Del in cylindrical and spherical coordinates

Operation Cartesian coordinates (x, y, z) Cylindrical coordinates (ρ, φ, z) Spherical coordinates (r, θ, φ),
where θ is the polar angle and φ is the azimuthal angle^α Vector field A Axx^+Ayy^+Azz^{displaystyle A_{x}{hat {mathbf {x} }}+A_{y}{hat {mathbf {y} }}+A_{z}{hat {mathbf {z} }}}

Aρρ^+Aφφ^+Azz^{displaystyle A_{rho }{hat {boldsymbol {rho }}}+A_{varphi }{hat {boldsymbol {varphi }}}+A_{z}{hat {mathbf {z} }}}

Arr^+Aθθ^+Aφφ^{displaystyle A_{r}{hat {mathbf {r} }}+A_{theta }{hat {boldsymbol {theta }}}+A_{varphi }{hat {boldsymbol {varphi }}}}

Gradient ∇f^[1] ∂f∂xx^+∂f∂yy^+∂f∂zz^{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∂ρρ^+1ρ∂f∂φφ^+∂f∂zz^{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∂rr^+1r∂f∂θθ^+1rsin⁡θ∂f∂φφ^{displaystyle {partial f over partial r}{hat {mathbf {r} }}+{1 over r}{partial f over partial theta }{hat {boldsymbol {theta }}}+{1 over rsin theta }{partial f over partial varphi }{hat {boldsymbol {varphi }}}}

Divergence ∇ ⋅ A^[1] ∂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_{varphi } over partial varphi }+{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_{varphi } over partial varphi }}

Curl ∇ × A^[1] (∂Az∂y−∂Ay∂z)x^+(∂Ax∂z−∂Az∂x)y^+(∂Ay∂x−∂Ax∂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}}}

(1ρ∂Az∂φ−∂Aφ∂z)ρ^+(∂Aρ∂z−∂Az∂ρ)φ^+1ρ(∂(ρAφ)∂ρ−∂Aρ∂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}}}

1rsin⁡θ(∂∂θ(Aφsin⁡θ)−∂Aθ∂φ)r^+1r(1sin⁡θ∂Ar∂φ−∂∂r(rAφ))θ^+1r(∂∂r(rAθ)−∂Ar∂θ)φ^{displaystyle {begin{aligned}{frac {1}{rsin 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] ∂2f∂x2+∂2f∂y2+∂2f∂z2{displaystyle {partial ^{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 varphi ^{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 varphi ^{2}}}

Vector Gradient ∇A ∂Ax∂xx^⊗x^+∂Ax∂yx^⊗y^+∂Ax∂zx^⊗z^+∂Ay∂xy^⊗x^+∂Ay∂yy^⊗y^+∂Ay∂zy^⊗z^+∂Az∂xz^⊗x^+∂Az∂yz^⊗y^+∂Az∂zz^⊗z^{displaystyle {begin{aligned}{}&{frac {partial A_{x}}{partial x}}{hat {mathbf {x} }}otimes {hat {mathbf {x} }}+{frac {partial A_{x}}{partial y}}{hat {mathbf {x} }}otimes {hat {mathbf {y} }}+{frac {partial A_{x}}{partial z}}{hat {mathbf {x} }}otimes {hat {mathbf {z} }}\{}+&{frac {partial A_{y}}{partial x}}{hat {mathbf {y} }}otimes {hat {mathbf {x} }}+{frac {partial A_{y}}{partial y}}{hat {mathbf {y} }}otimes {hat {mathbf {y} }}+{frac {partial A_{y}}{partial z}}{hat {mathbf {y} }}otimes {hat {mathbf {z} }}\{}+&{frac {partial A_{z}}{partial x}}{hat {mathbf {z} }}otimes {hat {mathbf {x} }}+{frac {partial A_{z}}{partial y}}{hat {mathbf {z} }}otimes {hat {mathbf {y} }}+{frac {partial A_{z}}{partial z}}{hat {mathbf {z} }}otimes {hat {mathbf {z} }}end{aligned}}}

∂Aρ∂ρρ^⊗ρ^+(1ρ∂Aρ∂φ−Aφρ)ρ^⊗φ^+∂Aρ∂zρ^⊗z^+∂Aφ∂ρφ^⊗ρ^+(1ρ∂Aφ∂φ+Aρρ)φ^⊗φ^+∂Aφ∂zφ^⊗z^+∂Az∂ρz^⊗ρ^+1ρ∂Az∂φz^⊗φ^+∂Az∂zz^⊗z^{displaystyle {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 }}}otimes {hat {boldsymbol {varphi }}}+{frac {partial A_{rho }}{partial z}}{hat {boldsymbol {rho }}}otimes {hat {mathbf {z} }}\{}+&{frac {partial A_{varphi }}{partial rho }}{hat {boldsymbol {varphi }}}otimes {hat {boldsymbol {rho }}}+left({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 {boldsymbol {varphi }}}otimes {hat {mathbf {z} }}\{}+&{frac {partial A_{z}}{partial rho }}{hat {mathbf {z} }}otimes {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} }}otimes {hat {mathbf {z} }}end{aligned}}}

