}{\partial z}}\right)&{\hat {\mathbf {y} }}\\+\left({\frac {\partial T_{xz}}{\partial x}}+{\frac {\partial T_{yz}}{\partial y}}+{\frac {\partial T_{zz}}{\partial z}}\right)&{\hat {\mathbf {z} }}\end{aligned}}}
[
∂
T
ρ
ρ
∂
ρ
+
1
ρ
∂
T
φ
ρ
∂
φ
+
∂
T
z
ρ
∂
z
+
1
ρ
(
T
ρ
ρ
−
T
φ
φ
)
]
ρ
^
+
[
∂
T
ρ
φ
∂
ρ
+
1
ρ
∂
T
φ
φ
∂
φ
+
∂
T
z
φ
∂
z
+
1
ρ
(
T
ρ
φ
+
T
φ
ρ
)
]
φ
^
+
[
∂
T
ρ
z
∂
ρ
+
1
ρ
∂
T
φ
z
∂
φ
+
∂
T
z
z
∂
z
+
T
ρ
z
ρ
]
z
^
{\displaystyle {\begin{aligned}\left[{\frac {\partial T_{\rho \rho }}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial T_{\varphi \rho }}{\partial \varphi }}+{\frac {\partial T_{z\rho }}{\partial z}}+{\frac {1}{\rho }}(T_{\rho \rho }-T_{\varphi \varphi })\right]&{\hat {\boldsymbol {\rho }}}\\+\left[{\frac {\partial T_{\rho \varphi }}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial T_{\varphi \varphi }}{\partial \varphi }}+{\frac {\partial T_{z\varphi }}{\partial z}}+{\frac {1}{\rho }}(T_{\rho \varphi }+T_{\varphi \rho })\right]&{\hat {\boldsymbol {\varphi }}}\\+\left[{\frac {\partial T_{\rho z}}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial T_{\varphi z}}{\partial \varphi }}+{\frac {\partial T_{zz}}{\partial z}}+{\frac {T_{\rho z}}{\rho }}\right]&{\hat {\mathbf {z} }}\end{aligned}}}
[
∂
T
r
r
∂
r
+
2
T
r
r
r
+
1
r
∂
T
θ
r
∂
θ
+
cot
θ
r
T
θ
r
+
1
r
sin
θ
∂
T
φ
r
∂
φ
−
1
r
(
T
θ
θ
+
T
φ
φ
)
]
r
^
+
[
∂
T
r
θ
∂
r
+
2
T
r
θ
r
+
1
r
∂
T
θ
θ
∂
θ
+
cot
θ
r
T
θ
θ
+
1
r
sin
θ
∂
T
φ
θ
∂
φ
+
T
θ
r
r
−
cot
θ
r
T
φ
φ
]
θ
^
+
[
∂
T
r
φ
∂
r
+
2
T
r
φ
r
+
1
r
∂
T
θ
φ
∂
θ
+
1
r
sin
θ
∂
T
φ
φ
∂
φ
+
T
φ
r
r
+
cot
θ
r
(
T
θ
φ
+
T
φ
θ
)
]
φ
^
{\displaystyle {\begin{aligned}\left[{\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_{\varphi r}}{\partial \varphi }}-{\frac {1}{r}}(T_{\theta \theta }+T_{\varphi \varphi })\right]&{\hat {\mathbf {r} }}\\+\left[{\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_{\varphi \theta }}{\partial \varphi }}+{\frac {T_{\theta r}}{r}}-{\frac {\cot \theta }{r}}T_{\varphi \varphi }\right]&{\hat {\boldsymbol {\theta }}}\\+\left[{\frac {\partial T_{r\varphi }}{\partial r}}+2{\frac {T_{r\varphi }}{r}}+{\frac {1}{r}}{\frac {\partial T_{\theta \varphi }}{\partial \theta }}+{\frac {1}{r\sin \theta }}{\frac {\partial T_{\varphi \varphi }}{\partial \varphi }}+{\frac {T_{\varphi r}}{r}}+{\frac {\cot \theta }{r}}(T_{\theta \varphi }+T_{\varphi \theta })\right]&{\hat {\boldsymbol {\varphi }}}\end{aligned}}}
