Del i cylindriska och sfäriska koordinater

Det här är en lista över några vektorkalkylformler för att arbeta med vanliga kurvlinjära koordinatsystem .

Anteckningar

Den här artikeln använder standardnotationen ISO 80000-2 , som ersätter ISO 31-11 , för sfäriska koordinater (andra källor kan ändra definitionerna av θ och φ ):
- Den polära vinkeln betecknas med $\theta \in [0,\pi ]$ : det är vinkeln mellan z -axeln och den radiella vektorn som förbinder origo med den aktuella punkten.
- Den azimutala vinkeln betecknas med $\varphi \in [0,2\pi ]$ : det är vinkeln mellan x -axeln och projektionen av den radiella vektorn på xy -planet .
Funktionen atan2 ( y , x ) kan användas istället för den matematiska funktionen arctan ( y / x ) på grund av dess domän och bild . Den klassiska arktanfunktionen har en bild av (−π/2, +π/2) , medan atan2 är definierad att ha en bild av (−π, π] .

Koordinera konverteringar

Omvandling mellan kartesiska, cylindriska och sfäriska koordinater
		Från
		kartesiska	Cylindrisk	Sfärisk
Till	kartesiska	—	${\begin{aligned}x&=\rho \cos \varphi \\y&=\rho \sin \varphi \\z&=z\end{ Justerat}}$	${\begin{aligned}x&=r\sin \theta \cos \varphi \\y&=r\ sin \theta \sin \varphi \\z&=r\cos \theta \end{aligned}}$
	Cylindrisk	${\begin{aligned}\rho &={\sqrt {x^{2}+y^{2}}}\\ \varphi &=\arctan \left({\frac {y}{x}}\right)\\z&=z\end{aligned}}$	—	${\begin{aligned}\rho &=r\sin \theta \\\varphi &=\varphi \\z&=r\cos \theta \end{aligned}}$
	Sfärisk	${\begin{aligned}r&={\sqrt {x^{2} +y^{2}+z^{2}}}\\\theta &=\arctan \left({\frac {\sqrt {x^{2}+y^{2}}}{z}}\ höger)\\\varphi &=\arctan \left({\frac {y}{x}}\right)\end{aligned}}$	${\begin{aligned}r&={\sqrt {\rho ^{2}+z^{2}}}\\\ theta &=\arctan {\left({\frac {\rho }{z}}\right)}\\\varphi &=\varphi \end{aligned}}$	—

FÖRSIKTIGHET: operationen $\arctan \left({\frac {A}{B}}\right)$ måste tolkas som den tvåargumentinversa tangenten, atan2 .

Enhetsvektoromvandlingar

, cylindriska och sfäriska koordinatsystem i termer av *destinationskoordinater*
	kartesiska	Cylindrisk	Sfärisk
kartesiska	—	${\begin{aligned}{\hat {\mathbf {x } }}&=\cos \varphi {\hat {\boldsymbol {\rho }}}-\sin \varphi {\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {y} } }&=\sin \varphi {\hat {\boldsymbol {\rho }}}+\cos \varphi {\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {z} }}& ={\hat {\mathbf {z} }}\end{aligned}}$	${\begin{aligned}{\hat {\mathbf {x} }}&=\sin \theta \cos \varphi {\hat { \mathbf {r} }}+\cos \theta \cos \varphi {\hat {\boldsymbol {\theta }}}-\sin \varphi {\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {y} }}&=\sin \theta \sin \varphi {\hat {\mathbf {r} }}+\cos \theta \sin \varphi {\hat {\boldsymbol {\theta }}} +\cos \varphi {\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {z} }}&=\cos \theta {\hat {\mathbf {r} }}-\sin \theta {\hat {\boldsymbol {\theta }}}\end{aligned}}$
Cylindrisk	${\begin{aligned}{\hat {\ fetstil {\rho }}}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y ^{2}}}}\\{\hat {\boldsymbol {\varphi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y } }}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end {Justerat}}$	—	${\begin{aligned}{\hat {\boldsymbol {\ rho }}}&=\sin \theta {\hat {\mathbf {r} }}+\cos \theta {\hat {\boldsymbol {\theta }}}\\{\hat {\boldsymbol {\varphi } }}&={\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {z} }}&=\cos \theta {\hat {\mathbf {r} }}-\sin \ theta {\hat {\boldsymbol {\theta }}}\end{aligned}}$
Sfärisk	${\begin{aligned}{\hat {\mathbf {r} }}&={\frac { x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}+z{\hat {\mathbf {z} }}}{\sqrt {x^{2}+y ^{2}+z^{2}}}}\\{\hat {\boldsymbol {\theta }}}&={\frac {\left(x{\hat {\mathbf {x}}}+y {\hat {\mathbf {y} }}\right)z-\left(x^{2}+y^{2}\right){\hat {\mathbf {z} }}}{{\sqrt { x^{2}+y^{2}+z^{2}}}{\sqrt {x^{2}+y^{2}}}}}\\{\hat {\boldsymbol {\varphi } }}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2} }}}\end{aligned}}$	${\begin{aligned}{\hat {\mathbf {r} }}&={\frac {\rho {\hat {\boldsymbol {\rho }}}+z{\hat {\mathbf {z} }}}{\sqrt {\rho ^{2}+ z^{2}}}}\\{\hat {\boldsymbol {\theta }}}&={\frac {z{\hat {\boldsymbol {\rho }}}-\rho {\hat {\mathbf {z} }}}{\sqrt {\rho ^{2}+z^{2}}}}\\{\hat {\boldsymbol {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\end{aligned}}$	—

