Дифференциальные операторы в различных системах координат

• Разделы:
  • Некоторые свойства
  • См. также

Здесь приведён список векторных дифференциальных операторов в некоторых системах координат.

Таблица операторов

Здесь используются стандартные физические обозначения. Для сферических координат, θ{displaystyle theta }

  обозначает угол между осью z и радиус-вектором точки. ϕ{displaystyle phi }  — угол между проекцией радиус-вектора на плоскость x-y и осью x.

Запись оператора Гамильтона в различных системах координат

Оператор	Прямоугольная система координат (x, y, z)	Цилиндрические координаты (ρ,φ,z)	Сферические координаты (r,θ,φ)	Параболические координаты (σ,τ,z)
Формулы преобразования координат	ρ=x2+y2ϕ=arctg⁡(y/x)z=z{displaystyle {begin{matrix}rho &=&{sqrt {x^{2}+y^{2}}}\phi &=&operatorname {arctg} (y/x)\z&=&zend{matrix}}}	x=ρcos⁡ϕy=ρsin⁡ϕz=z{displaystyle {begin{matrix}x&=&rho cos phi \y&=&rho sin phi \z&=&zend{matrix}}}	x=rsin⁡θcos⁡ϕy=rsin⁡θsin⁡ϕz=rcos⁡θ{displaystyle {begin{matrix}x&=&rsin theta cos phi \y&=&rsin theta sin phi \z&=&rcos theta end{matrix}}}	x=στy=12(τ2−σ2)z=z{displaystyle {begin{matrix}x&=&sigma tau \y&=&{frac {1}{2}}left(tau ^{2}-sigma ^{2}right)\z&=&zend{matrix}}}
	r=x2+y2+z2θ=arctg⁡(x2+y2x2+y2+z2)ϕ=arctg⁡(y/x){displaystyle {begin{matrix}r&=&{sqrt {x^{2}+y^{2}+z^{2}}}\theta &=&operatorname {arctg} {left({frac {sqrt {x^{2}+y^{2}}}{sqrt {x^{2}+y^{2}+z^{2}}}}right)}\phi &=&operatorname {arctg} (y/x)\end{matrix}}}	r=ρ2+z2θ=arctg⁡(ρ/z)ϕ=ϕ{displaystyle {begin{matrix}r&=&{sqrt {rho ^{2}+z^{2}}}\theta &=&operatorname {arctg} {(rho /z)}\phi &=&phi end{matrix}}}	ρ=rsin⁡(θ)ϕ=ϕz=rcos⁡(θ){displaystyle {begin{matrix}rho &=&rsin(theta )\phi &=&p hi \z&=&rcos(theta )end{matrix}}}	ρcos⁡ϕ=στρsin⁡ϕ=12(τ2−σ2)z=z{displaystyle {begin{matrix}rho cos phi &=&sigma tau \rho sin phi &=&{frac {1}{2}}left(tau ^{2}-sigma ^{2}right)\z&=&zend{matrix}}}
Радиус-вектор произвольной точки	xx^+yy^+zz^{displaystyle xmathbf {hat {x}} +ymathbf {hat {y}} +zmathbf {hat {z}} }	ρρ^+zz^{displaystyle rho {boldsymbol {hat {rho }}}+z{boldsymbol {hat {z}}}}	rr^{displaystyle r{boldsymbol {hat {r}}}}	?
Связь единичных векторов	ρ^=xρx^+yρy^ϕ^=−yρx^+xρy^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⁡ϕρ^−sin⁡ϕϕ^y^=sin⁡ϕρ^+cos⁡ϕϕ^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⁡θcos⁡ϕr^+cos⁡θcos⁡ϕθ^−sin⁡ϕϕ^y^=sin⁡θsin⁡ϕr^+cos⁡θsin⁡ϕθ^+cos⁡ϕϕ^z^=cos⁡θr^−sin⁡θθ^{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^=ττ2+σ2x^−στ2+σ2y^τ^=στ2+σ2x^+ττ2+σ2y^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^=xx^+yy^+zz^rθ^=xzx^+yzy^−ρ2z^rρϕ^=−yx^+xy^ρ{displaystyle {begin{matrix}mathbf {hat {r}} &=&{frac {xmathbf {hat {x}} +ymathbf {hat {y}} +zmathbf {hat {z}} }{r}}\{boldsymbol {hat {theta }}}&=&{frac {xzmathbf {hat {x}} +yzmathbf {hat {y}} -rho ^{2}mathbf {hat {z}} }{rrho }}\{boldsymbol {hat {phi }}}&=&{frac {-ymathbf {hat {x}} +xmathbf {hat {y}} }{rho }}end{matrix}}}	r^=ρrρ^+zrz^θ^=zrρ^−ρrz^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⁡θr^+cos⁡θθ^ϕ^=ϕ^z^=cos⁡θr^−sin⁡θθ^{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}}}	{displaystyle {begin{matrix}end{matrix}}}
