Nabla în coordonate cilindrice și sferice

În calculul vectorial este adesea util să știi să exprimi $\nabla$ ${\ displaystyle \ nabla}$ $\ nabla$ în alte sisteme de coordonate, altele decât cel cartezian .

Operator	Coordonatele carteziene (x, y, z)	Coordonate cilindrice (ρ, φ, z)	Coordonate sferice (r, θ, φ)
Definirea coordonatelor		${\begin{cases}x&=&\rho \cos \phi \\y&=&\rho \sin \phi \\z&=&z\end{cases}}$ ${\ displaystyle {\ begin {cases} x & = & \ rho \ cos \ phi \\ y & = & \ rho \ sin \ phi \\ z & = & z \ end {cases}}}$ ${\ displaystyle {\ begin {cases} x & = & \ rho \ cos \ phi \\ y & = & \ rho \ sin \ phi \\ z & = & z \ end {cases}}}$	${\begin{cases}x&=&r\sin \theta \cos \phi &0\leqslant \theta \leqslant \pi \\y&=&r\sin \theta \sin \phi &0\leqslant \phi <2\pi \\z&=&r\cos \theta &0\leqslant r<+\infty \\\end{cases}}$ ${\ displaystyle {\ begin {cases} x & = & r \ sin \ theta \ cos \ phi & 0 \ leqslant \ theta \ leqslant \ pi \\ y & = & r \ sin \ theta \ sin \ phi & 0 \ leqslant \ phi <2 \ pi \\ z & = & r \ cos \ theta & 0 \ leqslant r <+ \ infty \\\ end {cases}}}$ ${\ displaystyle {\ begin {cases} x & = & r \ sin \ theta \ cos \ phi & 0 \ leqslant \ theta \ leqslant \ pi \\ y & = & r \ sin \ theta \ sin \ phi & 0 \ leqslant \ phi <2 \ pi \\ z & = & r \ cos \ theta & 0 \ leqslant r <+ \ infty \\\ end {cases}}}$
		${\begin{cases}\rho &=&{\sqrt {x^{2}+y^{2}}}\\\phi &=&\arctan(y/x)\\z&=&z\end{cases}}$ ${\ displaystyle {\ begin {cases} \ rho & = & {\ sqrt {x ^ {2} + y ^ {2}}} \\\ phi & = & \ arctan (y / x) \\ z & = & z \ end {cases}}}$ ${\ displaystyle {\ begin {cases} \ rho & = & {\ sqrt {x ^ {2} + y ^ {2}}} \\\ phi & = & \ arctan (y / x) \\ z & = & z \ end {cases}}}$	${\begin{cases}r&=&{\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=&\arccos(z/r)\\\phi &=&\arctan(y/x)\end{cases}}$ ${\ displaystyle {\ begin {cases} r & = & {\ sqrt {x ^ {2} + y ^ {2} + z ^ {2}}} \\\ theta & = & \ arccos (z / r) \ \\ phi & = & \ arctan (y / x) \ end {cases}}}$ ${\ displaystyle {\ begin {cases} r & = & {\ sqrt {x ^ {2} + y ^ {2} + z ^ {2}}} \\\ theta & = & \ arccos (z / r) \ \\ phi & = & \ arctan (y / x) \ end {cases}}}$
Câmpul vector $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ $\ mathbf {A}$	$A_{x}\mathbf {\hat {x}} +A_{y}\mathbf {\hat {y}} +A_{z}\mathbf {\hat {z}}$ ${\ displaystyle A_ {x} \ mathbf {\ hat {x}} + A_ {y} \ mathbf {\ hat {y}} + A_ {z} \ mathbf {\ hat {z}}}$ $A_ {x} {\ mathbf {{\ hat x}}} + A_ {y} {\ mathbf {{\ hat y}}} + A_ {z} {\ mathbf {{\ hat z}}}$	$A_{\rho }{\boldsymbol {\hat {\rho }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}+A_{z}{\boldsymbol {\hat {z}}}$ ${\ displaystyle A _ {\ rho} {\ boldsymbol {\ hat {\ rho}}} + A _ {\ phi} {\ boldsymbol {\ hat {\ phi}}} + A_ {z} {\ boldsymbol {\ pălărie {z}}}}$ $A _ {\ rho} {\ boldsymbol {{\ hat \ rho}}} + A _ {\ phi} {\ boldsymbol {{\ hat \ phi}}} + A_ {z} {\ boldsymbol {{\ hat z }}}$	$A_{r}{\boldsymbol {\hat {r}}}+A_{\theta }{\boldsymbol {\hat {\theta }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}$ ${\ displaystyle A_ {r} {\ boldsymbol {\ hat {r}}} + A _ {\ theta} {\ boldsymbol {\ hat {\ theta}}} + A _ {\ phi} {\ boldsymbol {\ hat {\ phi}}}}$ $A_ {r} {\ boldsymbol {{\ hat r}}} + A _ {\ theta} {\ boldsymbol {{\ hat \ theta}}} + A _ {\ phi} {\ boldsymbol {{\ hat \ phi }}}$