, cylindriska och sfäriska koordinatsystem i termer av *källkoordinater*
	kartesiska	Cylindrisk	Sfärisk
kartesiska	—	${\begin{aligned}{\hat {\mathbf {x} }}&={\frac {x{\hat {\boldsymbol {\rho }}}-y{\hat {\boldsymbol {\varphi }}}}{\sqrt {x^{2}+y ^{2}}}}\\{\hat {\mathbf {y} }}&={\frac {y{\hat {\boldsymbol {\rho }}}+x{\hat {\boldsymbol {\varphi }}}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end {Justerat}}$	${\begin{aligned}{\hat {\mathbf {x} }}&={\frac {x\left({\sqrt {x^{2}+y^{2 }}}{\hat {\mathbf {r} }}+z{\hat {\boldsymbol {\theta }}}\right)-y{\sqrt {x^{2}+y^{2}+z ^{2}}}{\hat {\boldsymbol {\varphi }}}}{{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{ 2}+z^{2}}}}}\\{\hat {\mathbf {y} }}&={\frac {y\left({\sqrt {x^{2}+y^{2} }}{\hat {\mathbf {r} }}+z{\hat {\boldsymbol {\theta }}}\right)+x{\sqrt {x^{2}+y^{2}+z^ {2}}}{\hat {\boldsymbol {\varphi }}}}{{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2 }+z^{2}}}}}\\{\hat {\mathbf {z} }}&={\frac {z{\hat {\mathbf {r}}}-{\sqrt {x^{ 2}+y^{2}}}{\hat {\boldsymbol {\theta }}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}\end{ Justerat}}$
Cylindrisk	${\begin{aligned}{\hat {\boldsymbol { \rho }}}&=\cos \varphi {\hat {\mathbf {x} }}+\sin \varphi {\hat {\mathbf {y} }}\\{\hat {\boldsymbol {\varphi } }}&=-\sin \varphi {\hat {\mathbf {x} }}+\cos \varphi {\hat {\mathbf {y} }}\\{\hat {\mathbf {z} }}& ={\hat {\mathbf {z} }}\end{aligned}}$	—	${\begin{aligned}{\hat {\boldsymbol {\rho }}}&={\frac {\rho {\hat {\mathbf {r} }}+z{\hat {\boldsymbol {\theta }}}}{\sqrt {\rho ^{2} +z^{2}}}}\\{\hat {\boldsymbol {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {z} }} &={\frac {z{\hat {\mathbf {r} }}-\rho {\hat {\boldsymbol {\theta }}}}{\sqrt {\rho ^{2}+z^{2} }}}\end{aligned}}$
Sfärisk	${\begin{aligned}{\hat {\mathbf {r} }}&=\sin \theta \left(\cos \varphi {\ hat {\mathbf {x} }}+\sin \varphi {\hat {\mathbf {y} }}\right)+\cos \theta {\hat {\mathbf {z} }}\\{\hat { \boldsymbol {\theta }}}&=\cos \theta \left(\cos \varphi {\hat {\mathbf {x} }}+\sin \varphi {\hat {\mathbf {y} }}\right )-\sin \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\varphi }}}&=-\sin \varphi {\hat {\mathbf {x} }}+ \cos \varphi {\hat {\mathbf {y} }}\end{aligned}}$	${\begin{aligned}{\hat {\mathbf {r } }}&=\sin \theta {\hat {\boldsymbol {\rho }}}+\cos \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\theta }} }&=\cos \theta {\hat {\boldsymbol {\rho }}}-\sin \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\varphi }}}& ={\hat {\boldsymbol {\varphi }}}\end{aligned}}$	—