Векторное поле A{displaystyle mathbf {A} }	Axx^+Ayy^+Azz^{displaystyle A_{x}mathbf {hat {x}} +A_{y}mathbf {hat {y}} +A_{z}mathbf {hat {z}} }	Aρρ^+Aϕϕ^+Azz^{displaystyle A_{rho }{boldsymbol {hat {rho }}}+A_{phi }{boldsymbol {hat {phi }}}+A_{z}{boldsymbol {hat {z}}}}	Arr^+Aθθ^+Aϕϕ^{displaystyle A_{r}{boldsymbol {hat {r}}}+A_{theta }{boldsymbol {hat {theta }}}+A_{phi }{boldsymbol {hat {phi }}}}	Aσσ^+Aττ^+Aϕz^{displaystyle A_{sigma }{boldsymbol {hat {sigma }}}+A_{tau }{boldsymbol {hat {tau }}}+A_{phi }{boldsymbol {hat {z}}}}
Градиент ∇f{displaystyle nabla f}	∂f∂xx^+∂f∂yy^+∂f∂zz^{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∂ρρ^+1ρ∂f∂ϕϕ^+∂f∂zz^{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∂rr^+1r∂f∂θθ^+1rsin⁡θ∂f∂ϕϕ^{displaystyle {partial f over partial r}{boldsymbol {hat {r}}}+{1 over r}{partial f over partial theta }{boldsymbol {hat {theta }}}+{1 over rsin theta }{partial f over partial phi }{boldsymbol {hat {phi }}}}	1σ2+τ2∂f∂σσ^+1σ2+τ2∂f∂ττ^+∂f∂zz^{displaystyle {frac {1}{sqrt {sigma ^{2}+tau ^{2}}}}{partial f over partial sigma }{boldsymbol {hat {sigma }}}+{frac {1}{sqrt {sigma ^{2}+tau ^{2}}}}{partial f over partial tau }{boldsymbol {hat {tau }}}+{partial f over partial z}{boldsymbol {hat {z}}}}
Дивергенция ∇⋅A{displaystyle nabla cdot mathbf {A} }	∂Ax∂x+∂Ay∂y+∂Az∂z{displaystyle {partial A_{x} over partial x}+{partial A_{y} over partial y}+{partial A_{z} over partial z}}	1ρ∂(ρAρ)∂ρ+1ρ∂Aϕ∂ϕ+∂Az∂z{displaystyle {1 over rho }{partial left(rho A_{rho }right) over partial rho }+{1 over rho }{partial A_{phi } over partial phi }+{partial A_{z} over partial z}}	1r2∂(r2Ar)∂r+1rsin⁡θ∂∂θ(Aθsin⁡θ)+1rsin⁡θ∂Aϕ∂ϕ{displaystyle {1 over r^{2}}{partial left(r^{2}A_{r}right) over partial r}+{1 over rsin theta }{partial over partial theta }left(A_{theta }sin theta right)+{1 over rsin theta }{partial A_{phi } over partial phi }}	1σ2+τ2∂Aσ∂σ+1σ2+τ2∂Aτ∂τ+∂Az∂z{displaystyle {frac {1}{sigma ^{2}+tau ^{2}}}{partial A_{sigma } over partial sigma }+{frac {1}{sigma ^{2}+tau ^{2}}}{partial A_{tau } over partial tau }+{partial A_{z} over partial z}}