Gradient $\nabla f$ ${\ displaystyle \ nabla f}$ $\ nabla f$	${\partial f \over \partial x}\mathbf {\hat {x}} +{\partial f \over \partial y}\mathbf {\hat {y}} +{\partial f \over \partial z}\mathbf {\hat {z}}$ ${\ displaystyle {\ partial f \ over \ partial x} \ mathbf {\ hat {x}} + {\ partial f \ over \ partial y} \ mathbf {\ hat {y}} + {\ partial f \ over \ parțial z} \ mathbf {\ hat {z}}}$ ${\ partial f \ over \ partial x} {\ mathbf {{\ hat x}}} + {\ partial f \ over \ partial y} {\ mathbf {{\ hat y}}} + {\ partial f \ over \ partial z} {\ mathbf {{\ hat z}}}$	${\partial f \over \partial \rho }{\boldsymbol {\hat {\rho }}}+{1 \over \rho }{\partial f \over \partial \phi }{\boldsymbol {\hat {\phi }}}+{\partial f \over \partial z}{\boldsymbol {\hat {z}}}$ ${\ displaystyle {\ partial f \ over \ partial \ rho} {\ boldsymbol {\ hat {\ rho}}} + {1 \ over \ rho} {\ partial f \ over \ partial \ phi} {\ boldsymbol {\ hat {\ phi}}} + {\ partial f \ over \ partial z} {\ boldsymbol {\ hat {z}}}}$ ${\ partial f \ over \ partial \ rho} {\ boldsymbol {{\ hat \ rho}}} + {1 \ over \ rho} {\ partial f \ over \ partial \ phi} {\ boldsymbol {{\ hat \ phi}}} + {\ partial f \ over \ partial z} {\ boldsymbol {{\ hat z}}}$	${\partial f \over \partial r}{\boldsymbol {\hat {r}}}+{1 \over r}{\partial f \over \partial \theta }{\boldsymbol {\hat {\theta }}}+{1 \over r\sin \theta }{\partial f \over \partial \phi }{\boldsymbol {\hat {\phi }}}$ ${\ displaystyle {\ partial f \ over \ partial r} {\ boldsymbol {\ hat {r}}} + {1 \ over r} {\ partial f \ over \ partial \ theta} {\ boldsymbol {\ hat {\ theta}}} + {1 \ over r \ sin \ theta} {\ partial f \ over \ partial \ phi} {\ boldsymbol {\ hat {\ phi}}}}$ ${\ partial f \ over \ partial r} {\ boldsymbol {{\ hat r}}} + {1 \ over r} {\ partial f \ over \ partial \ theta} {\ boldsymbol {{\ hat \ theta}} } + {1 \ over r \ sin \ theta} {\ partial f \ over \ partial \ phi} {\ boldsymbol {{\ hat \ phi}}}$
Divergenţă $\nabla \cdot \mathbf {A}$ ${\ displaystyle \ nabla \ cdot \ mathbf {A}}$ $\ nabla \ cdot {\ mathbf {A}}$	${\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}$ ${\ displaystyle {\ partial A_ {x} \ over \ partial x} + {\ partial A_ {y} \ over \ partial y} + {\ partial A_ {z} \ over \ partial z}}$ ${\ partial A_ {x} \ over \ partial x} + {\ partial A_ {y} \ over \ partial y} + {\ partial A_ {z} \ over \ partial z}$	${1 \over \rho }{\partial (\rho A_{\rho }) \over \partial \rho }+{1 \over \rho }{\partial A_{\phi } \over \partial \phi }+{\partial A_{z} \over \partial z}$ ${\ displaystyle {1 \ over \ rho} {\ partial (\ rho A _ {\ rho}) \ over \ partial \ rho} + {1 \ over \ rho} {\ partial A _ {\ phi} \ over \ partial \ phi} + {\ partial A_ {z} \ over \ partial z}}$ ${1 \ over \ rho} {\ partial (\ rho A _ {\ rho}) \ over \ partial \ rho} + {1 \ over \ rho} {\ partial A _ {\ phi} \ over \ partial \ phi } + {\ partial A_ {z} \ over \ partial z}$	${1 \over r^{2}}{\partial (r^{2}A_{r}) \over \partial r}+{1 \over r\sin \theta }{\partial \over \partial \theta }(A_{\theta }\sin \theta )+{1 \over r\sin \theta }{\partial A_{\phi } \over \partial \phi }$ ${\ displaystyle {1 \ over r ^ {2}} {\ partial (r ^ {2} A_ {r}) \ over \ partial r} + {1 \ over r \ sin \ theta} {\ partial \ over \ partial \ theta} (A _ {\ theta} \ sin \ theta) + {1 \ over r \ sin \ theta} {\ partial A _ {\ phi} \ over \ partial \ phi}}$ ${1 \ over r ^ {2}} {\ partial (r ^ {2} A_ {r}) \ over \ partial r} + {1 \ over r \ sin \ theta} {\ partial \ over \ partial \ theta } (A _ {\ theta} \ sin \ theta) + {1 \ over r \ sin \ theta} {\ partial A _ {\ phi} \ over \ partial \ phi}$