Del formel

Tabell med deloperatorn i kartesiska, cylindriska och sfäriska koordinater
Drift	Kartesiska koordinater $(x, y, z)$	Cylindriska koordinater $(ρ, φ, z)$	Sfäriska koordinater $(r, θ, φ)$ , där $θ$ är den polära vinkeln och $φ$ är den azimutala vinkeln
Vektorfält $A$	$A_{x}{\hat {\mathbf {x} }}+A_{y}{\hat {\mathbf {y} }}+A_ {z}{\hat {\mathbf {z} }}$	$A_{\rho }{\hat {\boldsymbol {\rho }}}+A_{\varphi }{\hat {\boldsymbol {\varphi } }}+A_{z}{\hat {\mathbf {z} }}$	$A_{r}{\hat {\mathbf {r} }}+A_{\theta }{\hat {\boldsymbol {\theta }}} +A_{\varphi }{\hat {\boldsymbol {\varphi }}}$
Gradient $\nabla f$	${\partial f \over \partial x}{\hat {\mathbf {x} }}+{\partial f \ över \partial y}{\hat {\mathbf {y} }}+{\partial f \over \partial z}{\hat {\mathbf {z} }}$	${\partial f \over \partial \rho }{\hat {\boldsymbol {\rho }}}+{ 1 \over \rho }{\partial f \over \partial \varphi }{\hat {\boldsymbol {\varphi }}}+{\partial f \over \partial z}{\hat {\mathbf {z} } }$	${\partial f \over \partial r}{\hat {\mathbf {r} } }+{1 \over r}{\partial f \over \partial \theta }{\hat {\boldsymbol {\theta }}}+{1 \over r\sin \theta }{\partial f \over \partial \varphi }{\hat {\boldsymbol {\varphi }}}$
Divergens $\nabla \cdot A$	${\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\ partiell A_{z} \över \partial z}$	${1 \over \rho }{\partial \left(\rho A_{\rho }\right) \over \partial \rho }+{1 \over \rho }{\partial A_{\varphi } \over \partial \varphi }+{\partial A_{z} \over \partial z}$	$r ^$
Curl $\nabla \times A$	${\display {\begin{aligned}\left({\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\right)&{\ hatt {\mathbf {x} }}\\+\left({\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}} \right)&{\hat {\mathbf {y} }}\\+\left({\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}} {\partial y}}\right)&{\hat {\mathbf {z} }}\end{aligned}}}$	${\begin{aligned}\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}-{\frac { \partial A_{\varphi }}{\partial z}}\right)&{\hat {\boldsymbol {\rho }}}\\+\left({\frac {\partial A_{\rho }}{\ partiell z}}-{\frac {\partial A_{z}}{\partial \rho }}\right)&{\hat {\boldsymbol {\varphi }}}\\+{\frac {1}{\ rho }}\left({\frac {\partial \left(\rho A_{\varphi }\right)}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \ varphi }}\right)&{\hat {\mathbf {z} }}\end{aligned}}$	${\begin{aligned}{\frac {1}{r\sin \theta }}\left({\frac {\partial }{\partial \theta }}\left(A_{\varphi }\sin \theta \right)-{\frac {\partial A_{\theta}}{\partial \varphi }}\right)& {\hat {\mathbf {r} }}\\{}+{\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{ r}}{\partial \varphi }}-{\frac {\partial }{\partial r}}\left(rA_{\varphi}\right)\right)&{\hat {\boldsymbol {\theta }} }\\{}+{\frac {1}{r}}\left({\frac {\partial }{\partial r}}\left(rA_{\theta}\right)-{\frac {\partial A_{r}}{\partial \theta }}\right)&{\hat {\boldsymbol {\varphi }}}\end{aligned}}$
Laplace-operator $\nabla 2 f \equiv ∆ f$	${\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}$	${1 \over \rho }{\partial \over \}partial \rho vänster(\rho {\partial f \over \partial \rho }\right)+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \varphi ^{2}}+ {\partial ^{2}f \over \partial z^{2}}$	$_ {1 \over r^{2}}{\partial \over \partial r}\!\left(r^{2}{\partial f \over \partial r}\right)\!+\!{1 \ över r^{2}\!\sin \theta }{\partial \over \partial \theta }\!\left(\sin \theta {\partial f \over \partial \theta}\right)\!+\ !{1 \over r^{2}\!\sin ^{2}\theta }{\partial ^{2}f \over \partial \varphi ^{2}}}$