Ротор ∇×A{displaystyle nabla times mathbf {A} }	(∂Az∂y−∂Ay∂z)x^+(∂Ax∂z−∂Az∂x)y^+(∂Ay∂x−∂Ax∂y)z^ {displaystyle {begin{matrix}left({partial A_{z} over partial y}-{partial A_{y} over partial z}right)mathbf {hat {x}} &+\left({partial A_{x} over partial z}-{partial A_{z} over partial x}right)mathbf {hat {y}} &+\left({partial A_{y} over partial x}-{partial A_{x} over partial y}right)mathbf {hat {z}} & end{matrix}}}	(1ρ∂Az∂ϕ−∂Aϕ∂z)ρ^+(∂Aρ∂z−∂Az∂ρ)ϕ^+1ρ(∂(ρAϕ)∂ρ−∂Aρ∂ϕ)z^ {displaystyle {begin{matrix}left({frac {1}{rho }}{frac {partial A_{z}}{partial phi }}-{frac {partial A_{phi }}{partial z}}right){boldsymbol {hat {rho }}}&+\left({frac {partial A_{rho }}{partial z}}-{frac {partial A_{z}}{partial rho }}right){boldsymbol {hat {phi }}}&+\{frac {1}{rho }}left({frac {partial (rho A_{phi })}{partial rho }}-{frac {partial A_{rho }}{partial phi }}right){boldsymbol {hat {z}}}& end{matrix}}}	1rsin⁡θ(∂∂θ(Aϕsin⁡θ)−∂Aθ∂ϕ)r^+1r(1sin⁡θ∂Ar∂ϕ−∂∂r(rAϕ))θ^+1r(∂∂r(rAθ)−∂Ar∂θ)ϕ^ {displaystyle {begin{matrix}{1 over rsin theta }left({partial over partial theta }left(A_{phi }sin theta right)-{partial A_{theta } over partial phi }right){boldsymbol {hat {r}}}&+\{1 over r}left({1 over sin theta }{partial A_{r} over partial phi }-{partial over partial r}left(rA_{phi }right)right){boldsymbol {hat {theta }}}&+\{1 over r}left({partial over partial r}left(rA_{theta }right)-{partial A_{r} over partial theta }right){boldsymbol {hat {phi }}}& end{matrix}}}	(1σ2+τ2∂Az∂τ−∂Aτ∂z)σ^−(1σ2+τ2∂Az∂σ−∂Aσ∂z)τ^+1σ2+τ2(∂(sAϕ)∂s−∂As∂ϕ)z^ {displaystyle {begin{matrix}left({frac {1}{sqrt {sigma ^{2}+tau ^{2}}}}{partial A_{z} over partial tau }-{partial A_{tau } over partial z}right){boldsymbol {hat {sigma }}}&-\left({frac {1}{sqrt {sigma ^{2}+tau ^{2}}}}{partial A_{z} over partial sigma }-{partial A_{sigma } over partial z}right){boldsymbol {hat {tau }}}&+\{frac {1}{sqrt {sigma ^{2}+tau ^{2}}}}left({partial left(sA_{phi }right) over partial s}-{partial A_{s} over partial phi }right){boldsymbol {hat {z}}}& end{matrix}}}
Оператор Лапласа Δf=∇2f{displaystyle Delta f=nabla ^{2}f}	∂2f∂x2+∂2f∂y2+∂2f∂z2{displaystyle {par tial ^{2}f over partial x^{2}}+{partial ^{2}f over partial y^{2}}+{partial ^{2}f over partial z^{2}}}	1ρ∂∂ρ(ρ∂f∂ρ)+1ρ2∂2f∂ϕ2+∂2f∂z2{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 phi ^{2}}+{partial ^{2}f over partial z^{2}}}	1r2∂∂r(r2∂f∂r)+1r2sin⁡θ∂∂θ(sin⁡θ∂f∂θ)+1r2sin2⁡θ∂2f∂ϕ2{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 phi ^{2}}}	1σ2+τ2(∂2f∂σ2+∂2f∂τ2)+∂2f∂z2{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}}}}
шаблон не поддерживает такой синтаксис векторной функции ΔA=∇2A{displaystyle Delta mathbf {A} =nabla ^{2}mathbf {A} }	ΔAxx^+ΔAyy^+ΔAzz^{displaystyle Delta A_{x}mathbf {hat {x}} +Delta A_{y}mathbf {hat {y}} +Delta A_{z}mathbf {hat {z}} }	(ΔAρ−Aρρ2−2ρ2∂Aϕ∂ϕ)ρ^+(ΔAϕ−Aϕρ2+2ρ2∂Aρ∂ϕ)ϕ^+(ΔAz)z^ {displaystyle {begin{matrix}left(Delta A_{rho }-{A_{rho } over rho ^{2}}-{2 over rho ^{2}}{partial A_{phi } over partial phi }right){boldsymbol {hat {rho }}}&+\left(Delta A_{phi }-{A_{phi } over rho ^{2}}+{2 over rho ^{2}}{partial A_{rho } over partial phi }right){boldsymbol {hat {phi }}}&+\left(Delta A_{z}right){boldsymbol {hat {z}}}& end{matrix}}}	(ΔAr−2Arr2−2r2sin⁡θ∂(Aθsin⁡θ)∂θ−2r2sin⁡θ∂Aϕ∂ϕ)r^+(ΔAθ−Aθr2sin2⁡θ+2r2∂Ar∂θ−2cos⁡θr2sin2⁡θ∂Aϕ∂ϕ)θ^+(ΔAϕ−Aϕr2sin2⁡θ+2r2sin⁡θ∂Ar∂ϕ+2cos⁡θr2sin2⁡θ∂Aθ∂ϕ)ϕ^{displaystyle {begin{matrix}left(Delta A_{r}-{2A_{r} over r^{2}}-{2 over r^{2}sin theta }{partial left(A_{theta }sin theta right) over partial theta }-{2 over r^{2}sin theta }{partial A_{phi } over partial phi }right){boldsymbol {hat {r}}}&+\left(Delta A_{theta }-{A_{theta } over r^{2}sin ^{2}theta }+{2 over r^{2}}{partial A_{r} over partial theta }-{2cos theta over r^{2}sin ^{2}theta }{partial A_{phi } over partial phi }right){boldsymbol {hat {theta }}}&+\left(Delta A_{phi }-{A_{phi } over r^{2}sin ^{2}theta }+{2 over r^{2}sin theta }{partial A_{r} over partial phi }+{2cos theta over r^{2}sin ^{2}theta }{partial A_{theta } over partial phi }right){boldsymbol {hat {phi }}}&end{matrix}}}	?
Элемент длины	dl=dxx^+dyy^+dzz^{displaystyle dmathbf {l} =dxmathbf {hat {x}} +dymathbf {hat {y}} +dzmathbf {hat {z}} }	dl=dρρ^+ρdϕϕ^+dzz^{displaystyle dmathbf {l} =drho {boldsymbol {hat {rho }}}+rho dphi {boldsymbol {hat {phi }}}+dz{boldsymbol {hat {z}}}}	dl=drr^+rdθθ^+rsin⁡θdϕϕ^{displaystyle dmathbf {l} =drmathbf {hat {r}} +rdtheta {boldsymbol {hat {theta }}}+rsin theta dphi {boldsymbol {hat {phi }}}}	dl=σ2+τ2dσσ^+σ2+τ2dττ^+dzz^{dis playstyle dmathbf {l} ={sqrt {sigma ^{2}+tau ^{2}}}dsigma {boldsymbol {hat {sigma }}}+{sqrt {sigma ^{2}+tau ^{2}}}dtau {boldsymbol {hat {tau }}}+dz{boldsymbol {hat {z}}}}
Элемент ориентированной площади	dS=dydzx^+dxdzy^+dxdyz^{displaystyle {begin{matrix}dmathbf {S} =&dy,dz,mathbf {hat {x}} +\&dx,dz,mathbf {hat {y}} +\&dx,dy,mathbf {hat {z}} end{matrix}}}	dS=ρdϕdzρ^+dρdzϕ^+ρdρdϕz^{displaystyle {begin{matrix}dmathbf {S} =&rho ,dphi ,dz,{boldsymbol {hat {rho }}}+\&drho ,dz,{boldsymbol {hat {phi }}}+\&rho ,drho dphi ,mathbf {hat {z}} end{matrix}}}	dS=r2sin⁡θdθdϕr^+rsin⁡θdrdϕθ^+rdrdθϕ^{displaystyle {begin{matrix}dmathbf {S} =&r^{2}sin theta ,dtheta ,dphi ,mathbf {hat {r}} +\&rsin theta ,dr,dphi ,{boldsymbol {hat {theta }}}+\&r,dr,dtheta ,{boldsymbol {hat {phi }}}end{matrix}}}	dS=σ2+τ2,dτdzσ^+σ2+τ2dσdzτ^+σ2+τ2dσ,dτz^{displaystyle {begin{matrix}dmathbf {S} =&{sqrt {sigma ^{2}+tau ^{2}}},dtau ,dz,{boldsymbol {hat {sigma }}}+\&{sqrt {sigma ^{2}+tau ^{2}}}dsigma ,dz,{boldsymbol {hat {tau }}}+\&sigma ^{2}+tau ^{2}dsigma ,dtau ,mathbf {hat {z}} end{matrix}}}
Элемент объёма	dτ=dxdydz{displaystyle dtau =dx,dy,dz,}	dτ=ρdρdϕdz{displaystyle dtau =rho ,drho ,dphi ,dz,}	dτ=r2sin⁡θdrdθdϕ{displaystyle dtau =r^{2}sin theta ,dr,dtheta ,dphi ,}	dτ=(σ2+τ2)dσdτdz,{displaystyle dtau =left(sigma ^{2}+tau ^{2}right)dsigma dtau dz,}