∂Ar∂rr^⊗r^+(1r∂Ar∂θ−Aθr)r^⊗θ^+(1rsin⁡θ∂Ar∂φ−Aφr)r^⊗φ^+∂Aθ∂rθ^⊗r^+(1r∂Aθ∂θ+Arr)θ^⊗θ^+(1rsin⁡θ∂Aθ∂φ−cot⁡θAφr)θ^⊗φ^+∂Aφ∂rφ^⊗r^+1r∂Aφ∂θφ^⊗θ^+(1rsin⁡θ∂Aφ∂φ+cot⁡θAθr+Arr)φ^⊗φ^{displaystyle {begin{aligned}{}&{frac {partial A_{r}}{partial r}}{hat {mathbf {r} }}otimes {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}{rsin theta }}{frac {partial A_{r}}{partial varphi }}-{frac {A_{varphi }}{r}}right){hat {mathbf {r} }}otimes {hat {boldsymbol {varphi }}}\{}+&{frac {partial A_{theta }}{partial r}}{hat {boldsymbol {theta }}}otimes {hat {mathbf {r} }}+left({frac {1}{r}}{frac {partial A_{theta }}{partial theta }}+{frac {A_{r}}{r}}right){hat {boldsymbol {theta }}}otimes {hat {boldsymbol {theta }}}+left({frac {1}{rsin theta }}{frac {partial A_{theta }}{partial varphi }}-cot theta {frac {A_{varphi }}{r}}right){hat {boldsymbol {theta }}}otimes {hat {boldsymbol {varphi }}}\{}+&{frac {partial A_{varphi }}{partial r}}{hat {boldsymbol {varphi }}}otimes {hat {mathbf {r} }}+{frac {1}{r}}{frac {partial A_{varphi }}{partial theta }}{hat {boldsymbol {varphi }}}otimes {hat {boldsymbol {theta }}}+left({frac {1}{rsin 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}}}

Vector Laplacian ∇²A ≡ ∆A^[2] ∇2Axx^+∇2Ayy^+∇2Azz^{displaystyle nabla ^{2}A_{x}{hat {mathbf {x} }}+nabla ^{2}A_{y}{hat {mathbf {y} }}+nabla ^{2}A_{z}{hat {mathbf {z} }}}

(∇2Aρ−Aρρ2−2ρ2∂Aφ∂φ)ρ^+(∇2Aφ−Aφρ2+2ρ2∂Aρ∂φ)φ^+∇2Azz^{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}}}

(∇2Ar−2Arr2−2r2sin⁡θ∂(Aθsin⁡θ)∂θ−2r2sin⁡θ∂Aφ∂φ)r^+(∇2Aθ−Aθr2sin2⁡θ+2r2∂Ar∂θ−2cos⁡θr2sin2⁡θ∂Aφ∂φ)θ^+(∇2Aφ−Aφr2sin2⁡θ+2r2sin⁡θ∂Ar∂φ+2cos⁡θr2sin2⁡θ∂Aθ∂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 {2cos 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 {2cos theta }{r^{2}sin ^{2}theta }}{frac {partial A_{theta }}{partial varphi }}right)&{hat {boldsymbol {varphi }}}end{aligned}}}

Material derivative^α[3](A ⋅ ∇)B A⋅∇Bxx^+A⋅∇Byy^+A⋅∇Bzz^{displaystyle 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} }}}

(Aρ∂Bρ∂ρ+Aφρ∂Bρ∂φ+Az∂Bρ∂z−AφBφρ)ρ^+(Aρ∂Bφ∂ρ+Aφρ∂Bφ∂φ+Az∂Bφ∂z+AφBρρ)φ^+(Aρ∂Bz∂ρ+Aφρ∂Bz∂φ+Az∂Bz∂z)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}}}

(Ar∂Br∂r+Aθr∂Br∂θ+Aφrsin⁡θ∂Br∂φ−AθBθ+AφBφr)r^+(Ar∂Bθ∂r+Aθr∂Bθ∂θ+Aφrsin⁡θ∂Bθ∂φ+AθBrr−AφBφcot⁡θr)θ^+(Ar∂Bφ∂r+Aθr∂Bφ∂θ+Aφrsin⁡θ∂Bφ∂φ+AφBrr+AφBθcot⁡θr)φ^{displaystyle {begin{aligned}left(A_{r}{frac {partial B_{r}}{partial r}}+{frac {A_{theta }}{r}}{frac {partial B_{r}}{partial theta }}+{frac {A_{varphi }}{rsin 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 }}{rsin 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 }}{rsin 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 ∇ ⋅ T (not to be confused with 2nd order tensor divergence)

