Operator nabla w różnych układach współrzędnych

Poniżej zestawiono listę formuł analizy wektorowej, gdy prowadzi się obliczenia w układach współrzędnych krzywoliniowych. W przypadkach szczególnych, np. we współrzędnych kartezjańskich, poniższe wzory upraszczają się.

Uwagi

  • Zastosowano tu typowe oznaczenia współrzędnych stosowane w fizyce. Np. dla współrzędnych sferycznych:
  1. θ {\displaystyle \theta } oznacza kąt między osią z {\displaystyle z} a wektorem wodzącym łączącym początek układu z rozpatrywanym punktem
  2. ϕ {\displaystyle \phi } oznacza kąt pomiędzy rzutem wektora wodzącego na płaszczyznę x y {\displaystyle xy} a osią x . {\displaystyle x.}
  3. (W niektórych źródłach definicje θ {\displaystyle \theta } i ϕ {\displaystyle \phi } są zamienione, więc znaczenie należy wywnioskować z kontekstu.)
  • Zamiast symbolu funkcji arctg ⁡ ( y / x ) {\displaystyle \operatorname {arctg} (y/x)} używa się symbolu arctg2 ⁡ ( y , x ) {\displaystyle \operatorname {arctg2} (y,x)} dla wskazania, że funkcja arctg2 ⁡ ( y , x ) {\displaystyle \operatorname {arctg2} (y,x)} ma przeciwdziedzinę ( − π , π ] {\displaystyle (-\pi ,\pi ]} (podczas gdy funkcji arctg ⁡ ( y / x ) {\displaystyle \operatorname {arctg} (y/x)} ma przeciwdziedzinę ( − π / 2 , + π / 2 ) {\displaystyle (-\pi /2,+\pi /2)} )
  • Wyrażenia na operator nabla we współrzędnych sferycznych mogą wymagać poprawy.

UWAGA: Niektóre symbole użyte w tabeli powtarzają się, mimo że odnoszą się do innych wielkości (ich znaczenie można odczytać z kontekstu)

Tabela z operatorem nabla we współrzędnych walcowych, sferycznych oraz parabolicznych walcowych
Operacja Współrzędne kartezjańskie ( x , y , z ) {\displaystyle (x,y,z)} Współrzędne walcowe ( ρ , ϕ , z ) {\displaystyle (\rho ,\phi ,z)} Współrzędne sferyczne ( r , θ , ϕ ) {\displaystyle (r,\theta ,\phi )} Współrzędne paraboliczne walcowe ( σ , τ , z ) {\displaystyle (\sigma ,\tau ,z)}
Definicje
współrzędnych 		ρ = x 2 + y 2 ϕ = arctg ⁡ ( y / x ) z = z {\displaystyle {\begin{matrix}\rho &=&{\sqrt {x^{2}+y^{2}}}\\\phi &=&\operatorname {arctg} (y/x)\\z&=&z\end{matrix}}} x = ρ cos ⁡ ϕ y = ρ sin ⁡ ϕ z = z {\displaystyle {\begin{matrix}x&=&\rho \cos \phi \\y&=&\rho \sin \phi \\z&=&z\end{matrix}}} x = r sin ⁡ θ cos ⁡ ϕ y = r sin ⁡ θ sin ⁡ ϕ z = r cos ⁡ θ {\displaystyle {\begin{matrix}x&=&r\sin \theta \cos \phi \\y&=&r\sin \theta \sin \phi \\z&=&r\cos \theta \end{matrix}}} x = σ τ y = 1 2 ( τ 2 − σ 2 ) z = z {\displaystyle {\begin{matrix}x&=&\sigma \tau \\y&=&{\frac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=&z\end{matrix}}}