Desplaçament diferencial dℓ
d
x
x
^
+
d
y
y
^
+
d
z
z
^
{\displaystyle dx\,{\hat {\mathbf {x} }}+dy\,{\hat {\mathbf {y} }}+dz\,{\hat {\mathbf {z} }}}
d
ρ
ρ
^
+
ρ
d
φ
φ
^
+
d
z
z
^
{\displaystyle d\rho \,{\hat {\boldsymbol {\rho }}}+\rho \,d\varphi \,{\hat {\boldsymbol {\varphi }}}+dz\,{\hat {\mathbf {z} }}}
d
r
r
^
+
r
d
θ
θ
^
+
r
sin
θ
d
φ
φ
^
{\displaystyle dr\,{\hat {\mathbf {r} }}+r\,d\theta \,{\hat {\boldsymbol {\theta }}}+r\,\sin \theta \,d\varphi \,{\hat {\boldsymbol {\varphi }}}}
Normal d'àrea diferencial d S
d
y
d
z
x
^
+
d
x
d
z
y
^
+
d
x
d
y
z
^
{\displaystyle {\begin{aligned}dy\,dz&\,{\hat {\mathbf {x} }}\\{}+dx\,dz&\,{\hat {\mathbf {y} }}\\{}+dx\,dy&\,{\hat {\mathbf {z} }}\end{aligned}}}
ρ
d
φ
d
z
ρ
^
+
d
ρ
d
z
φ
^
+
ρ
d
ρ
d
φ
z
^
{\displaystyle {\begin{aligned}\rho \,d\varphi \,dz&\,{\hat {\boldsymbol {\rho }}}\\{}+d\rho \,dz&\,{\hat {\boldsymbol {\varphi }}}\\{}+\rho \,d\rho \,d\varphi &\,{\hat {\mathbf {z} }}\end{aligned}}}
r
2
sin
θ
d
θ
d
φ
r
^
+
r
sin
θ
d
r
d
φ
θ
^
+
r
d
r
d
θ
φ
^
{\displaystyle {\begin{aligned}r^{2}\sin \theta \,d\theta \,d\varphi &\,{\hat {\mathbf {r} }}\\{}+r\sin \theta \,dr\,d\varphi &\,{\hat {\boldsymbol {\theta }}}\\{}+r\,dr\,d\theta &\,{\hat {\boldsymbol {\varphi }}}\end{aligned}}}
Volum diferencial dV
d
x
d
y
d
z
{\displaystyle dx\,dy\,dz}
ρ
d
ρ
d
φ
d
z
{\displaystyle \rho \,d\rho \,d\varphi \,dz}
r
2
sin
θ
d
r
d
θ
d
φ
{\displaystyle r^{2}\sin \theta \,dr\,d\theta \,d\varphi }
^α
θ
{\displaystyle \theta }
φ
{\displaystyle \varphi }
θ
{\displaystyle \theta }
azimut i
φ
{\displaystyle \varphi }
θ
{\displaystyle \theta }
φ
{\displaystyle \varphi }
Normes de càlcul no trivials [ modifica ]
div
grad
f
≡
∇
⋅
∇
f
≡
∇
2
f
{\displaystyle \operatorname {div} \,\operatorname {grad} f\equiv \nabla \cdot \nabla f\equiv \nabla ^{2}f}
Operador laplacià )
curl
grad
f
≡
∇
×
∇
f
=
0
{\displaystyle \operatorname {curl} \,\operatorname {grad} f\equiv \nabla \times \nabla f=\mathbf {0} }
div
curl
A
≡
∇
⋅
(
∇
×
A
)
=
0
{\displaystyle \operatorname {div} \,\operatorname {curl} \mathbf {A} \equiv \nabla \cdot (\nabla \times \mathbf {A} )=0}
curl
curl
A
≡
∇
×
(
∇
×
A
)
=
∇
(
∇
⋅
A
)
−
∇
2
A
{\displaystyle \operatorname {curl} \,\operatorname {curl} \mathbf {A} \equiv \nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} }
∇
2
(
f
g
)
=
f
∇
2
g
+
2
∇
f
⋅
∇
g