(∂Txx∂x+∂Tyx∂y+∂Tzx∂z)x^+(∂Txy∂x+∂Tyy∂y+∂Tzy∂z)y^+(∂Txz∂x+∂Tyz∂y+∂Tzz∂z)z^{displaystyle {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}}}

[∂Tρρ∂ρ+1ρ∂Tφρ∂φ+∂Tzρ∂z+1ρ(Tρρ−Tφφ)]ρ^+[∂Tρφ∂ρ+1ρ∂Tφφ∂φ+∂Tzφ∂z+1ρ(Tρφ+Tφρ)]φ^+[∂Tρz∂ρ+1ρ∂Tφz∂φ+∂Tzz∂z+Tρzρ]z^{displaystyle {begin{aligned}left[{frac {partial T_{rho rho }}{partial rho }}+{frac {1}{rho }}{frac {partial T_{varphi rho }}{partial varphi }}+{frac {partial T_{zrho }}{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_{zvarphi }}{partial z}}+{frac {1}{rho }}(T_{rho varphi }+T_{varphi rho })right]&{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}}}

[∂Trr∂r+2Trrr+1r∂Tθr∂θ+cot⁡θrTθr+1rsin⁡θ∂Tφr∂φ−1r(Tθθ+Tφφ)]r^+[∂Trθ∂r+2Trθr+1r∂Tθθ∂θ+cot⁡θrTθθ+1rsin⁡θ∂Tφθ∂φ+Tθrr−cot⁡θrTφφ]θ^+[∂Trφ∂r+2Trφr+1r∂Tθφ∂θ+1rsin⁡θ∂Tφφ∂φ+Tφrr+cot⁡θr(Tθφ+Tφθ)]φ^{displaystyle {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}{rsin theta }}{frac {partial T_{varphi r}}{partial varphi }}-{frac {1}{r}}(T_{theta theta }+T_{varphi varphi })right]&{hat {mathbf {r} }}\+left[{frac {partial T_{rtheta }}{partial r}}+2{frac {T_{rtheta }}{r}}+{frac {1}{r}}{frac {partial T_{theta theta }}{partial theta }}+{frac {cot theta }{r}}T_{theta theta }+{frac {1}{rsin 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_{rvarphi }}{partial r}}+2{frac {T_{rvarphi }}{r}}+{frac {1}{r}}{frac {partial T_{theta varphi }}{partial theta }}+{frac {1}{rsin 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}}}

Differential displacement dℓ^[1] dxx^+dyy^+dzz^{displaystyle dx,{hat {mathbf {x} }}+dy,{hat {mathbf {y} }}+dz,{hat {mathbf {z} }}}

dρρ^+ρdφφ^+dzz^{displaystyle drho ,{hat {boldsymbol {rho }}}+rho ,dvarphi ,{hat {boldsymbol {varphi }}}+dz,{hat {mathbf {z} }}}

drr^+rdθθ^+rsin⁡θdφφ^{displaystyle dr,{hat {mathbf {r} }}+r,dtheta ,{hat {boldsymbol {theta }}}+r,sin theta ,dvarphi ,{hat {boldsymbol {varphi }}}}

Differential normal area dS dydzx^+dxdzy^+dxdyz^{displaystyle {begin{aligned}dy,dz&,{hat {mathbf {x} }}\{}+dx,dz&,{hat {mathbf {y} }}\{}+dx,dy&,{hat {mathbf {z} }}end{aligned}}}

ρdφdzρ^+dρdzφ^+ρdρdφz^{displaystyle {begin{aligned}rho ,dvarphi ,dz&,{hat {boldsymbol {rho }}}\{}+drho ,dz&,{hat {boldsymbol {varphi }}}\{}+rho ,drho ,dvarphi &,{hat {mathbf {z} }}end{aligned}}}

r2sin⁡θdθdφr^+rsin⁡θdrdφθ^+rdrdθφ^{displaystyle {begin{aligned}r^{2}sin theta ,dtheta ,dvarphi &,{hat {mathbf {r} }}\{}+rsin theta ,dr,dvarphi &,{hat {boldsymbol {theta }}}\{}+r,dr,dtheta &,{hat {boldsymbol {varphi }}}end{aligned}}}

Differential volume dV^[1] dxdydz{displaystyle dx,dy,dz}

ρdρdφdz{displaystyle rho ,drho ,dvarphi ,dz}

r2sin⁡θdrdθdφ{displaystyle r^{2}sin theta ,dr,dtheta ,dvarphi }