Rotor $\nabla \times \mathbf {A}$ ${\ displaystyle \ nabla \ times \ mathbf {A}}$ $\ nabla \ times \ mathbf {A}$	${\begin{matrix}\displaystyle {\bigg (}{\partial A_{z} \over \partial y}-{\partial A_{y} \over \partial z}{\bigg )}\mathbf {\hat {x}} &+\\\displaystyle {\bigg (}{\partial A_{x} \over \partial z}-{\partial A_{z} \over \partial x}{\bigg )}\mathbf {\hat {y}} &+\\\displaystyle {\bigg (}{\partial A_{y} \over \partial x}-{\partial A_{x} \over \partial y}{\bigg )}\mathbf {\hat {z}} &\ \end{matrix}}$ ${\ displaystyle {\ begin {matrix} \ displaystyle {\ bigg (} {\ partial A_ {z} \ over \ partial y} - {\ partial A_ {y} \ over \ partial z} {\ bigg)} \ mathbf {\ hat {x}} & + \\\ displaystyle {\ bigg (} {\ partial A_ {x} \ over \ partial z} - {\ partial A_ {z} \ over \ partial x} {\ bigg)} \ mathbf {\ hat {y}} & + \\\ displaystyle {\ bigg (} {\ partial A_ {y} \ over \ partial x} - {\ partial A_ {x} \ over \ partial y} {\ bigg )} \ mathbf {\ hat {z}} & \ \ end {matrix}}}$ ${\ displaystyle {\ begin {matrix} \ displaystyle {\ bigg (} {\ partial A_ {z} \ over \ partial y} - {\ partial A_ {y} \ over \ partial z} {\ bigg)} \ mathbf {\ hat {x}} & + \\\ displaystyle {\ bigg (} {\ partial A_ {x} \ over \ partial z} - {\ partial A_ {z} \ over \ partial x} {\ bigg)} \ mathbf {\ hat {y}} & + \\\ displaystyle {\ bigg (} {\ partial A_ {y} \ over \ partial x} - {\ partial A_ {x} \ over \ partial y} {\ bigg )} \ mathbf {\ hat {z}} & \ \ end {matrix}}}$	${\begin{matrix}\displaystyle {\bigg (}{1 \over \rho }{\partial A_{z} \over \partial \phi }-{\partial A_{\phi } \over \partial z}{\bigg )}{\boldsymbol {\hat {\rho }}}&+\\\displaystyle {\bigg (}{\partial A_{\rho } \over \partial z}-{\partial A_{z} \over \partial \rho }{\bigg )}{\boldsymbol {\hat {\phi }}}&+\\\displaystyle {1 \over \rho }{\bigg (}{\partial (\rho A_{\phi }) \over \partial \rho }-{\partial A_{\rho } \over \partial \phi }{\bigg )}{\boldsymbol {\hat {z}}}&\ \end{matrix}}$ ${\ displaystyle {\ begin {matrix} \ displaystyle {\ bigg (} {1 \ over \ rho} {\ partial A_ {z} \ over \ partial \ phi} - {\ partial A _ {\ phi} \ over \ partial z} {\ bigg)} {\ boldsymbol {\ hat {\ rho}}} & + \\\ displaystyle {\ bigg (} {\ partial A _ {\ rho} \ over \ partial z} - {\ partial A_ {z} \ over \ partial \ rho} {\ bigg)} {\ boldsymbol {\ hat {\ phi}}} & + \\\ displaystyle {1 \ over \ rho} {\ bigg (} {\ partial ( \ rho A _ {\ phi}) \ over \ partial \ rho} - {\ partial A _ {\ rho} \ over \ partial \ phi} {\ bigg)} {\ boldsymbol {\ hat {z}}} & \ \ end {matrix}}}$ ${\ displaystyle {\ begin {matrix} \ displaystyle {\ bigg (} {1 \ over \ rho} {\ partial A_ {z} \ over \ partial \ phi} - {\ partial A _ {\ phi} \ over \ partial z} {\ bigg)} {\ boldsymbol {\ hat {\ rho}}} & + \\\ displaystyle {\ bigg (} {\ partial A _ {\ rho} \ over \ partial z} - {\ partial A_ {z} \ over \ partial \ rho} {\ bigg)} {\ boldsymbol {\ hat {\ phi}}} & + \\\ displaystyle {1 \ over \ rho} {\ bigg (} {\ partial ( \ rho A _ {\ phi}) \ over \ partial \ rho} - {\ partial A _ {\ rho} \ over \ partial \ phi} {\ bigg)} {\ boldsymbol {\ hat {z}}} & \ \ end {matrix}}}$	${\begin{matrix}\displaystyle {1 \over r\sin \theta }{\bigg (}{\partial \over \partial \theta }(A_{\phi }\sin \theta )-{\partial A_{\theta } \over \partial \phi }{\bigg )}{\boldsymbol {\hat {r}}}&+\\\displaystyle {1 \over r}{\bigg (}{1 \over \sin \theta }{\partial A_{r} \over \partial \phi }-{\partial \over \partial r}(rA_{\phi }){\bigg )}{\boldsymbol {\hat {\theta }}}&+\\\displaystyle {1 \over