Vektorgradient $\nabla A$	${$	${\begin{aligned}{}&{\frac {\partial A_{\rho }}{\partial \rho }}{\hat {\boldsymbol {\rho }}}\otimes {\hat {\boldsymbol {\rho }}}+\left({\frac {1}{\rho }}{\frac {\partial A_{\rho }} {\partial \varphi }}-{\frac {A_{\varphi }}{\rho }}\right){\hat {\boldsymbol {\rho }}}\ ibland {\hat {\boldsymbol {\varphi } }}+{\frac {\partial A_{\rho }}{\partial z}}{\hat {\boldsymbol {\rho }}}\ibland {\hat {\mathbf {z} }}\\{} +&{\frac {\partial A_{\varphi }}{\partial \rho }}{\hat {\boldsymbol {\varphi }}}\ibland {\hat {\boldsymbol {\rho }}}+\vänster ({\frac {1}{\rho }}{\frac {\partial A_{\varphi }}{\partial \varphi }}+{\frac {A_{\rho }}{\rho }}\right) {\hat {\boldsymbol {\varphi }}}\otimes {\hat {\boldsymbol {\varphi }}}+{\frac {\partial A_{\varphi }}{\partial z}}{\hat {\ fetsymbol {\varphi }}}\otimes {\hat {\mathbf {z} }}\\{}+&{\frac {\partial A_{z}}{\partial \rho }}{\hat {\mathbf {z} }}\ ibland {\hat {\boldsymbol {\rho }}}+{\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}{ \hat {\mathbf {z} }}\otimes {\hat {\boldsymbol {\varphi }}}+{\frac {\partial A_{z}}{\partial z}}{\hat {\mathbf {z } }}\ibland {\hat {\mathbf {z} }}\end{aligned}}$	${\begin{aligned}{}&{\frac {\partial A_{r}}{\partial r}}{\hat {\mathbf {r} }}\ ibland {\hat {\mathbf {r} }}+\left({\frac {1}{r}}{\frac {\partial A_{r}}{\partial \theta }}-{\frac {A_ {\theta }}{r}}\right){\hat {\mathbf {r} }}\otimes {\hat {\boldsymbol {\theta }}}+\left({\frac {1}{r\ sin \theta }}{\frac {\partial A_{r}}{\partial \varphi }}-{\frac {A_{\varphi }}{r}}\right){\hat {\mathbf {r} }}\ibland {\hat {\boldsymbol {\varphi }}}\\{}+&{\frac {\partial A_{\theta }}{\partial r}}{\hat {\boldsymbol {\theta } }}\ibland {\hat {\mathbf {r} }}+\left({\frac {1}{r}}{\frac {\partial A_{\theta }}{\partial \theta }}+{ \frac {A_{r}}{r}}\right){\hat {\boldsymbol {\theta }}}\ibland {\hat {\boldsymbol {\theta }}}+\left({\frac {1 }{r\sin \theta }}{\frac {\partial A_{\theta }}{\partial \varphi }}-\cot \theta {\frac {A_{\varphi }}{r}}\right) {\hat {\boldsymbol {\theta }}}\ibland {\hat {\boldsymbol {\varphi }}}\\{}+&{\frac {\partial A_{\varphi }}{\partial r}} {\hat {\boldsymbol {\varphi }}}\otimes {\hat {\mathbf {r} }}+{\frac {1}{r}}{\frac {\partial A_{\varphi }}{\ partiell \theta }}{\hat {\boldsymbol {\varphi }}}\otimes {\hat {\boldsymbol {\theta }}}+\left({\frac {1}{r\sin \theta }}{ \frac {\partial A_{\varphi }}{\partial \varphi }}+\cot \theta {\frac {A_{\theta }}{r}}+{\frac {A_{r}}{r} }\right){\hat {\boldsymbol {\varphi }}}\otimes {\hat {\boldsymbol {\varphi }}}\end{aligned}}$
Vektor Laplacian $\nabla 2 A \equiv ∆ A$	$\nabla ^{2}A_{x}{\hat {\mathbf {x} }}+\nabla ^{2 }A_{y}{\hat {\mathbf {y} }}+\nabla ^{2}A_{z}{\hat {\mathbf {z} }}$	${\begin{aligned}{\mathopen {}}\left(\nabla ^{2}A_{\rho }-{\frac {A_{\rho }}{\rho ^{2}} }-{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\varphi }}{\partial \varphi }}\right){\mathclose {}}&{\hat { \boldsymbol {\rho }}}\\+{\mathopen {}}\left(\nabla ^{2}A_{\varphi }-{\frac {A_{\varphi }}{\rho ^{2}} }+{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\rho }}{\partial \varphi }}\right){\mathclose {}}&{\hat { \boldsymbol {\varphi }}}\\{}+\nabla ^{2}A_{z}&{\hat {\mathbf {z} }}\end{aligned}}$	${\begin{aligned} ^{2}\na A_{r}-{\frac {2A_{r}}{r^{2}}}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial \left( A_{\theta }\sin \theta \right)}{\partial \theta }}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{\varphi } }{\partial \varphi }}\right)&{\hat {\mathbf {r} }}\\+\left(\nabla ^{2}A_{\theta }-{\frac {A_{\theta } }{r^{2}\sin ^{2}\theta }}+{\frac {2}{r^{2}}}{\frac {\partial A_{r}}{\partial \theta }} -{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial A_{\varphi }}{\partial \varphi }}\right)& {\hat {\boldsymbol {\theta }}}\\+\left(\nabla ^{2}A_{\varphi }-{\frac {A_{\varphi }}{r^{2}\sin ^{ 2}\theta }}+{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{r}}{\partial \varphi }}+{\frac {2\ cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial A_{\theta }}{\partial \varphi }}\right)&{\hat {\boldsymbol { \varphi }}}\end{aligned}}$
Materialderivat $(A \cdot \nabla) B$	$\mathbf {A} \cdot \nabla B_{x}{\hat {\mathbf {x} }} +\mathbf {A} \cdot \nabla B_{y}{\hat {\mathbf {y} }}+\mathbf {A} \cdot \nabla B_{z}{\hat {\mathbf {z} }}$	$_ displaystyle {\begin{aligned}\left(A_{\rho }{\frac {\partial B_{\rho }}{\partial \rho }}+{\frac {A_{\varphi }}{\rho }} {\frac {\partial B_{\rho }}{\partial \varphi }}+A_{z}{\frac {\partial B_{\rho }}{\partial z}}-{\frac {A_{\ varphi }B_{\varphi }}{\rho }}\right)&{\hat {\boldsymbol {\rho }}}\\+\left(A_{\rho }{\frac {\partial B_{\varphi }}{\partial \rho }}+{\frac {A_{\varphi }}{\rho }}{\frac {\partial B_{\varphi }}{\partial \varphi }}+A_{z}{ \frac {\partial B_{\varphi }}{\partial z}}+{\frac {A_{\varphi }B_{\rho }}{\rho }}\right)&{\hat {\boldsymbol {\ varphi }}}\\+\left(A_{\rho }{\frac {\partial B_{z}}{\partial \rho }}+{\frac {A_{\varphi }}{\rho }}{ \frac {\partial B_{z}}{\partial \varphi }}+A_{z}{\frac {\partial B_{z}}{\partial z}}\right)&{\hat {\mathbf { z} }}\end{aligned}}}$	${\displaystyle}\begin{ (A_{r}{\frac {\partial B_{r}}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{r}}{\ partial \theta }}+{\frac {A_{\varphi }}{r\sin \theta }}{\frac {\partial B_{r}}{\partial \varphi }}-{\frac {A_{\ theta }B_{\theta }+A_{\varphi }B_{\varphi }}{r}}\right)&{\hat {\mathbf {r}}}\\+\left(A_{r}{\ frac {\partial B_{\theta }}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{\theta }}{\partial \theta }} +{\frac {A_{\varphi }}{r\sin \theta }}{\frac {\partial B_{\theta }}{\partial \varphi }}+{\frac {A_{\theta }B_{ r}}{r}}-{\frac {A_{\varphi }B_{\varphi }\cot \theta }{r}}\right)&{\hat {\boldsymbol {\theta }}}\\+ \left(A_{r}{\frac {\partial B_{\varphi }}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{\varphi }}{\partial \theta }}+{\frac {A_{\varphi }}{r\sin \theta }}{\frac {\partial B_{\varphi }}{\partial \varphi }}+{\ frac {A_{\varphi }B_{r}}{r}}+{\frac {A_{\varphi }B_{\theta }\cot \theta }{r}}\right)&{\hat {\boldsymbol {\varphi }}}\end{aligned}}}$