Некоторые свойства

Выражения для операторов второго порядка:

1. div grad ⁡f=∇⋅(∇f)=∇2f=Δf{displaystyle operatorname {div grad } f=nabla cdot (nabla f)=nabla ^{2}f=Delta f}  (Оператор Лапласа)
2. rot grad ⁡f=∇×(∇f)=0{displaystyle operatorname {rot grad } f=nabla times (nabla f)=0} 
3. div rot ⁡A=∇⋅(∇×A)=0{displaystyle operatorname {div rot } mathbf {A} =nabla cdot (nabla times mathbf {A} )=0} 
4. rot rot ⁡A=∇×(∇×A)=∇(∇⋅A)−∇2A{displaystyle operatorname {rot rot } mathbf {A} =nabla times (nabla times mathbf {A} )=nabla (nabla cdot mathbf {A} )-nabla ^{2}mathbf {A} } 

(используя формулу Лагранжа для двойного векторного произведения)

1. Δfg=fΔg+2∇f⋅∇g+gΔf{displaystyle Delta fg=fDelta g+2nabla fcdot nabla g+gDelta f} 

См. также

• < a href="/wiki/%25D0%259E%25D1%2580%25D1%2582%25D0%25BE%25D0%25B3%25D0%25BE%25D0%25BD%25D0%25B0%25D0%25BB%25D1%258C%25D0%25BD%25D1%258B%25D0%25B5_%25D0%25BA%25D0%25BE%25D0%25BE%25D1%2580%25D0%25B4%25D0%25B8%25D0%25BD%25D0%25B0%25D1%2582%25D1%258B/" class="mw-redirect" title="Ортогональные координаты">Ортогональные координаты
• Криволинейные координаты
• Коэффициенты Ламе
• Метод координат
• шаблон не поддерживает такой синтаксис