Nabla in coordinate cilindriche e sferiche

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Nel calcolo vettoriale è spesso utile conoscere come esprimere ∇ {\displaystyle \nabla } in altri sistemi di coordinate diversi da quello cartesiano.

Operatore Coordinate cartesiane (x,y,z) Coordinate cilindriche (ρ,φ,z) Coordinate sferiche (r,θ,φ)
Definizione delle coordinate   { x = ρ cos ⁡ ϕ y = ρ sin ⁡ ϕ z = z {\displaystyle {\begin{cases}x&=&\rho \cos \phi \\y&=&\rho \sin \phi \\z&=&z\end{cases}}} { x = r sin ⁡ θ cos ⁡ ϕ 0 ⩽ θ ⩽ π y = r sin ⁡ θ sin ⁡ ϕ 0 ⩽ ϕ < 2 π z = r cos ⁡ θ 0 ⩽ r < + ∞ {\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}}}
{ ρ = x 2 + y 2 ϕ = arctan ⁡ ( y / x ) z = z {\displaystyle {\begin{cases}\rho &=&{\sqrt {x^{2}+y^{2}}}\\\phi &=&\arctan(y/x)\\z&=&z\end{cases}}} { r = x 2 + y 2 + z 2 θ = arccos ⁡ ( z / r ) ϕ = arctan ⁡ ( y / x ) {\displaystyle {\begin{cases}r&=&{\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=&\arccos(z/r)\\\phi &=&\arctan(y/x)\end{cases}}}
Campo vettoriale 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 }}}}
Gradiente ∇ f {\displaystyle \nabla f} ∂ f ∂ x x ^ + ∂ f ∂ y y ^ + ∂ f ∂ z z ^ {\displaystyle {\partial f \over \partial x}\mathbf {\hat {x}} +{\partial f \over \partial y}\mathbf {\hat {y}} +{\partial f \over \partial z}\mathbf {\hat {z}} } ∂ f ∂ ρ ρ ^ + 1 ρ ∂ f ∂ ϕ ϕ ^ + ∂ f ∂ z 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}}}} ∂ f ∂ r r ^ + 1 r ∂ f ∂ θ θ ^ + 1 r sin ⁡ θ ∂ f ∂ ϕ ϕ ^ {\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 }}}}
Divergenza ∇ ⋅ A {\displaystyle \nabla \cdot \mathbf {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 (\rho A_{\rho }) \over \partial \rho }+{1 \over \rho }{\partial A_{\phi } \over \partial \phi }+{\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 (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 }}
Rotore ∇ × 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 {\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}}} ( 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 {\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}}} 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 {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}}}
Laplaciano ∇ 2 f {\displaystyle \nabla ^{2}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 }{\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 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}{\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}}}
Laplaciano di un vettore ∇ 2 A {\displaystyle \nabla ^{2}\mathbf {A} } ∇ 2 A x x ^ + ∇ 2 A y y ^ + ∇ 2 A z z ^ {\displaystyle \nabla ^{2}A_{x}\mathbf {\hat {x}} +\nabla ^{2}A_{y}\mathbf {\hat {y}} +\nabla ^{2}A_{z}\mathbf {\hat {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{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}}} ( ∇ 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{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}}}
Lunghezza infinitesima 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 }}}}
Area infinitesima d S = d y d z x ^ + d x d z y ^ + d x d y z ^ {\displaystyle {\begin{matrix}d\mathbf {S} =&dydz\mathbf {\hat {x}} +\\&dxdz\mathbf {\hat {y}} +\\&dxdy\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 drd\phi {\boldsymbol {\hat {\theta }}}+\\&rdrd\theta {\boldsymbol {\hat {\phi }}}\end{matrix}}}
Volume infinitesimo d v = d x d y d z {\displaystyle dv=dxdydz} d v = ρ d ρ d ϕ d z {\displaystyle dv=\rho d\rho d\phi dz} d v = ρ 2 sin ⁡ θ d ρ d θ d ϕ {\displaystyle dv=\rho ^{2}\sin \theta d\rho d\theta d\phi }

Relazioni notevoli (valgono in tutti i sistemi di riferimento)

  • div grad ⁡ f = ∇ ⋅ ( ∇ f ) = ∇ 2 f {\displaystyle \operatorname {div} \,\operatorname {grad} f=\nabla \cdot (\nabla f)=\nabla ^{2}f} (Laplaciano)
  • rot grad ⁡ f = ∇ × ( ∇ f ) = 0 {\displaystyle \operatorname {rot} \,\operatorname {grad} f=\nabla \times (\nabla f)=0}
  • div rot ⁡ A = ∇ ⋅ ( ∇ × A ) = 0 {\displaystyle \operatorname {div} \,\operatorname {rot} \mathbf {A} =\nabla \cdot (\nabla \times \mathbf {A} )=0}
  • 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} }
  • ∇ 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}

Formula di Lagrange per il prodotto vettoriale: A × ( B × C ) = B ( A ⋅ C ) − C ( A ⋅ B ) {\displaystyle \mathbf {A} \times (\mathbf {B} \times \mathbf {C} )=\mathbf {B} (\mathbf {A} \cdot \mathbf {C} )-\mathbf {C} (\mathbf {A} \cdot \mathbf {B} )}

  • ∇ ⋅ ( f A ) = f ∇ ⋅ A + A ⋅ ∇ f {\displaystyle \nabla \cdot (f\mathbf {A} )=f\nabla \cdot \mathbf {A} +\mathbf {A} \cdot \nabla f}
  • ∇ × f A = f ∇ × A − A × ∇ f {\displaystyle \nabla \times f\mathbf {A} =f\nabla \times \mathbf {A} -\mathbf {A} \times \nabla f}
  • ∇ ( A ⋅ B ) = ( A ⋅ ∇ ) B + ( B ⋅ ∇ ) A + A × ( ∇ × B ) + B × ( ∇ × 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} ),}
che insieme a A = B = v {\displaystyle \mathbf {A} =\mathbf {B} =\mathbf {v} } segue immediatamente la chiave per il fluido di trasformazione meccanica Weber[senza fonte]:
( v ⋅ ∇ ) v = ∇ ‖ v ‖ 2 2 − v × ( ∇ × v ) {\displaystyle (\mathbf {v} \cdot \nabla )\mathbf {v} =\nabla {\frac {\left\|\mathbf {v} \right\|^{2}}{2}}-\mathbf {v} \times (\nabla \times \mathbf {v} )}
  • ∇ × ( A × B ) = A ( ∇ ⋅ B ) − B ( ∇ ⋅ A ) + ( B ⋅ ∇ ) A − ( A ⋅ ∇ ) 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} }
  • ∇ ⋅ ( A × B ) = B ⋅ ( ∇ × A ) − A ⋅ ( ∇ × B ) {\displaystyle \nabla \cdot (\mathbf {A} \times \mathbf {B} )=\mathbf {B} \cdot (\nabla \times \mathbf {A} )-\mathbf {A} \cdot (\nabla \times \mathbf {B} )}

Nota

  • La funzione atan2(y,x) è usata al posto di a r c t a n ( y / x ) {\displaystyle \mathrm {arctan} (y/x)} per il suo dominio. La funzione a r c t a n ( y / x ) {\displaystyle \mathrm {arctan} (y/x)} ha immagine in ( − π / 2 , π / 2 ) , {\displaystyle (-\pi /2,\pi /2),} mentre a t a n 2 ( y , x ) {\displaystyle \mathrm {atan2} (y,x)} ha immagine in ( − π , π ] . {\displaystyle (-\pi ,\pi ].}

Voci correlate

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