+
g
∇
2
f
{\displaystyle \nabla ^{2}(fg)=f\nabla ^{2}g+2\nabla f\cdot \nabla g+g\nabla ^{2}f}
Derivació cartesiana [ modifica ] Element infinitesimal en coordenades cartesianes
div
A
=
lim
V
→
0
∬
∂
V
A
⋅
d
S
∭
V
d
V
=
A
x
(
x
+
d
x
)
d
y
d
z
−
A
x
(
x
)
d
y
d
z
+
A
y
(
y
+
d
y
)
d
x
d
z
−
A
y
(
y
)
d
x
d
z
+
A
z
(
z
+
d
z
)
d
x
d
y
−
A
z
(
z
)
d
x
d
y
d
x
d
y
d
z
=
∂
A
x
∂
x
+
∂
A
y
∂
y
+
∂
A
z
∂
z
{\displaystyle {\begin{aligned}\operatorname {div} \mathbf {A} =\lim _{V\to 0}{\frac {\iint _{\partial V}\mathbf {A} \cdot d\mathbf {S} }{\iiint _{V}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{aligned}}}
(
curl
A
)
x
=
lim
S
⊥
x
^
→
0
∫
∂
S
A
⋅
d
ℓ
∬
S
d
S
=
A
z
(
z
+
d
z
)
d
z
−
A
z
(
z
)
d
z
+
A
y
(
y
)
d
y
−
A
y
(
y
+
d
y
)
d
y
d
y
d
z
=
∂
A
z
∂
y
−
∂
A
y
∂
z
{\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{x}=\lim _{S^{\perp \mathbf {\hat {x}} }\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}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{aligned}}}
Les expressions per
(
curl
A
)
y
{\displaystyle (\operatorname {curl} \mathbf {A} )_{y}}
(
curl
A
)
z
{\displaystyle (\operatorname {curl} \mathbf {A} )_{z}}
Derivació cilíndrica [ modifica ] Element infinitesimal en coordenades cilíndriques
div
A
=
lim
V
→
0
∬
∂
V
A
⋅
d
S
∭
V
d
V
=
A
ρ
(
ρ
+
d
ρ
)
(
ρ
+
d
ρ
)
d
ϕ
d
z
−
A
ρ
(
ρ
)
ρ
d
ϕ
d
z
+
A
ϕ
(
ϕ
+
d
ϕ
)
d
ρ
d
z
−
A
ϕ
(
ϕ
)
d
ρ
d
z
+
A
z
(
z
+
d
z
)
d
ρ
(
ρ
+
d
ρ
/
2
)
d
ϕ
−
A
z
(
z
)
d
ρ
(
ρ
+
d
ρ
/
2
)
d
ϕ
(
ρ
+
d
ρ
/
2
)
d
ϕ
d
ρ
d
z
=
1
ρ
∂
(
ρ
A
ρ
)
∂
ρ
+
1
ρ
∂
A
ϕ
∂
ϕ
+
∂
A
z
∂
z
{\textstyle {\begin{aligned}\operatorname {div} \mathbf {A} =\lim _{V\to 0}{\frac {\iint _{\partial V}\mathbf {A} \cdot d\mathbf {S} }{\iiint _{V}dV}}&={\frac {A_{\rho }(\rho +d\rho )(\rho +d\rho )d\phi \,dz-A_{\rho }(\rho )\rho d\phi \,dz+A_{\phi }(\phi +d\phi )d\rho \,dz-A_{\phi }(\phi )d\rho dz+A_{z}(z+dz)d\rho \,(\rho +d\rho /2)d\phi -A_{z}(z)d\rho \,(\rho +d\rho /2)d\phi }{(\rho +d\rho /2)\,d\phi \,d\rho \,dz}}\\&={\frac {1}{\rho }}{\frac {\partial (\rho A_{\rho })}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial A_{\phi }}{\partial \phi }}+{\frac {\partial A_{z}}{\partial z}}\end{aligned}}}
(
curl
A
)
ρ
=
lim
S
⊥
ρ
^
→
0
∫
∂
S
A
⋅
d
ℓ
∬
S
d
S
=
A
ϕ
(
z
)
(
ρ
+
d
ρ
)
d
ϕ
−
A
ϕ
(
z
+
d
z
)
(
ρ
+
d
ρ
)
d
ϕ
+
A
z
(
ϕ
+
d
ϕ
)
d
z
−
A
z
(
ϕ
)
d
z
(
ρ
+
d
ρ
)
d
ϕ
d
z
=
−
∂
A
ϕ
∂
z
+
1
ρ
∂
A
z
∂
ϕ
