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Del in cylindrical and spherical coordinates

This is a list of some vector calculus formulae of general use in working with various coordinate systems.

Note

This page uses standard physics notation. For spherical coordinates, $theta$ is the angle between the z axis and the radius vector connecting the origin to the point in question. $phi$ is the angle between the projection of the radius vector onto the x-y plane and the x axis. Some (American mathematics) sources reverse this definition.
The function atan2(y, x) is used instead of the mathematical function arctan(y/x) due to its domain and image. The classical arctan(y/x) has an image of (-π/2, +π/2), whereas atan2(y, x) is defined to have an image of (-π, π]. (The expressions for the Nabla in spherical coordinates may need to be corrected)

**Table with the del operator in cylindrical, spherical and parabolic cylindrical coordinates**
Operation	Cartesian coordinates (x,y,z)	Cylindrical coordinates (s,φ,z)	Spherical coordinates (r,θ,φ)	Parabolic cylindrical coordinates (ο,τ,z)
Definition of coordinates	$begin{matrix} s & = & sqrt{x^2+y^2}$ phi & = & arctan(y/x) z & = & z end{matrix}	$begin{matrix}$ x & = & scosphi y & = & ssinphi z & = & z end{matrix}	$begin{matrix}$ x & = & rsinthetacosphi y & = & rsinthetasinphi z & = & rcostheta end{matrix}	$begin{matrix}$ x & = & sigma tau y & = & frac{1}{2} left(tau^{2} - sigma^{2} right) z & = & z end{matrix}
Definition of coordinates	$begin{matrix} r & = & sqrt{x^2+y^2+z^2} theta & = & arctan{left(frac{sqrt{x^2+y^2}}{z}right)} phi & = & arctan(y/x) end{matrix}$	$begin{matrix} r & = & sqrt{s^2 + z^2} theta & = & arctan{(s/z)} phi & = & phi end{matrix}$	$begin{matrix}$ s & = & rsin(theta) phi & = & phi z & = & rcos(theta) end{matrix}	$begin{matrix}$ scosphi & = & sigma tau ssinphi & = & frac{1}{2} left(tau^{2} - sigma^{2} right) z & = & z end{matrix}
Definition of unit vectors	$begin{matrix} boldsymbol{hat s} & = & frac{x}{s}mathbf{hat x}+frac{y}{s}mathbf{hat y} boldsymbol{hatphi} & = & -frac{y}{s}mathbf{hat x}+frac{x}{s}mathbf{hat y} mathbf{hat z} & = & mathbf{hat z} end{matrix}$	$begin{matrix} mathbf{hat x} & = & cosphiboldsymbol{hat s}-sinphiboldsymbol{hatphi} mathbf{hat y} & = & sinphiboldsymbol{hat s}+cosphiboldsymbol{hatphi} mathbf{hat z} & = & mathbf{hat z} end{matrix}$	$begin{matrix} mathbf{hat x} & = & sinthetacosphiboldsymbol{hat r}+costhetacosphiboldsymbol{hattheta}-sinphiboldsymbol{hatphi} mathbf{hat y} & = & sinthetasinphiboldsymbol{hat r}+costhetasinphiboldsymbol{hattheta}+cosphiboldsymbol{hatphi} mathbf{hat z} & = & costheta boldsymbol{hat r}-sintheta boldsymbol{hattheta} end{matrix}$	$begin{matrix} boldsymbol{hat sigma} & = & frac{tau}{sqrt{tau^2+sigma^2}}mathbf{hat x}-frac{sigma}{sqrt{tau^2+sigma^2}}mathbf{hat y} boldsymbol{hattau} & = & frac{sigma}{sqrt{tau^2+sigma^2}}mathbf{hat x}+frac{tau}{sqrt{tau^2+sigma^2}}mathbf{hat y} mathbf{hat z} & = & mathbf{hat z} end{matrix}$
Definition of unit vectors	$begin{matrix} mathbf{hat r} & = & frac{xmathbf{hat x}+ymathbf{hat y}+zmathbf{hat z}}{r} boldsymbol{hattheta} & = & frac{xzmathbf{hat x}+yzmathbf{hat y}-s^2mathbf{hat z}}{r s} boldsymbol{hatphi} & = & frac{-ymathbf{hat x}+xmathbf{hat y}}{s} end{matrix}$	$begin{matrix} mathbf{hat r} & = & frac{s}{r}boldsymbol{hat s}+frac{ z}{r}mathbf{hat z} boldsymbol{hattheta} & = & frac{z }{r}boldsymbol{hat s}-frac{s}{r}mathbf{hat z} boldsymbol{hatphi} & = & boldsymbol{hatphi} end{matrix}$	$begin{matrix} boldsymbol{hat s} & = & sinthetamathbf{hat r}+costhetaboldsymbol{hattheta} boldsymbol{hatphi} & = & boldsymbol{hatphi} mathbf{hat z} & = & costhetamathbf{hat r}-sinthetaboldsymbol{hattheta} end{matrix}$	$begin{matrix} end{matrix}$