r}{\bigg (}{\partial \over \partial r}(rA_{\theta })-{\partial A_{r} \over \partial \theta }{\bigg )}{\boldsymbol {\hat {\phi }}}&\ \end{matrix}}$ ${\ displaystyle {\ begin {matrix} \ displaystyle {1 \ over r \ sin \ theta} {\ bigg (} {\ partial \ over \ partial \ theta} (A _ {\ phi} \ sin \ theta) - { \ partial A _ {\ theta} \ over \ partial \ phi} {\ bigg)} {\ boldsymbol {\ hat {r}}} & + \\\ displaystyle {1 \ over r} {\ bigg (} {1 \ over \ sin \ theta} {\ partial A_ {r} \ over \ partial \ phi} - {\ partial \ over \ partial r} (rA _ {\ phi}) {\ bigg)} {\ boldsymbol {\ hat {\ theta}}} & + \\\ displaystyle {1 \ over r} {\ bigg (} {\ partial \ over \ partial r} (rA _ {\ theta}) - {\ partial A_ {r} \ over \ partial \ theta} {\ bigg)} {\ boldsymbol {\ hat {\ phi}}} & \ \ end {matrix}}}$ ${\ displaystyle {\ begin {matrix} \ displaystyle {1 \ over r \ sin \ theta} {\ bigg (} {\ partial \ over \ partial \ theta} (A _ {\ phi} \ sin \ theta) - { \ partial A _ {\ theta} \ over \ partial \ phi} {\ bigg)} {\ boldsymbol {\ hat {r}}} & + \\\ displaystyle {1 \ over r} {\ bigg (} {1 \ over \ sin \ theta} {\ partial A_ {r} \ over \ partial \ phi} - {\ partial \ over \ partial r} (rA _ {\ phi}) {\ bigg)} {\ boldsymbol {\ hat {\ theta}}} & + \\\ displaystyle {1 \ over r} {\ bigg (} {\ partial \ over \ partial r} (rA _ {\ theta}) - {\ partial A_ {r} \ over \ partial \ theta} {\ bigg)} {\ boldsymbol {\ hat {\ phi}}} & \ \ end {matrix}}}$
Laplacian $\nabla ^{2}f$ ${\ displaystyle \ nabla ^ {2} f}$ $\ nabla ^ {2} f$	${\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}$ ${\ displaystyle {\ partial ^ {2} f \ over \ partial x ^ {2}} + {\ partial ^ {2} f \ over \ partial y ^ {2}} + {\ partial ^ {2} f \ peste \ partial z ^ {2}}}$ ${\ partial ^ {2} f \ over \ partial x ^ {2}} + {\ partial ^ {2} f \ over \ partial y ^ {2}} + {\ partial ^ {2} f \ over \ partial z ^ {2}}$	${1 \over \rho }{\partial \over \partial \rho }{\bigg (}\rho {\partial f \over \partial \rho }{\bigg )}+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \phi ^{2}}+{\partial ^{2}f \over \partial z^{2}}$ ${\ displaystyle {1 \ over \ rho} {\ partial \ over \ partial \ rho} {\ bigg (} \ rho {\ partial f \ over \ partial \ rho} {\ bigg)} + {1 \ over \ rho ^ {2}} {\ partial ^ {2} f \ over \ partial \ phi ^ {2}} + {\ partial ^ {2} f \ over \ partial z ^ {2}}}$ ${\ displaystyle {1 \ over \ rho} {\ partial \ over \ partial \ rho} {\ bigg (} \ rho {\ partial f \ over \ partial \ rho} {\ bigg)} + {1 \ over \ rho ^ {2}} {\ partial ^ {2} f \ over \ partial \ phi ^ {2}} + {\ partial ^ {2} f \ over \ partial z ^ {2}}}$	${1 \over r^{2}}{\partial \over \partial r}{\bigg (}r^{2}{\partial f \over \partial r}{\bigg )}+{1 \over r^{2}\sin \theta }{\partial \over \partial \theta }{\bigg (}\sin \theta {\partial f \over \partial \theta }{\bigg )}+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2}f \over \partial \phi ^{2}}$ ${\ displaystyle {1 \ over r ^ {2}} {\ partial \ over \ partial r} {\ bigg (} r ^ {2} {\ partial f \ over \ partial r} {\ bigg)} + {1 \ over r ^ {2} \ sin \ theta} {\ partial \ over \ partial \ theta} {\ bigg (} \ sin \ theta {\ partial f \ over \ partial \ theta} {\ bigg)} + {1 \ over r ^ {2} \ sin ^ {2} \ theta} {\ partial ^ {2} f \ over \ partial \ phi ^ {2}}}$ ${\ displaystyle {1 \ over r ^ {2}} {\ partial \ over \ partial r} {\ bigg (} r ^ {2} {\ partial f \ over \ partial r} {\ bigg)} + {1 \ over r ^ {2} \ sin \ theta} {\ partial \ over \ partial \ theta} {\ bigg (} \ sin \ theta {\ partial f \ over \ partial \ theta} {\ bigg)} + {1 \ over r ^ {2} \ sin ^ {2} \ theta} {\ partial ^ {2} f \ over \ partial \ phi ^ {2}}}$