{\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\rho }=\lim _{S^{\perp {\boldsymbol {\hat {\rho }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}&={\frac {A_{\phi }(z)(\rho +d\rho )d\phi -A_{\phi }(z+dz)(\rho +d\rho )d\phi +A_{z}(\phi +d\phi )dz-A_{z}(\phi )dz}{(\rho +d\rho )d\phi dz}}\\&=-{\frac {\partial A_{\phi }}{\partial z}}+{\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}\end{aligned}}}
(
curl
A
)
ϕ
=
lim
S
⊥
ϕ
^
→
0
∫
∂
S
A
⋅
d
ℓ
∬
S
d
S
=
A
z
(
ρ
)
d
z
−
A
z
(
ρ
+
d
ρ
)
d
z
+
A
ρ
(
z
+
d
z
)
d
ρ
−
A
ρ
(
z
)
d
ρ
d
ρ
d
z
=
−
∂
A
z
∂
ρ
+
∂
A
ρ
∂
z
{\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\phi }=\lim _{S^{\perp {\boldsymbol {\hat {\phi }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}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{aligned}}}
(
curl
A
)
z
=
lim
S
⊥
z
^
→
0
∫
∂
S
A
⋅
d
ℓ
∬
S
d
S
=
A
ρ
(
ϕ
)
d
ρ
−
A
ρ
(
ϕ
+
d
ϕ
)
d
ρ
+
A
ϕ
(
ρ
+
d
ρ
)
(
ρ
+
d
ρ
)
d
ϕ
−
A
ϕ
(
ρ
)
ρ
d
ϕ
(
ρ
+
d
ρ
/
2
)
d
ρ
d
ϕ
=
−
1
ρ
∂
A
ρ
∂
ϕ
+
1
ρ
∂
(
ρ
A
ϕ
)
∂
ρ
{\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{z}=\lim _{S^{\perp {\boldsymbol {\hat {z}}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}&={\frac {A_{\rho }(\phi )d\rho -A_{\rho }(\phi +d\phi )d\rho +A_{\phi }(\rho +d\rho )(\rho +d\rho )d\phi -A_{\phi }(\rho )\rho d\phi }{(\rho +d\rho /2)d\rho d\phi }}\\&=-{\frac {1}{\rho }}{\frac {\partial A_{\rho }}{\partial \phi }}+{\frac {1}{\rho }}{\frac {\partial (\rho A_{\phi })}{\partial \rho }}\end{aligned}}}
curl
A
=
(
curl
A
)
ρ
ρ
^
+
(
curl
A
)
ϕ
ϕ
^
+
(
curl
A
)
z
z
^
=
(
1
ρ
∂
A
z
∂
ϕ
−
∂
A
ϕ
∂
z
)
ρ
^
+
(
∂
A
ρ
∂
z
−
∂
A
z
∂
ρ
)
ϕ
^
+
1
ρ
(
∂
(
ρ
A
ϕ
)
∂
ρ
−
∂
A
ρ
∂
ϕ
)
z
^
{\displaystyle \operatorname {curl} \mathbf {A} =(\operatorname {curl} \mathbf {A} )_{\rho }\,{\hat {\boldsymbol {\rho }}}+(\operatorname {curl} \mathbf {A} )_{\phi }\,{\hat {\boldsymbol {\phi }}}+(\operatorname {curl} \mathbf {A} )_{z}\,{\hat {\boldsymbol {z}}}=\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}-{\frac {\partial A_{\phi }}{\partial z}}\right){\hat {\boldsymbol {\rho }}}+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right){\hat {\boldsymbol {\phi }}}+{\frac {1}{\rho }}\left({\frac {\partial (\rho A_{\phi })}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \phi }}\right){\hat {\boldsymbol {z}}}}
Derivació esfèrica [ modifica ] Element infinitesimal en coordenades esfèriques.