A vector field $mathbf{A}$	$A_xmathbf{hat x} + A_ymathbf{hat y} + A_zmathbf{hat z}$	$A_sboldsymbol{hat s} + A_phiboldsymbol{hat phi} + A_zboldsymbol{hat z}$	$A_rboldsymbol{hat r} + A_thetaboldsymbol{hat theta} + A_phiboldsymbol{hat phi}$	$A_sigmaboldsymbol{hat sigma} + A_tauboldsymbol{hat tau} + A_phiboldsymbol{hat z}$
Gradient $nabla f$	${partial f over partial x}mathbf{hat x} + {partial f over partial y}mathbf{hat y} + {partial f over partial z}mathbf{hat z}$	${partial f over partial s}boldsymbol{hat s} + {1 over s}{partial f over partial phi}boldsymbol{hat phi} + {partial f over partial z}boldsymbol{hat z}$	${partial f over partial r}boldsymbol{hat r} + {1 over r}{partial f over partial theta}boldsymbol{hat theta} + {1 over rsintheta}{partial f over partial phi}boldsymbol{hat phi}$	$frac{1}{sqrt{sigma^{2} + tau^{2}}} {partial f over partial sigma}boldsymbol{hat sigma} + frac{1}{sqrt{sigma^{2} + tau^{2}}} {partial f over partial tau}boldsymbol{hat tau} + {partial f over partial z}boldsymbol{hat z}$
Divergence $nabla cdot mathbf{A}$	${partial A_x over partial x} + {partial A_y over partial y} + {partial A_z over partial z}$	${1 over s}{partial left(s A_s right) over partial s} + {1 over s}{partial A_phi over partial phi} + {partial A_z over partial z}$	${1 over r^2}{partial left(r^2 A_r right) over partial r} + {1 over rsintheta}{partial over partial theta} left( A_thetasintheta right) + {1 over rsintheta}{partial A_phi over partial phi}$	$frac{1}{sigma^{2} + tau^{2}}{partial A_sigma over partial sigma} + frac{1}{sigma^{2} + tau^{2}}{partial A_tau over partial tau} + {partial A_z over partial z}$
Curl $nabla times mathbf{A}$	$begin{matrix} displaystyleleft({partial A_z over partial y} - {partial A_y over partial z}right) mathbf{hat x} & + displaystyleleft({partial A_x over partial z} - {partial A_z over partial x}right) mathbf{hat y} & + displaystyleleft({partial A_y over partial x} - {partial A_x over partial y}right) mathbf{hat z} & end{matrix}$	$begin{matrix} displaystyleleft({1 over s}{partial A_z over partial phi} - {partial A_phi over partial z}right) boldsymbol{hat s} & + displaystyleleft({partial A_s over partial z} - {partial A_z over partial s}right) boldsymbol{hat phi} & + displaystyle{1 over s}left({partial left(s A_phi right) over partial s} - {partial A_s over partial phi}right) boldsymbol{hat z} & end{matrix}$	$begin{matrix} displaystyle{1 over rsintheta}left({partial over partial theta} left(A_phisintheta right) - {partial A_theta over partial phi}right) boldsymbol{hat r} & + displaystyle{1 over r}left({1 over sintheta}{partial A_r over partial phi} - {partial over partial r} left(r A_phi right) right) boldsymbol{hat theta} & + displaystyle{1 over r}left({partial over partial r} left(r A_theta right) - {partial A_r over partial theta}right) boldsymbol{hat phi} & end{matrix}$	$begin{matrix} displaystyleleft(frac{1}{sqrt{sigma^{2} + tau^{2}}}{partial A_z over partial tau} - {partial A_tau over partial z}right) boldsymbol{hat sigma} & - displaystyleleft(frac{1}{sqrt{sigma^{2} + tau^{2}}}{partial A_z over partial sigma}- {partial A_sigma over partial z}right) boldsymbol{hat tau} & + displaystylefrac{1}{sqrt{sigma^{2} + tau^{2}}}left({partial left(s A_phi right) over partial s} - {partial A_s over partial phi}right) boldsymbol{hat z} & end{matrix}$