r = x 2 + y 2 + z 2 θ = arccos ⁡ ( z / r ) ϕ = arctg ⁡ ( y / x ) {\displaystyle {\begin{matrix}r&=&{\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=&\arccos(z/r)\\\phi &=&\operatorname {arctg} (y/x)\end{matrix}}} r = ρ 2 + z 2 θ = arctg ⁡ ( ρ / z ) ϕ = ϕ {\displaystyle {\begin{matrix}r&=&{\sqrt {\rho ^{2}+z^{2}}}\\\theta &=&\operatorname {arctg} {(\rho /z)}\\\phi &=&\phi \end{matrix}}} ρ = r sin ⁡ ( θ ) ϕ = ϕ z = r cos ⁡ ( θ ) {\displaystyle {\begin{matrix}\rho &=&r\sin(\theta )\\\phi &=&\phi \\z&=&r\cos(\theta )\end{matrix}}} ρ cos ⁡ ϕ = σ τ ρ sin ⁡ ϕ = 1 2 ( τ 2 − σ 2 ) z = z {\displaystyle {\begin{matrix}\rho \cos \phi &=&\sigma \tau \\\rho \sin \phi &=&{\frac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=&z\end{matrix}}}
Definicje
wersorów 		ρ ^ = x ρ x ^ + y ρ y ^ ϕ ^ = − y ρ x ^ + x ρ y ^ z ^ = z ^ {\displaystyle {\begin{matrix}{\boldsymbol {\hat {\rho }}}&=&{\frac {x}{\rho }}\mathbf {\hat {x}} +{\frac {y}{\rho }}\mathbf {\hat {y}} \\{\boldsymbol {\hat {\phi }}}&=&-{\frac {y}{\rho }}\mathbf {\hat {x}} +{\frac {x}{\rho }}\mathbf {\hat {y}} \\\mathbf {\hat {z}} &=&\mathbf {\hat {z}} \end{matrix}}} x ^ = cos ⁡ ϕ ρ ^ − sin ⁡ ϕ ϕ ^ y ^ = sin ⁡ ϕ ρ ^ + cos ⁡ ϕ ϕ ^ z ^ = z ^ {\displaystyle {\begin{matrix}\mathbf {\hat {x}} &=&\cos \phi {\boldsymbol {\hat {\rho }}}-\sin \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {y}} &=&\sin \phi {\boldsymbol {\hat {\rho }}}+\cos \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=&\mathbf {\hat {z}} \end{matrix}}} 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{matrix}\mathbf {\hat {x}} &=&\sin \theta \cos \phi {\boldsymbol {\hat {r}}}+\cos \theta \cos \phi {\boldsymbol {\hat {\theta }}}-\sin \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {y}} &=&\sin \theta \sin \phi {\boldsymbol {\hat {r}}}+\cos \theta \sin \phi {\boldsymbol {\hat {\theta }}}+\cos \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=&\cos \theta {\boldsymbol {\hat {r}}}-\sin \theta {\boldsymbol {\hat {\theta }}}\end{matrix}}} σ ^ = τ τ 2 + σ 2 x ^ − σ τ 2 + σ 2 y ^ τ ^ = σ τ 2 + σ 2 x ^ + τ τ 2 + σ 2 y ^ z ^ = z ^ {\displaystyle {\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}}}
r ^ = x x ^ + y y ^ + z z ^ r θ ^ = x z x ^ + y z y ^ − ρ 2 z ^ r ρ ϕ ^ = − y x ^ + x y ^ ρ {\displaystyle {\begin{matrix}\mathbf {\hat {r}} &=&{\frac {x\mathbf {\hat {x}} +y\mathbf {\hat {y}} +z\mathbf {\hat {z}} }{r}}\\{\boldsymbol {\hat {\theta }}}&=&{\frac {xz\mathbf {\hat {x}} +yz\mathbf {\hat {y}} -\rho ^{2}\mathbf {\hat {z}} }{r\rho }}\\{\boldsymbol {\hat {\phi }}}&=&{\frac {-y\mathbf {\hat {x}} +x\mathbf {\hat {y}} }{\rho }}\end{matrix}}} r ^ = ρ r ρ ^ + z r z ^ θ ^ = z r ρ ^ − ρ r z ^ ϕ ^ = ϕ ^ {\displaystyle {\begin{matrix}\mathbf {\hat {r}} &=&{\frac {\rho }{r}}{\boldsymbol {\hat {\rho }}}+{\frac {z}{r}}\mathbf {\hat {z}} \\{\boldsymbol {\hat {\theta }}}&=&{\frac {z}{r}}{\boldsymbol {\hat {\rho }}}-{\frac {\rho }{r}}\mathbf {\hat {z}} \\{\boldsymbol {\hat {\phi }}}&=&{\boldsymbol {\hat {\phi }}}\end{matrix}}} ρ ^ = sin ⁡ θ r ^ + cos ⁡ θ θ ^ ϕ ^ = ϕ ^ z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ {\displaystyle {\begin{matrix}{\boldsymbol {\hat {\rho }}}&=&\sin \theta \mathbf {\hat {r}} +\cos \theta {\boldsymbol {\hat {\theta }}}\\{\boldsymbol {\hat {\phi }}}&=&{\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=&\cos \theta \mathbf {\hat {r}} -\sin \theta {\boldsymbol {\hat {\theta }}}\end{matrix}}}  
Pole wektorowe A {\displaystyle \mathbf {A} } A x x ^ + A y y ^ + A z z ^ {\displaystyle A_{x}\mathbf {\hat {x}} +A_{y}\mathbf {\hat {y}} +A_{z}\mathbf {\hat {z}} } A ρ ρ ^ + A ϕ ϕ ^ + A z z ^ {\displaystyle A_{\rho }{\boldsymbol {\hat {\rho }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}+A_{z}{\boldsymbol {\hat {z}}}} A r r ^ + A θ θ ^ + A ϕ ϕ ^ {\displaystyle A_{r}{\boldsymbol {\hat {r}}}+A_{\theta }{\boldsymbol {\hat {\theta }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}} A σ σ ^ + A τ τ ^ + A ϕ z ^ {\displaystyle A_{\sigma }{\boldsymbol {\hat {\sigma }}}+A_{\tau }{\boldsymbol {\hat {\tau }}}+A_{\phi }{\boldsymbol {\hat {z}}}}
Gradient ∇ f {\displaystyle \nabla f} ∂ f ∂ x x ^ + ∂ f ∂ y y ^ + ∂ f ∂ z z ^ {\displaystyle {\frac {\partial f}{\partial x}}\mathbf {\hat {x}} +{\frac {\partial f}{\partial y}}\mathbf {\hat {y}} +{\frac {\partial f}{\partial z}}\mathbf {\hat {z}} } ∂ f ∂ ρ ρ ^ + 1 ρ ∂ f ∂ ϕ ϕ ^ + ∂ f ∂ z z ^ {\displaystyle {\frac {\partial f}{\partial \rho }}{\boldsymbol {\hat {\rho }}}+{\frac {1}{\rho }}{\frac {\partial f}{\partial \phi }}{\boldsymbol {\hat {\phi }}}+{\frac {\partial f}{\partial z}}{\boldsymbol {\hat {z}}}} ∂ f ∂ r r ^ + 1 r ∂ f ∂ θ θ ^ + 1 r sin ⁡ θ ∂ f ∂ ϕ ϕ ^ {\displaystyle {\frac {\partial f}{\partial r}}{\boldsymbol {\hat {r}}}+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}{\boldsymbol {\hat {\theta }}}+{\frac {1}{r\sin \theta }}{\frac {\partial f}{\partial \phi }}{\boldsymbol {\hat {\phi }}}} 1 σ 2 + τ 2 ∂ f ∂ σ σ ^ + 1 σ 2 + τ 2 ∂ f ∂ τ τ ^ + ∂ f ∂ z z ^ {\displaystyle {\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial f}{\partial \sigma }}{\boldsymbol {\hat {\sigma }}}+{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial f}{\partial \tau }}{\boldsymbol {\hat {\tau }}}+{\frac {\partial f}{\partial z}}{\boldsymbol {\hat {z}}}}