div
A
=
lim
V
→
0
∬
∂
V
A
⋅
d
S
∭
V
d
V
=
A
r
(
r
+
d
r
)
(
r
+
d
r
)
d
θ
(
r
+
d
r
)
sin
θ
d
ϕ
−
A
r
(
r
)
r
d
θ
r
sin
θ
d
ϕ
+
A
θ
(
θ
+
d
θ
)
sin
(
θ
+
d
θ
)
r
d
r
d
ϕ
−
A
θ
(
θ
)
sin
(
θ
)
r
d
r
d
ϕ
+
A
ϕ
(
ϕ
+
d
ϕ
)
(
r
+
d
r
/
2
)
d
r
d
θ
−
A
ϕ
(
ϕ
)
(
r
+
d
r
/
2
)
d
r
d
θ
d
r
r
d
θ
r
sin
θ
d
ϕ
=
1
r
2
∂
(
r
2
A
r
)
∂
r
+
1
r
sin
θ
∂
(
A
θ
sin
θ
)
∂
θ
+
1
r
sin
θ
∂
A
ϕ
∂
ϕ
{\displaystyle {\begin{aligned}\operatorname {div} \mathbf {A} &=\lim _{V\to 0}{\frac {\iint _{\partial V}\mathbf {A} \cdot d\mathbf {S} }{\iiint _{V}dV}}\\&={\frac {A_{r}(r+dr)(r+dr)d\theta \,(r+dr)\sin \theta d\phi -A_{r}(r)rd\theta \,r\sin \theta d\phi +A_{\theta }(\theta +d\theta )\sin(\theta +d\theta )\,rdrd\phi -A_{\theta }(\theta )\sin(\theta )\,rdrd\phi +A_{\phi }(\phi +d\phi )(r+dr/2)drd\theta -A_{\phi }(\phi )(r+dr/2)drd\theta }{dr\,rd\theta \,r\sin \theta d\phi }}\\&={\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_{\phi }}{\partial \phi }}\end{aligned}}}
(
curl
A
)
r
=
lim
S
⊥
r
^
→
0
∫
∂
S
A
⋅
d
ℓ
∬
S
d
S
=
A
θ
(
ϕ
)
r
d
θ
+
A
ϕ
(
θ
+
d
θ
)
r
sin
(
θ
+
d
θ
)
d
ϕ
−
A
θ
(
ϕ
+
d
ϕ
)
r
d
θ
−
A
ϕ
(
θ
)
r
sin
(
θ
)
d
ϕ
r
d
θ
r
sin
θ
d
ϕ
=
1
r
sin
θ
∂
(
A
ϕ
sin
θ
)
∂
θ
−
1
r
sin
θ
∂
A
θ
∂
ϕ
{\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{r}=\lim _{S^{\perp {\boldsymbol {\hat {r}}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}&={\frac {A_{\theta }(\phi )\,rd\theta +A_{\phi }(\theta +d\theta )\,r\sin(\theta +d\theta )d\phi -A_{\theta }(\phi +d\phi )\,rd\theta -A_{\phi }(\theta )\,r\sin(\theta )d\phi }{rd\theta \,r\sin \theta d\phi }}\\&={\frac {1}{r\sin \theta }}{\frac {\partial (A_{\phi }\sin \theta )}{\partial \theta }}-{\frac {1}{r\sin \theta }}{\frac {\partial A_{\theta }}{\partial \phi }}\end{aligned}}}
(
curl
A
)
θ
=
lim
S
⊥
θ
^
→
0
∫
∂
S
A
⋅
d
ℓ
∬
S
d
S
=
A
ϕ
(
r
)
r
sin
θ
d
ϕ
+
A
r
(
ϕ
+
d
ϕ
)
d
r
−
A
ϕ
(
r
+
d
r
)
(
r
+
d
r
)
sin
θ
d
ϕ
−
A
r
(
ϕ
)
d
r
d
r
r
sin
θ
d
ϕ
=
1
r
sin
θ
∂
A
r
∂
ϕ
−
1
r
∂
(
r
A
ϕ
)
∂
r
{\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\theta }=\lim _{S^{\perp {\boldsymbol {\hat {\theta }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}&={\frac {A_{\phi }(r)\,r\sin \theta d\phi +A_{r}(\phi +d\phi )dr-A_{\phi }(r+dr)(r+dr)\sin \theta d\phi -A_{r}(\phi )dr}{dr\,r\sin \theta d\phi }}\\&={\frac {1}{r\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {1}{r}}{\frac {\partial (rA_{\phi })}{\partial r}}\end{aligned}}}
(
curl
A
)
ϕ
=
lim
S
⊥
ϕ
^
→
0
∫
∂
S
A
⋅
d
ℓ
∬
S
d
S
=
A
r
(
θ
)
d
r
+
A
θ
(
r
+
d
r
)
(
r
+
d
r
)
d
θ
−
A
r
(
θ
+
d
θ
)
d
r
−
A
θ
(
r
)
r
d
θ
(
r
+
d
r
/
2
)
d
r
d
θ
=
1
r
∂
(
r
A
θ
)
∂
r
−
1
r
∂
A
r
∂
θ
{\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\phi }=\lim _{S^{\perp {\boldsymbol {\hat {\phi }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}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 (rA_{\theta })}{\partial r}}-{\frac {1}{r}}{\frac {\partial A_{r}}{\partial \theta }}\end{aligned}}}