Laplace operator $Delta f = nabla^2 f$	${partial^2 f over partial x^2} + {partial^2 f over partial y^2} + {partial^2 f over partial z^2}$	${1 over s}{partial over partial s}left(s {partial f over partial s}right) + {1 over s^2}{partial^2 f over partial phi^2} + {partial^2 f over partial z^2}$	${1 over r^2}{partial over partial r}!left(r^2 {partial f over partial r}right) !+!{1 over r^2!sintheta}{partial over partial theta}!left(sintheta {partial f over partial theta}right) !+!{1 over r^2!sin^2theta}{partial^2 f over partial phi^2}$	$frac{1}{sigma^{2} + tau^{2}} left( frac{partial^{2} f}{partial sigma^{2}} + frac{partial^{2} f}{partial tau^{2}} right) + frac{partial^{2} f}{partial z^{2}}$
Vector Laplacian $Delta mathbf{A} = nabla^2 mathbf{A}$	$Delta A_x mathbf{hat x} + Delta A_y mathbf{hat y} + Delta A_z mathbf{hat z}$	$begin{matrix} displaystyleleft(Delta A_s - {A_s over s^2} - {2 over s^2}{partial A_phi over partial phi}right) boldsymbol{hat s} & + displaystyleleft(Delta A_phi - {A_phi over s^2} + {2 over s^2}{partial A_s over partial phi}right) boldsymbol{hatphi} & + displaystyleleft(Delta A_z right) boldsymbol{hat z} & end{matrix}$	$begin{matrix} left(Delta A_r - {2 A_r over r^2} - {2 over r^2sintheta}{partial left(A_theta sinthetaright) over partialtheta} - {2 over r^2sintheta}{partial A_phi over partial phi}right) boldsymbol{hat r} & + left(Delta A_theta - {A_theta over r^2sin^2theta} + {2 over r^2}{partial A_r over partial theta} - {2 costheta over r^2sin^2theta}{partial A_phi over partial phi}right) boldsymbol{hattheta} & + left(Delta A_phi - {A_phi over r^2sin^2theta} + {2 over r^2sintheta}{partial A_r over partial phi} + {2 costheta over r^2sin^2theta}{partial A_theta over partial phi}right) boldsymbol{hatphi} & end{matrix}$
Differential displacement	$dmathbf{l} = dxmathbf{hat x} + dymathbf{hat y} + dzmathbf{hat z}$	$dmathbf{l} = dsboldsymbol{hat s} + s dphiboldsymbol{hat phi} + dzboldsymbol{hat z}$	$dmathbf{l} = drmathbf{hat r} + rdthetaboldsymbol{hat theta} + rsintheta dphiboldsymbol{hat phi}$	$dmathbf{l} = sqrt{sigma^{2} + tau^{2}} dsigmaboldsymbol{hat sigma} + sqrt{sigma^{2} + tau^{2}} dtauboldsymbol{hat tau} + dzboldsymbol{hat z}$
Differential normal area	$begin{matrix}dmathbf{S} = &dy,dz,mathbf{hat x} + &dx,dz,mathbf{hat y} + &dx,dy,mathbf{hat z}end{matrix}$	$begin{matrix} dmathbf{S} = & s, dphi, dz,boldsymbol{hat s} + & ds ,dz,boldsymbol{hat phi} + & s ,ds dphi ,mathbf{hat z} end{matrix}$	$begin{matrix} dmathbf{S} = & r^2 sintheta ,dtheta ,dphi ,mathbf{hat r} + & rsintheta ,dr,dphi ,boldsymbol{hat theta} + & r,dr,dtheta,boldsymbol{hat phi} end{matrix}$	$begin{matrix} dmathbf{S} = & sqrt{sigma^{2} + tau^{2}}, dtau, dz,boldsymbol{hat sigma} + & sqrt{sigma^{2} + tau^{2}} dsigma,dz,boldsymbol{hat tau} + & sigma^{2} + tau^{2} dsigma, dtau ,mathbf{hat z} end{matrix}$
Differential volume	$dtau = dx,dy,dz ,$	$dtau = s, ds, dphi, dz,$	$dtau = r^2sintheta ,dr,dtheta, dphi,$	$dtau = left(sigma^{2} + tau^{2} right) dsigma dtau dz,$
Non-trivial calculation rules: $operatorname{div grad } f = nabla cdot (nabla f) = nabla^2 f = Delta f$ (Laplacian) $operatorname{curl grad } f = nabla times (nabla f) = mathbf{0}$ $operatorname{div curl } mathbf{A} = nabla cdot (nabla times mathbf{A}) = 0$ $operatorname{curl curl } mathbf{A} = nabla times (nabla times mathbf{A}) = nabla (nabla cdot mathbf{A}) - nabla^2 mathbf{A}$ (using Lagrange's formula for the cross product) $Delta f g = f Delta g + 2 nabla f cdot nabla g + g Delta f$

Del in cylindrical and spherical coordinates

Note

See also