Dywergencja ∇ ⋅ A {\displaystyle \nabla \cdot \mathbf {A} } ∂ A x ∂ x + ∂ A y ∂ y + ∂ A z ∂ z {\displaystyle {\frac {\partial A_{x}}{\partial x}}+{\frac {\partial A_{y}}{\partial y}}+{\frac {\partial A_{z}}{\partial z}}} 1 ρ ∂ ( ρ A ρ ) ∂ ρ + 1 ρ ∂ A ϕ ∂ ϕ + ∂ A z ∂ z {\displaystyle {\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}}} 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 {\frac {1}{r^{2}}}{\frac {\partial \left(r^{2}A_{r}\right)}{\partial r}}+{\frac {1}{r\sin \theta }}{\frac {\partial }{\partial \theta }}\left(A_{\theta }\sin \theta \right)+{\frac {1}{r\sin \theta }}{\frac {\partial A_{\phi }}{\partial \phi }}} 1 σ 2 + τ 2 ∂ ( A σ σ 2 + τ 2 ) ∂ σ + 1 σ 2 + τ 2 ∂ ( A τ σ 2 + τ 2 ) ∂ τ + ∂ A z ∂ z {\displaystyle {\frac {1}{\sigma ^{2}+\tau ^{2}}}{\frac {\partial \left(A_{\sigma }{\sqrt {\sigma ^{2}+\tau ^{2}}}\right)}{\partial \sigma }}+{\frac {1}{\sigma ^{2}+\tau ^{2}}}{\frac {\partial \left(A_{\tau }{\sqrt {\sigma ^{2}+\tau ^{2}}}\right)}{\partial \tau }}+{\frac {\partial A_{z}}{\partial z}}}
Rotacja ∇ × A {\displaystyle \nabla \times \mathbf {A} } ( ∂ A z ∂ y − ∂ A y ∂ z ) x ^ + ( ∂ A x ∂ z − ∂ A z ∂ x ) y ^ + ( ∂ A y ∂ x − ∂ A x ∂ y ) z ^   {\displaystyle {\begin{matrix}\displaystyle \left({\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\right)\mathbf {\hat {x}} &+\\\displaystyle \left({\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}}\right)\mathbf {\hat {y}} &+\\\displaystyle \left({\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}\right)\mathbf {\hat {z}} &\ \end{matrix}}} ( 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{matrix}\displaystyle \left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}-{\frac {\partial A_{\phi }}{\partial z}}\right){\boldsymbol {\hat {\rho }}}&+\\\displaystyle \left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right){\boldsymbol {\hat {\phi }}}&+\\\displaystyle {\frac {1}{\rho }}\left({\frac {\partial \left(\rho A_{\phi }\right)}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \phi }}\right){\boldsymbol {\hat {z}}}&\ \end{matrix}}} 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{matrix}\displaystyle {\frac {1}{r\sin \theta }}\left({\frac {\partial }{\partial \theta }}\left(A_{\phi }\sin \theta \right)-{\frac {\partial A_{\theta }}{\partial \phi }}\right){\boldsymbol {\hat {r}}}&+\\\displaystyle {\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {\partial }{\partial r}}\left(rA_{\phi }\right)\right){\boldsymbol {\hat {\theta }}}&+\\\displaystyle {\frac {1}{r}}\left({\frac {\partial }{\partial r}}\left(rA_{\theta }\right)-{\frac {\partial A_{r}}{\partial \theta }}\right){\boldsymbol {\hat {\phi }}}&\ \end{matrix}}} ( 1 σ 2 + τ 2 ∂ A z ∂ τ − ∂ A τ ∂ z ) σ ^ − ( 1 σ 2 + τ 2 ∂ A z ∂ σ − ∂ A σ ∂ z ) τ ^ + 1 σ 2 + τ 2 ( ∂ ( ρ A ϕ ) ∂ ρ − ∂ A ρ ∂ ϕ ) z ^   {\displaystyle {\begin{matrix}\displaystyle \left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial A_{z}}{\partial \tau }}-{\frac {\partial A_{\tau }}{\partial z}}\right){\boldsymbol {\hat {\sigma }}}&-\\\displaystyle \left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial A_{z}}{\partial \sigma }}-{\frac {\partial A_{\sigma }}{\partial z}}\right){\boldsymbol {\hat {\tau }}}&+\\\displaystyle {\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}\left({\frac {\partial \left(\rho A_{\phi }\right)}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \phi }}\right){\boldsymbol {\hat {z}}}&\ \end{matrix}}}
Operator Laplace’a Δ f = ∇ 2 f {\displaystyle \Delta f=\nabla ^{2}f} ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2 {\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}} 1 ρ ∂ ∂ ρ ( ρ ∂ f ∂ ρ ) + 1 ρ 2 ∂ 2 f ∂ ϕ 2 + ∂ 2 f ∂ z 2 {\displaystyle {\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}}}} 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 {\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 }}\!\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)\!+\!{\frac {1}{r^{2}\!\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \phi ^{2}}}} 1 σ 2 + τ 2 ( ∂ 2 f ∂ σ 2 + ∂ 2 f ∂ τ 2 ) + ∂ 2 f ∂ z 2 {\displaystyle {\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}}}}
Laplasjan wektorowy Δ A = ∇ 2 A {\displaystyle \Delta \mathbf {A} =\nabla ^{2}\mathbf {A} } Δ A x x ^ + Δ A y y ^ + Δ A z z ^ {\displaystyle \Delta A_{x}\mathbf {\hat {x}} +\Delta A_{y}\mathbf {\hat {y}} +\Delta A_{z}\mathbf {\hat {z}} } ( Δ A ρ − A ρ ρ 2 − 2 ρ 2 ∂ A ϕ ∂ ϕ ) ρ ^ + ( Δ A ϕ − A ϕ ρ 2 + 2 ρ 2 ∂ A ρ ∂ ϕ ) ϕ ^ + ( Δ A z ) z ^   {\displaystyle {\begin{matrix}\displaystyle \left(\Delta A_{\rho }-{\frac {A_{\rho }}{\rho ^{2}}}-{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\phi }}{\partial \phi }}\right){\boldsymbol {\hat {\rho }}}&+\\\displaystyle \left(\Delta A_{\phi }-{\frac {A_{\phi }}{\rho ^{2}}}+{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\rho }}{\partial \phi }}\right){\boldsymbol {\hat {\phi }}}&+\\\displaystyle \left(\Delta A_{z}\right){\boldsymbol {\hat {z}}}&\ \end{matrix}}} ( Δ 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 ^ + ( Δ 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 ^ + ( Δ 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{matrix}\left(\Delta 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_{\phi }}{\partial \phi }}\right){\boldsymbol {\hat {r}}}&+\\\left(\Delta 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 }}\right){\boldsymbol {\hat {\theta }}}&+\\\left(\Delta 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 }}\right){\boldsymbol {\hat {\phi }}}\end{matrix}}}