Tensor $\nabla \cdot T$ (inte att förväxla med 2:a ordningens tensordivergens )	${\begin{aligned}\left({\frac {\partial T_{xx}}{\partial x}}+{\frac {\partial T_{yx}}{\partial y}}+{\frac {\partial T_{zx}}{\partial z}}\right)&{\hat {\mathbf { x} }}\\+\left({\frac {\partial T_{xy}}{\partial x}}+{\frac {\partial T_{yy}}{\partial y}}+{\frac { \partial T_{zy}}{\partial z}}\right)&{\hat {\mathbf {y} }}\\+\left({\frac {\partial T_{xz}}{\partial x} }+{\frac {\partial T_{yz}}{\partial y}}+{\frac {\partial T_{zz}}{\partial z}}\right)&{\hat {\mathbf {z} }}\end{aligned}}$	${\begin{aligned}\left[{\frac {\partial T_{\rho \rho }}{\partial \rho }}+{\frac {1}{ \rho }}{\frac {\partial T_{\varphi \rho }}{\partial \varphi }}+{\frac {\partial T_{z\rho }}{\partial z}}+{\frac { 1}{\rho }}(T_{\rho \rho }-T_{\varphi \varphi })\right]&{\hat {\boldsymbol {\rho }}}\\+\left[{\frac { \partial T_{\rho \varphi }}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial T_{\varphi \varphi }}{\partial \varphi }} +{\frac {\partial T_{z\varphi }}{\partial z}}+{\frac {1}{\rho }}(T_{\rho \varphi }+T_{\varphi \rho })\ höger]&{\hat {\boldsymbol {\varphi }}}\\+\left[{\frac {\partial T_{\rho z}}{\partial \rho }}+{\frac {1}{\ rho }}{\frac {\partial T_{\varphi z}}{\partial \varphi }}+{\frac {\partial T_{zz}}{\partial z}}+{\frac {T_{\rho z}}{\rho }}\right]&{\hat {\mathbf {z} }}\end{aligned}}$	${\begin{aligned}\left[{\frac {\partial T_{rr}}{\partial r }}+2{\frac {T_{rr}}{r}}+{\frac {1}{r}}{\frac {\partial T_{\theta r}}{\partial \theta }}+{ \frac {\cot \theta }{r}}T_{\theta r}+{\frac {1}{r\sin \theta }}{\frac {\partial T_{\varphi r}}{\partial \ varphi }}-{\frac {1}{r}}(T_{\theta \theta }+T_{\varphi \varphi })\right]&{\hat {\mathbf {r} }}\\+\ vänster[{\frac {\partial T_{r\theta }}{\partial r}}+2{\frac {T_{r\theta }}{r}}+{\frac {1}{r}}{ \frac {\partial T_{\theta \theta }}{\partial \theta }}+{\frac {\cot \theta }{r}}T_{\theta \theta }+{\frac {1}{r \sin \theta }}{\frac {\partial T_{\varphi \theta }}{\partial \varphi }}+{\frac {T_{\theta r}}{r}}-{\frac {\cot \theta }{r}}T_{\varphi \varphi }\right]&{\hat {\boldsymbol {\theta }}}\\+\left[{\frac {\partial T_{r\varphi }}{ \partial r}}+2{\frac {T_{r\varphi }}{r}}+{\frac {1}{r}}{\frac {\partial T_{\theta \varphi }}{\partial \theta }}+{\frac {1}{r\sin \theta }}{\frac {\partial T_{\varphi \varphi }}{\partial \varphi }}+{\frac {T_{\varphi r }}{r}}+{\frac {\cot \theta }{r}}(T_{\theta \varphi }+T_{\varphi \theta })\right]&{\hat {\boldsymbol {\varphi }}}\end{aligned}}$
Differentialförskjutning $d ℓ$	$dx\,{\hat {\mathbf {x} }}+dy\,{\hat {\mathbf {y} }}+dz\, {\hat {\mathbf {z} }}$	$d\rho \,{\hat {\boldsymbol {\rho }}}+\rho \,d\varphi \,{\hat {\ fetstil {\varphi }}}+dz\,{\hat {\mathbf {z} }}$	$dr\,{\hat {\mathbf {r} }}+r\,d\theta \,{\hat {\ fetsymbol {\theta }}}+r\,\sin \theta \,d\varphi \,{\hat {\boldsymbol {\varphi }}}$
Differentialnormalarea $d S$	${\begin{aligned}dy\,dz&\,{\hat {\mathbf {x} }}\\{} +dx\,dz&\,{\hat {\mathbf {y} }}\\{}+dx\,dy&\,{\hat {\mathbf {z} }}\end{aligned}}$	${\begin{aligned}\rho \,d\varphi \,dz&\,{\hat {\boldsymbol { \rho }}}\\{}+d\rho \,dz&\,{\hat {\boldsymbol {\varphi }}}\\{}+\rho \,d\rho \,d\varphi &\, {\hat {\mathbf {z} }}\end{aligned}}$	${\begin{aligned}r^{2}\sin \theta \, d\theta \,d\varphi &\,{\hat {\mathbf {r}}}\\{}+r\sin \theta \,dr\,d\varphi &\,{\hat {\boldsymbol { \theta }}}\\{}+r\,dr\,d\theta &\,{\hat {\boldsymbol {\varphi }}}\end{aligned}}$
Differentialvolym $dV$	$dx\,dy\,dz$	$\rho \,d\rho \,d\varphi \,dz$	$r^{2}\sin \theta \,dr\,d\theta \,d\varphi$