curl
A
=
(
curl
A
)
r
r
^
+
(
curl
A
)
θ
θ
^
+
(
curl
A
)
ϕ
ϕ
^
=
1
r
sin
θ
(
∂
(
A
ϕ
sin
θ
)
∂
θ
−
∂
A
θ
∂
ϕ
)
r
^
+
1
r
(
1
sin
θ
∂
A
r
∂
ϕ
−
∂
(
r
A
ϕ
)
∂
r
)
θ
^
+
1
r
(
∂
(
r
A
θ
)
∂
r
−
∂
A
r
∂
θ
)
ϕ
^
{\displaystyle \operatorname {curl} \mathbf {A} =(\operatorname {curl} \mathbf {A} )_{r}\,{\hat {\boldsymbol {r}}}+(\operatorname {curl} \mathbf {A} )_{\theta }\,{\hat {\boldsymbol {\theta }}}+(\operatorname {curl} \mathbf {A} )_{\phi }\,{\hat {\boldsymbol {\phi }}}={\frac {1}{r\sin \theta }}\left({\frac {\partial (A_{\phi }\sin \theta )}{\partial \theta }}-{\frac {\partial A_{\theta }}{\partial \phi }}\right){\hat {\boldsymbol {r}}}+{\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {\partial (rA_{\phi })}{\partial r}}\right){\hat {\boldsymbol {\theta }}}+{\frac {1}{r}}\left({\frac {\partial (rA_{\theta })}{\partial r}}-{\frac {\partial A_{r}}{\partial \theta }}\right){\hat {\boldsymbol {\phi }}}}
![{\displaystyle {\begin{aligned}\left[{\frac {\partial T_{\rho \rho }}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial T_{\varphi \rho }}{\partial \varphi }}+{\frac {\partial T_{z\rho }}{\partial z}}+{\frac {1}{\rho }}(T_{\rho \rho }-T_{\varphi \varphi })\right]&{\hat {\boldsymbol {\rho }}}\\+\left[{\frac {\partial T_{\rho \varphi }}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial T_{\varphi \varphi }}{\partial \varphi }}+{\frac {\partial T_{z\varphi }}{\partial z}}+{\frac {1}{\rho }}(T_{\rho \varphi }+T_{\varphi \rho })\right]&{\hat {\boldsymbol {\varphi }}}\\+\left[{\frac {\partial T_{\rho z}}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial T_{\varphi z}}{\partial \varphi }}+{\frac {\partial T_{zz}}{\partial z}}+{\frac {T_{\rho z}}{\rho }}\right]&{\hat {\mathbf {z} }}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3116bd7e75151c8d599b5e090d2433daffa21069)
![{\displaystyle {\begin{aligned}\left[{\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_{\varphi r}}{\partial \varphi }}-{\frac {1}{r}}(T_{\theta \theta }+T_{\varphi \varphi })\right]&{\hat {\mathbf {r} }}\\+\left[{\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_{\varphi \theta }}{\partial \varphi }}+{\frac {T_{\theta r}}{r}}-{\frac {\cot \theta }{r}}T_{\varphi \varphi }\right]&{\hat {\boldsymbol {\theta }}}\\+\left[{\frac {\partial T_{r\varphi }}{\partial r}}+2{\frac {T_{r\varphi }}{r}}+{\frac {1}{r}}{\frac {\partial T_{\theta \varphi }}{\partial \theta }}+{\frac {1}{r\sin \theta }}{\frac {\partial T_{\varphi \varphi }}{\partial \varphi }}+{\frac {T_{\varphi r}}{r}}+{\frac {\cot \theta }{r}}(T_{\theta \varphi }+T_{\varphi \theta })\right]&{\hat {\boldsymbol {\varphi }}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89522d9531233ee5f0a581ec441bb20c337a13b8)