Różniczka przesunięcia d l = d x x ^ + d y y ^ + d z z ^ {\displaystyle d\mathbf {l} =dx\mathbf {\hat {x}} +dy\mathbf {\hat {y}} +dz\mathbf {\hat {z}} } d l = d ρ ρ ^ + ρ d ϕ ϕ ^ + d z z ^ {\displaystyle d\mathbf {l} =d\rho {\boldsymbol {\hat {\rho }}}+\rho d\phi {\boldsymbol {\hat {\phi }}}+dz{\boldsymbol {\hat {z}}}} d l = d r r ^ + r d θ θ ^ + r sin ⁡ θ d ϕ ϕ ^ {\displaystyle d\mathbf {l} =dr\mathbf {\hat {r}} +rd\theta {\boldsymbol {\hat {\theta }}}+r\sin \theta d\phi {\boldsymbol {\hat {\phi }}}} d l = σ 2 + τ 2 d σ σ ^ + σ 2 + τ 2 d τ τ ^ + d z z ^ {\displaystyle 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}}}}
Różniczki powierzchni d S = d y d z x ^ + d x d z y ^ + d x d y z ^ {\displaystyle {\begin{matrix}d\mathbf {S} =&dy\,dz\,\mathbf {\hat {x}} +\\&dx\,dz\,\mathbf {\hat {y}} +\\&dx\,dy\,\mathbf {\hat {z}} \end{matrix}}} d S = ρ d ϕ d z ρ ^ + d ρ d z ϕ ^ + ρ d ρ d ϕ z ^ {\displaystyle {\begin{matrix}d\mathbf {S} =&\rho \,d\phi \,dz\,{\boldsymbol {\hat {\rho }}}+\\&d\rho \,dz\,{\boldsymbol {\hat {\phi }}}+\\&\rho \,d\rho d\phi \,\mathbf {\hat {z}} \end{matrix}}} d S = 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{matrix}d\mathbf {S} =&r^{2}\sin \theta \,d\theta \,d\phi \,\mathbf {\hat {r}} +\\&r\sin \theta \,dr\,d\phi \,{\boldsymbol {\hat {\theta }}}+\\&r\,dr\,d\theta \,{\boldsymbol {\hat {\phi }}}\end{matrix}}} d S = σ 2 + τ 2 , d τ d z σ ^ + σ 2 + τ 2 d σ d z τ ^ + σ 2 + τ 2 d σ , d τ z ^ {\displaystyle {\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}}}
Różniczka objętości d V = d x d y d z {\displaystyle dV=dx\,dy\,dz} d V = ρ d ρ d ϕ d z {\displaystyle dV=\rho \,d\rho \,d\phi \,dz} d V = r 2 sin ⁡ θ d r d θ d ϕ {\displaystyle dV=r^{2}\sin \theta \,dr\,d\theta \,d\phi } d V = ( σ 2 + τ 2 ) d σ d τ d z , {\displaystyle dV=\left(\sigma ^{2}+\tau ^{2}\right)d\sigma d\tau dz,}
Nietrywialne reguły rachunkowe:
  1. div grad ⁡ f ≡ ∇ ⋅ ∇ f = ∇ 2 f ≡ Δ f {\displaystyle \operatorname {div} \,\operatorname {grad} f\equiv \nabla \cdot \nabla f=\nabla ^{2}f\equiv \Delta f} (Laplasjan)
  2. rot grad ⁡ f ≡ ∇ × ∇ f = 0 {\displaystyle \operatorname {rot} \,\operatorname {grad} f\equiv \nabla \times \nabla f=\mathbf {0} }
  3. div rot ⁡ A ≡ ∇ ⋅ ( ∇ × A ) = 0 {\displaystyle \operatorname {div} \,\operatorname {rot} \mathbf {A} \equiv \nabla \cdot (\nabla \times \mathbf {A} )=0}
  4. rot rot ⁡ A = ∇ × ( ∇ × A ) = ∇ ( ∇ ⋅ A ) − ∇ 2 A {\displaystyle \operatorname {rot} \,\operatorname {rot} \mathbf {A} =\nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} } (stosując formułę Lagrange’a na iloczyn wektorowy)
  5. Δ ( f g ) = f Δ g + 2 ∇ f ⋅ ∇ g + g Δ f {\displaystyle \Delta (fg)=f\Delta g+2\nabla f\cdot \nabla g+g\Delta f}

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