^α Denna sida använder

\theta

för den polära vinkeln och

\varphi

för den azimutala vinkeln, vilket är vanlig notation inom fysik. Källan som används för dessa formler använder

\theta

för azimutvinkeln och

\varphi

för den polära vinkeln, vilket är vanlig matematisk notation. För att få matematiska formler, byt

\theta

och

\varphi

i formlerna som visas i tabellen ovan.

Beräkningsregler

$\operatörsnamn {div} \,\operatörsnamn {grad} f\equiv \nabla \cdot \nabla f\equiv \nabla ^{2}f$
$\operatörsnamn {curl} \,\operatörsnamn {grad} f\equiv \nabla \times \nabla f=\mathbf {0}$
$\operatörsnamn {div} \,\operatörsnamn {curl} \mathbf {A} \equiv \nabla \cdot (\nabla \times \mathbf {A} ) =0$
${\displaystyle \operatorname {curl} \,\operatörsnamn {curl} \mathbf {A} \equiv \nabla \times ( \nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} } ( Lagranges formel$ för del )
$\nabla ^{2}(fg)=f\nabla ^{2}g+2\nabla f\ cdot \nabla g+g\nabla ^{2}f$

Kartesisk härledning

{\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)\,dy\,dz-A_{ x}(x)\,dy\,dz+A_{y}(y+dy)\,dx\,dz-A_{y}(y)\,dx\,dz+A_{z}(z+dz )\,dx\,dy-A_{z}(z)\,dx\,dy}{dx\,dy\,dz}}\\&={\frac {\partial A_{x}}{\partial x}}+{\frac {\partial A_{y}}{\partial y}}+{\frac {\partial A_{z}}{\partial z}}\end{aligned}}

{\begin{aligned}(\operatörsnamn {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}(y+dy)\,dz-A_{z}(y)\,dz+A_{y}(z )\,dy-A_{y}(z+dz)\,dy}{dy\,dz}}\\&={\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\end{aligned}}

Uttrycken för $(\operatorname {curl} \mathbf {A} )_{y}$ och $(\operatorname {curl} \mathbf {A} )_{z}$ hittas på samma sätt.