Laplacian al unui vector $\nabla ^{2}\mathbf {A}$ ${\ displaystyle \ nabla ^ {2} \ mathbf {A}}$ $\ nabla ^ {2} {\ mathbf {A}}$	$\nabla ^{2}A_{x}\mathbf {\hat {x}} +\nabla ^{2}A_{y}\mathbf {\hat {y}} +\nabla ^{2}A_{z}\mathbf {\hat {z}}$ ${\ displaystyle \ nabla ^ {2} A_ {x} \ mathbf {\ hat {x}} + \ nabla ^ {2} A_ {y} \ mathbf {\ hat {y}} + \ nabla ^ {2} A_ {z} \ mathbf {\ hat {z}}}$ $\ nabla ^ {2} A_ {x} {\ mathbf {{\ hat x}}} + \ nabla ^ {2} A_ {y} {\ mathbf {{\ hat y}}} + \ nabla ^ {2} A_ {z} {\ mathbf {{\ hat z}}}$	${\begin{matrix}\displaystyle {\bigg (}\nabla ^{2}A_{\rho }-{A_{\rho } \over \rho ^{2}}-{2 \over \rho ^{2}}{\partial A_{\phi } \over \partial \phi }{\bigg )}{\boldsymbol {\hat {\rho }}}&+\\\displaystyle {\bigg (}\nabla ^{2}A_{\phi }-{A_{\phi } \over \rho ^{2}}+{2 \over \rho ^{2}}{\partial A_{\rho } \over \partial \phi }{\bigg )}{\boldsymbol {\hat {\phi }}}&+\\\displaystyle (\nabla ^{2}A_{z}){\boldsymbol {\hat {z}}}&\ \end{matrix}}$ ${\ displaystyle {\ begin {matrix} \ displaystyle {\ bigg (} \ nabla ^ {2} A _ {\ rho} - {A _ {\ rho} \ over \ rho ^ {2}} - {2 \ over \ rho ^ {2}} {\ partial A _ {\ phi} \ over \ partial \ phi} {\ bigg)} {\ boldsymbol {\ hat {\ rho}}} & + \\\ displaystyle {\ bigg ( } \ nabla ^ {2} A _ {\ phi} - {A _ {\ phi} \ over \ rho ^ {2}} + {2 \ over \ rho ^ {2}} {\ partial A _ {\ rho } \ over \ partial \ phi} {\ bigg)} {\ boldsymbol {\ hat {\ phi}}} & + \\\ displaystyle (\ nabla ^ {2} A_ {z}) {\ boldsymbol {\ hat { z}}} & \ \ end {matrix}}}$ ${\ displaystyle {\ begin {matrix} \ displaystyle {\ bigg (} \ nabla ^ {2} A _ {\ rho} - {A _ {\ rho} \ over \ rho ^ {2}} - {2 \ over \ rho ^ {2}} {\ partial A _ {\ phi} \ over \ partial \ phi} {\ bigg)} {\ boldsymbol {\ hat {\ rho}}} & + \\\ displaystyle {\ bigg ( } \ nabla ^ {2} A _ {\ phi} - {A _ {\ phi} \ over \ rho ^ {2}} + {2 \ over \ rho ^ {2}} {\ partial A _ {\ rho } \ over \ partial \ phi} {\ bigg)} {\ boldsymbol {\ hat {\ phi}}} & + \\\ displaystyle (\ nabla ^ {2} A_ {z}) {\ boldsymbol {\ hat { z}}} & \ \ end {matrix}}}$	${\begin{matrix}{\bigg (}\nabla ^{2}A_{r}-{2A_{r} \over r^{2}}-{2 \over r^{2}\sin \theta }{\partial (A_{\theta }\sin \theta ) \over \partial \theta }-{2 \over r^{2}\sin \theta }{\partial A_{\phi } \over \partial \phi }{\bigg )}{\boldsymbol {\hat {r}}}&+\\{\bigg (}\nabla ^{2}A_{\theta }-{A_{\theta } \over r^{2}\sin ^{2}\theta }+{2 \over r^{2}}{\partial A_{r} \over \partial \theta }-{2\cos \theta \over r^{2}\sin ^{2}\theta }{\partial A_{\phi } \over \partial \phi }{\bigg )}{\boldsymbol {\hat {\theta }}}&+\\{\bigg (}\nabla ^{2}A_{\phi }-{A_{\phi } \over r^{2}\sin ^{2}\theta }+{2 \over r^{2}\sin \theta }{\partial A_{r} \over \partial \phi }+{2\cos \theta \over r^{2}\sin ^{2}\theta }{\partial A_{\theta } \over \partial \phi }{\bigg )}{\boldsymbol {\hat {\phi }}}&\end{matrix}}$ ${\ displaystyle {\ begin {matrix} {\ bigg (} \ nabla ^ {2} A_ {r} - {2A_ {r} \ over r ^ {2}} - {2 \ over r ^ {2} \ sin \ theta} {\ partial (A _ {\ theta} \ sin \ theta) \ over \ partial \ theta} - {2 \ over r ^ {2} \ sin \ theta} {\ partial A _ {\ phi} \ over \ partial \ phi} {\ bigg)} {\ boldsymbol {\ hat {r}}} & + \\ {\ bigg (} \ nabla ^ {2} A _ {\ theta} - {A _ {\ theta } \ over r ^ {2} \ sin ^ {2} \ theta} + {2 \ over r ^ {2}} {\ partial A_ {r} \ over \ partial \ theta} - {2 \ cos \ theta \ over r ^ {2} \ sin ^ {2} \ theta} {\ partial A _ {\ phi} \ over \ partial \ phi} {\ bigg)} {\ boldsymbol {\ hat {\ theta}}} & + \\ {\ bigg (} \ nabla ^ {2} A _ {\ phi} - {A _ {\ phi} \ over r ^ {2} \ sin ^ {2} \ theta} + {2 \ over r ^ {2} \ sin \ theta} {\ partial A_ {r} \ over \ partial \ phi} + {2 \ cos \ theta \ over r ^ {2} \ sin ^ {2} \ theta} {\ partial A _ {\ theta} \ over \ partial \ phi} {\ bigg)} {\ boldsymbol {\ hat {\ phi}}} & \ end {matrix}}}$ ${\ displaystyle {\ begin {matrix} {\ bigg (} \ nabla ^ {2} A_ {r} - {2A_ {r} \ over r ^ {2}} - {2 \ over r ^ {2} \ sin \ theta} {\ partial (A _ {\ theta} \ sin \ theta) \ over \ partial \ theta} - {2 \ over r ^ {2} \ sin \ theta} {\ partial A _ {\ phi} \ over \ partial \ phi} {\ bigg)} {\ boldsymbol {\ hat {r}}} & + \\ {\ bigg (} \ nabla ^ {2} A _ {\ theta} - {A _ {\ theta } \ over r ^ {2} \ sin ^ {2} \ theta} + {2 \ over r ^ {2}} {\ partial A_ {r} \ over \ partial \ theta} - {2 \ cos \ theta \ over r ^ {2} \ sin ^ {2} \ theta} {\ partial A _ {\ phi} \ over \ partial \ phi} {\ bigg)} {\ boldsymbol {\ hat {\ theta}}} & + \\ {\ bigg (} \ nabla ^ {2} A _ {\ phi} - {A _ {\ phi} \ over r ^ {2} \ sin ^ {2} \ theta} + {2 \ over r ^ {2} \ sin \ theta} {\ partial A_ {r} \ over \ partial \ phi} + {2 \ cos \ theta \ over r ^ {2} \ sin ^ {2} \ theta} {\ partial A _ {\ theta} \ over \ partial \ phi} {\ bigg)} {\ boldsymbol {\ hat {\ phi}}} & \ end {matrix}}}$
Lungime infinitesimală	$d\mathbf {l} =dx\mathbf {\hat {x}} +dy\mathbf {\hat {y}} +dz\mathbf {\hat {z}}$ ${\ displaystyle d \ mathbf {l} = dx \ mathbf {\ hat {x}} + dy \ mathbf {\ hat {y}} + dz \ mathbf {\ hat {z}}}$ $d {\ mathbf {l}} = dx {\ mathbf {{\ hat x}}} + dy {\ mathbf {{\ hat y}}} + dz {\ mathbf {{\ hat z}}}$	$d\mathbf {l} =d\rho {\boldsymbol {\hat {\rho }}}+\rho d\phi {\boldsymbol {\hat {\phi }}}+dz{\boldsymbol {\hat {z}}}$ ${\ displaystyle d \ mathbf {l} = d \ rho {\ boldsymbol {\ hat {\ rho}}} + \ rho d \ phi {\ boldsymbol {\ hat {\ phi}}} + dz {\ boldsymbol {\ pălărie {z}}}}$ $d {\ mathbf {l}} = d \ rho {\ boldsymbol {{\ hat \ rho}}} + \ rho d \ phi {\ boldsymbol {{\ hat \ phi}}} + dz {\ boldsymbol {{\ pălărie z}}}$	$d\mathbf {l} =dr\mathbf {\hat {r}} +rd\theta {\boldsymbol {\hat {\theta }}}+r\sin \theta d\phi {\boldsymbol {\hat {\phi }}}$ ${\ displaystyle d \ mathbf {l} = dr \ mathbf {\ hat {r}} + rd \ theta {\ boldsymbol {\ hat {\ theta}}} + r \ sin \ theta d \ phi {\ boldsymbol {\ pălărie {\ phi}}}}$ $d {\ mathbf {l}} = dr {\ mathbf {{\ hat r}}} + rd \ theta {\ boldsymbol {{\ hat \ theta}}} + r \ sin \ theta d \ phi {\ boldsymbol { {\ hat \ phi}}}$
Zonele infinitesimale	${\begin{matrix}d\mathbf {S} =&dydz\mathbf {\hat {x}} +\\&dxdz\mathbf {\hat {y}} +\\&dxdy\mathbf {\hat {z}} \end{matrix}}$ ${\ displaystyle {\ begin {matrix} d \ mathbf {S} = & dydz \ mathbf {\ hat {x}} + \\ & dxdz \ mathbf {\ hat {y}} + \\ & dxdy \ mathbf {\ pălărie {z}} \ end {matrix}}}$ ${\ begin {matrix} d {\ mathbf {S}} = & dydz {\ mathbf {{\ hat x}}} + \\ & dxdz {\ mathbf {{\ hat y}}} + \\ & dxdy { \ mathbf {{\ hat z}}} \ end {matrix}}$	${\begin{matrix}d\mathbf {S} =&\rho d\phi dz{\boldsymbol {\hat {\rho }}}+\\&d\rho dz{\boldsymbol {\hat {\phi }}}+\\&\rho d\rho d\phi \mathbf {\hat {z}} \end{matrix}}$ ${\ displaystyle {\ begin {matrix} d \ mathbf {S} = & \ rho d \ phi dz {\ boldsymbol {\ hat {\ rho}}} + \\ & d \ rho dz {\ boldsymbol {\ hat { \ phi}}} + \\ & \ rho d \ rho d \ phi \ mathbf {\ hat {z}} \ end {matrix}}}$ ${\ begin {matrix} d {\ mathbf {S}} = & \ rho d \ phi dz {\ boldsymbol {{\ hat \ rho}}} + \\ & d \ rho dz {\ boldsymbol {{\ hat \ phi}}} + \\ & \ rho d \ rho d \ phi {\ mathbf {{\ hat z}}} \ end {matrix}}$	${\begin{matrix}d\mathbf {S} =&r^{2}\sin \theta d\theta d\phi \mathbf {\hat {r}} +\\&r\sin \theta drd\phi {\boldsymbol {\hat {\theta }}}+\\&rdrd\theta {\boldsymbol {\hat {\phi }}}\end{matrix}}$ ${\ displaystyle {\ begin {matrix} d \ mathbf {S} = & r ^ {2} \ sin \ theta d \ theta d \ phi \ mathbf {\ hat {r}} + \\ & r \ sin \ theta drd \ phi {\ boldsymbol {\ hat {\ theta}}} + \\ & rdrd \ theta {\ boldsymbol {\ hat {\ phi}}} \ end {matrix}}}$ ${\ begin {matrix} d {\ mathbf {S}} = & r ^ {2} \ sin \ theta d \ theta d \ phi {\ mathbf {{\ hat r}}} + \\ & r \ sin \ theta drd \ phi {\ boldsymbol {{\ hat \ theta}}} + \\ & rdrd \ theta {\ boldsymbol {{\ hat \ phi}}} \ end {matrix}}$
Volumul infinitesimal	$dv=dxdydz$ ${\ displaystyle dv = dxdydz}$ $dv = dxdydz$	$dv=\rho d\rho d\phi dz$ ${\ displaystyle dv = \ rho d \ rho d \ phi dz}$ $dv = \ rho d \ rho d \ phi dz$	$dv=r^{2}\sin \theta drd\theta d\phi$ ${\ displaystyle dv = r ^ {2} \ sin \ theta drd \ theta d \ phi}$ $dv = r ^ {2} \ sin \ theta drd \ theta d \ phi$
Relații notabile (valabile în toate sistemele de referință): $\operatorname {div} \,\operatorname {grad} f=\nabla \cdot (\nabla f)=\nabla ^{2}f$ ${\ displaystyle \ operatorname {div} \, \ operatorname {grad} f = \ nabla \ cdot (\ nabla f) = \ nabla ^ {2} f}$ $\ operatorname {div} \, \ operatorname {grad} f = \ nabla \ cdot (\ nabla f) = \ nabla ^ {2} f$ ( Laplacian ) $\operatorname {rot} \,\operatorname {grad} f=\nabla \times (\nabla f)=0$ ${\ displaystyle \ operatorname {rot} \, \ operatorname {grad} f = \ nabla \ times (\ nabla f) = 0}$ $\ operatorname {rot} \, \ operatorname {grad} f = \ nabla \ times (\ nabla f) = 0$ $\operatorname {div} \,\operatorname {rot} \mathbf {A} =\nabla \cdot (\nabla \times \mathbf {A} )=0$ ${\ displaystyle \ operatorname {div} \, \ operatorname {rot} \ mathbf {A} = \ nabla \ cdot (\ nabla \ times \ mathbf {A}) = 0}$ $\ operatorname {div} \, \ operatorname {rot} {\ mathbf {A}} = \ nabla \ cdot (\ nabla \ times {\ mathbf {A}}) = 0$ $\operatorname {rot} \,\operatorname {rot} \mathbf {A} =\nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A}$ ${\ displaystyle \ operatorname {rot} \, \ operatorname {rot} \ mathbf {A} = \ nabla \ times (\ nabla \ times \ mathbf {A}) = \ nabla (\ nabla \ cdot \ mathbf {A}) - \ nabla ^ {2} \ mathbf {A}}$ $\ operatorname {rot} \, \ operatorname {rot} {\ mathbf {A}} = \ nabla \ times (\ nabla \ times {\ mathbf {A}}) = \ nabla (\ nabla \ cdot {\ mathbf {A }}) - \ nabla ^ {2} {\ mathbf {A}}$ $\nabla ^{2}fg=f\nabla ^{2}g+2\nabla f\cdot \nabla g+g\nabla ^{2}f$ ${\ displaystyle \ nabla ^ {2} fg = f \ nabla ^ {2} g + 2 \ nabla f \ cdot \ nabla g + g \ nabla ^ {2} f}$ $\ nabla ^ {2} fg = f \ nabla ^ {2} g + 2 \ nabla f \ cdot \ nabla g + g \ nabla ^ {2} f$ Formula Lagrange pentru produsul vector : $\mathbf {A} \times (\mathbf {B} \times \mathbf {C} )=\mathbf {B} (\mathbf {A} \cdot \mathbf {C} )-\mathbf {C} (\mathbf {A} \cdot \mathbf {B} )$ ${\ displaystyle \ mathbf {A} \ times (\ mathbf {B} \ times \ mathbf {C}) = \ mathbf {B} (\ mathbf {A} \ cdot \ mathbf {C}) - \ mathbf {C} (\ mathbf {A} \ cdot \ mathbf {B})}$ ${\ mathbf {A}} \ times ({\ mathbf {B}} \ times {\ mathbf {C}}) = {\ mathbf {B}} ({\ mathbf {A}} \ cdot {\ mathbf {C }}) - {\ mathbf {C}} ({\ mathbf {A}} \ cdot {\ mathbf {B}})$ $\nabla \cdot (f\mathbf {A} )=f\nabla \cdot \mathbf {A} +\mathbf {A} \cdot \nabla f$ ${\ displaystyle \ nabla \ cdot (f \ mathbf {A}) = f \ nabla \ cdot \ mathbf {A} + \ mathbf {A} \ cdot \ nabla f}$ $\ nabla \ cdot (f {\ mathbf A}) = f \ nabla \ cdot {\ mathbf A} + {\ mathbf A} \ cdot \ nabla f$ $\nabla \times f\mathbf {A} =f\nabla \times \mathbf {A} -\mathbf {A} \times \nabla f$ ${\ displaystyle \ nabla \ times f \ mathbf {A} = f \ nabla \ times \ mathbf {A} - \ mathbf {A} \ times \ nabla f}$ $\ nabla \ times f {\ mathbf A} = f \ nabla \ times {\ mathbf A} - {\ mathbf A} \ times \ nabla f$ $\nabla (\mathbf {A} \cdot \mathbf {B} )=(\mathbf {A} \cdot \nabla )\mathbf {B} +(\mathbf {B} \cdot \nabla )\mathbf {A} +\mathbf {A} \times (\nabla \times \mathbf {B} )+\mathbf {B} \times (\nabla \times \mathbf {A} ),$ ${\ displaystyle \ nabla (\ mathbf {A} \ cdot \ mathbf {B}) = (\ mathbf {A} \ cdot \ nabla) \ mathbf {B} + (\ mathbf {B} \ cdot \ nabla) \ mathbf {A} + \ mathbf {A} \ times (\ nabla \ times \ mathbf {B}) + \ mathbf {B} \ times (\ nabla \ times \ mathbf {A}),}$ $\ nabla ({\ mathbf {A}} \ cdot {\ mathbf {B}}) = ({\ mathbf {A}} \ cdot \ nabla) {\ mathbf {B}} + ({\ mathbf {B}} \ cdot \ nabla) {\ mathbf {A}} + {\ mathbf {A}} \ times (\ nabla \ times {\ mathbf {B}}) + {\ mathbf {B}} \ times (\ nabla \ times {\ mathbf {A}}),$ care împreună cu $\mathbf {A} =\mathbf {B} =\mathbf {v}$ ${\ displaystyle \ mathbf {A} = \ mathbf {B} = \ mathbf {v}}$ ${\ mathbf {A}} = {\ mathbf {B}} = {\ mathbf {v}}$ Cheia fluidului de transformare mecanică Weber urmează imediat: $(\mathbf {v} \cdot \nabla )\mathbf {v} =\nabla {\frac {\mathbf {v} ^{2}}{2}}-\mathbf {v} \times (\nabla \times \mathbf {v} )$ ${\ displaystyle (\ mathbf {v} \ cdot \ nabla) \ mathbf {v} = \ nabla {\ frac {\ mathbf {v} ^ {2}} {2}} - \ mathbf {v} \ times (\ nabla \ times \ mathbf {v})}$ $({\ mathbf {v}} \ cdot \ nabla) {\ mathbf {v}} = \ nabla {\ frac {{\ mathbf {v}} ^ {2}} {2}} - {\ mathbf {v} } \ times (\ nabla \ times {\ mathbf {v}})$ $\nabla \times (\mathbf {A} \times \mathbf {B} )=\mathbf {A} \,(\nabla \cdot \mathbf {B} )-\mathbf {B} \,(\nabla \cdot \mathbf {A} )+(\mathbf {B} \cdot \nabla )\mathbf {A} -(\mathbf {A} \cdot \nabla )\mathbf {B}$ ${\ displaystyle \ nabla \ times (\ mathbf {A} \ times \ mathbf {B}) = \ mathbf {A} \, (\ nabla \ cdot \ mathbf {B}) - \ mathbf {B} \, (\ nabla \ cdot \ mathbf {A}) + (\ mathbf {B} \ cdot \ nabla) \ mathbf {A} - (\ mathbf {A} \ cdot \ nabla) \ mathbf {B}}$ $\ nabla \ times ({\ mathbf A} \ times {\ mathbf B}) = {\ mathbf A} \, (\ nabla \ cdot {\ mathbf B}) - {\ mathbf B} \, (\ nabla \ cdot {\ mathbf A}) + ({\ mathbf B} \ cdot \ nabla) {\ mathbf A} - ({\ mathbf A} \ cdot \ nabla) {\ mathbf B}$ $\nabla \cdot (\mathbf {A} \times \mathbf {B} )=\mathbf {B} \cdot (\nabla \times \mathbf {A} )-\mathbf {A} \cdot (\nabla \times \mathbf {B} )$ ${\ displaystyle \ nabla \ cdot (\ mathbf {A} \ times \ mathbf {B}) = \ mathbf {B} \ cdot (\ nabla \ times \ mathbf {A}) - \ mathbf {A} \ cdot (\ nabla \ times \ mathbf {B})}$ $\ nabla \ cdot ({\ mathbf A} \ times {\ mathbf B}) = {\ mathbf B} \ cdot (\ nabla \ times {\ mathbf A}) - {\ mathbf A} \ cdot (\ nabla \ times {\ mathbf B})$

Notă

Funcția atan2 (y, x) este utilizată în locul arctan (y / x) pentru domeniul său. Funcția arctan (y / x) are imagine în (-π / 2, + π / 2), în timp ce atan2 (y, x) are imagine în (-π, π].

Elemente conexe

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