Nabla in verschillende assenstelsels

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

Conversies tussen stelsels

conversies tussen cartesische, cilinder- en bolcoördinaten[1]
van
cartesisch cilindrisch bol
naar cartesisch x = x y = y z = z {\displaystyle {\begin{aligned}x&=x\\y&=y\\z&=z\end{aligned}}} 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}}}
cilindrisch 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}}} ρ = ρ φ = φ z = z {\displaystyle {\begin{aligned}\rho &=\rho \\\varphi &=\varphi \\z&=z\end{aligned}}} 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}}}
bol r = x 2 + y 2 + z 2 θ = arctan ⁡ ( x 2 + y 2 z ) φ = arctan ⁡ ( y x ) {\displaystyle {\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arctan \left({\frac {\sqrt {x^{2}+y^{2}}}{z}}\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}}} r = r φ = φ θ = θ {\displaystyle {\begin{aligned}r&=r\\\varphi &=\varphi \\\theta &=\theta \\\end{aligned}}}

Conversies tussen eenheidsvectoren

conversies tussen eenheidsvectoren in cartesische, cilindrische en bolcoördinaten in termen van bestemmingscoördinaten[1]
cartesische cilindrische bol
cartesische 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}}}
cilindrische 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}}} 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}}}
bol 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}}}
conversies tussen eenheidsvectoren in cartesische, cilindrische en bolcoördinaten in termen van oorsprongscoördinaten
cartesische cilindrische bol
cartesische 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}}}
cilindrische 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}}} 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}}}
bol 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}}}

Formules met de gradiënt

Tabel met nabla-operator in cartesische, cilindrische en bolcoördinaten
operatie cartesische coördinaten (x, y, z) cilindercoördinaten (ρ, φ, z) bolcoördinaten (r, θ, φ), waar φ de azimutale en θ de polaire hoek is[2]
vectorveld 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 }}}}
gradiënt ∇f[1] ∂ 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 }}}}
divergentie ∇ ⋅ A[1] ∂ 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 }}
rotatie ∇ × A[1] ( ∂ 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}}}
laplace-operator ∇2f ≡ ∆f[1] ∂ 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 Laplaciaan ∇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}}}

Rekenregels

  1. div ⁡ grad ⁡ f ≡ ∇ ⋅ ∇ f ≡ ∇ 2 f {\displaystyle \operatorname {div} \operatorname {grad} f\equiv \nabla \cdot \nabla f\equiv \nabla ^{2}f}
  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} \equiv \nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} } , Lagrange's formule voor de gradiënt
  5. ∇ 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}
Voetnoten
  1. ↑ a b c d e f DJ Griffiths. Introduction to Electrodynamics, 2012. isbn 978-0-321-85656-2
  2. ↑ Deze pagina gebruikt θ {\displaystyle \theta } voor de polaire hoek en φ {\displaystyle \varphi } voor de azimutale hoek. Dat is de gebruikelijke notatie voor natuurkunde. De bron voor deze formules gebruikt θ {\displaystyle \theta } voor de azimutale hoek en φ {\displaystyle \varphi } voor de polaire hoek, dat is de gebruikelijke wiskundige notatie. Om de wiskundige variant te krijgen, verwissel θ {\displaystyle \theta } en φ {\displaystyle \varphi } in de bovenstaande tabel.