Cylindrisk härledning

börja {\{displaystyle {\ }\operatörsnamn {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\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}}}

{\begin{aligned}(\operatörsnamn {curl} \mathbf {A} )_{\rho }&=\lim _{S^{\perp {\boldsymbol {\hat {\rho }}}}\ till 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}}

{\begin{aligned}(\operatörsnamn {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}}

{\

{\begin{name aligned}\} mathbf {A} &=(\operatörsnamn {curl} \mathbf {A} )_{\rho }{\hat {\boldsymbol {\rho }}}+(\operatörsnamn {curl} \mathbf {A} )_{ \phi }{\hat {\boldsymbol {\phi }}}+(\operatörsnamn {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}}}\end{aligned}}

Sfärisk härledning

{\begin{aligned}\operatörsnamn {div} \mathbf {A} &=\lim _{ V\till 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)r\,d\theta \, r\sin \theta \,d\phi +A_{\theta }(\theta +d\theta )\sin(\theta +d\theta )r\,dr\,d\phi -A_{\theta }( \theta )\sin(\theta )r\,dr\,d\phi +A_{\phi}(\phi +d\phi )r\,dr\,d\theta -A_{\phi}(\phi )r\,dr\,d\theta }{dr\,r\,d\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}}

{\begin{aligned}(\operatörsnamn {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 )r\,d\theta +A_{\phi}(\theta +d\theta )r\sin(\theta +d\theta )\ ,d\phi -A_{\theta }(\phi +d\phi )r\,d\theta -A_{\phi}(\theta )r\sin(\theta )\,d\phi }{r\ ,d\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} }

{\begin{aligned}(\operatörsnamn {curl} \mathbf {A} )_{\theta }=\lim _{S^{\perp {\boldsymbol {\hat {\theta }}}}\till 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}}

{\begin{ aligned}(\operatörsnamn {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)r\,d \theta }{r\,dr\,d\theta }}\\&={\frac {1}{r}}{\frac {\partial (rA_{\theta })}{\partial r}}- {\frac {1}{r}}{\frac {\partial A_{r}}{\partial \theta }}\end{aligned}}

\operatörsnamn {curl} \mathbf {A} =(\operatörsnamn {curl} \mathbf {A} )_{r}\,{\hat {\boldsymbol {r}}}+(\operatörsnamn { curl} \mathbf {A} )_{\theta }\,{\hat {\boldsymbol {\theta }}}+(\operatörsnamn {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 }}}

Enhetsvektoromvandlingsformel

Enhetsvektorn för en koordinatparameter u definieras på så sätt att en liten positiv förändring i u gör att positionsvektorn $\mathbf {r}$ ändras i $\mathbf {u}$ riktning.

Därför,

{\frac {\partial {\mathbf {r} }}{\partial u}}={\frac {\partial {s}}{\partial u}} \mathbf {u}

där

s

är båglängdsparametern.

För två uppsättningar koordinatsystem $u_{i}$ och ${\displaystyle v_{j}} ,$ enligt kedjeregeln ,

d\mathbf {r} =\summa _{i}{\frac {\partial \mathbf {r} }{\partial u_{i}}}\,du_{i}=\summa _{i} {\frac {\partial s}{\partial u_{i}}}{\hat {\mathbf {u} }}_{i}du_{i}=\summa _{j}{\frac {\partial s }{\partial v_{j}}}{\hat {\mathbf {v} }}_{j}\,dv_{j}=\summa _{j}{\frac {\partial s}{\partial v_ {j}}}{\hat {\mathbf {v} }}_{j}\sum _{i}{\frac {\partial v_{j}}{\partial u_{i}}}\,du_{ i}=\summa _{i}\summa _{j}{\frac {\partial s}{\partial v_{j}}}{\frac {\partial v_{j}}{\partial u_{i} }}{\hat {\mathbf {v} }}_{j}\,du_{i}.

Nu isolerar vi den ${\displaystyle i}:$ ^e komponenten. För $i{\neq }k$ , låt $\mathrm {d} u_{k}=0$ . Dela sedan på båda sidor med $\mathrm {d} u_{i}$ för att få:

{\frac {\partial s}{\partial u_{i}}}{\hat {\mathbf {u} }}_{i}=\summa _{j}{\frac {\partial s} {\partial v_{j}}}{\frac {\partial v_{j}}{\partial u_{i}}}{\hat {\mathbf {v} }}_{j}.

Se även

externa länkar

Maxima Computer Algebra-system skript för att generera några av dessa operatorer i cylindriska och sfäriska koordinater.