NationMaster - Encyclopedia: Nabla in cylindrical and spherical coordinates

This is a list of some vector calculus formulae of general use in working with standard coordinate systems. Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ...

**Table with the del operator in cylindrical and spherical coordinates**
Operation	Cartesian coordinates (x,y,z)	Cylindrical coordinates (ρ,φ,z)	Spherical coordinates (r,θ,φ)
Definition of coordinates		$left[begin{matrix} x & = & rhocosphi y & = & rhosinphi z & = & z end{matrix}right.$	$left[begin{matrix} x & = & rsinthetacosphi y & = & rsinthetasinphi z & = & rcostheta end{matrix}right.$
Definition of coordinates		$left[begin{matrix} rho & = & sqrt{x^2 + y^2} phi & = & operatorname{atan2}(y, x) z & = & z end{matrix}right.$	$left[begin{matrix} r & = & sqrt{x^2 + y^2 + z^2} theta & = & arccos(z / r) phi & = & operatorname{atan2}(y, x) end{matrix}right.$
$mathbf{A}$	$A_xmathbf{hat x} + A_ymathbf{hat y} + A_zmathbf{hat z}$	$A_rhoboldsymbol{hat rho} + A_phiboldsymbol{hat phi} + A_zboldsymbol{hat z}$	$A_rboldsymbol{hat r} + A_thetaboldsymbol{hat theta} + A_phiboldsymbol{hat phi}$
$nabla f$	${partial f over partial x}mathbf{hat x} + {partial f over partial y}mathbf{hat y} + {partial f over partial z}mathbf{hat z}$	${partial f over partial rho}boldsymbol{hat rho} + {1 over rho}{partial f over partial phi}boldsymbol{hat phi} + {partial f over partial z}boldsymbol{hat z}$	${partial f over partial r}boldsymbol{hat r} + {1 over r}{partial f over partial theta}boldsymbol{hat theta} + {1 over rsintheta}{partial f over partial phi}boldsymbol{hat phi}$
$nabla cdot mathbf{A}$	${partial A_x over partial x} + {partial A_y over partial y} + {partial A_z over partial z}$	${1 over rho}{partial ( rho A_rho ) over partial rho} + {1 over rho}{partial A_phi over partial phi} + {partial A_z over partial z}$	${1 over r^2}{partial ( r^2 A_r ) over partial r} + {1 over rsintheta}{partial over partial theta} ( A_thetasintheta ) + {1 over rsintheta}{partial A_phi over partial phi}$
$nabla times mathbf{A}$	$begin{matrix} ({partial A_z over partial y} - {partial A_y over partial z}) mathbf{hat x} & + ({partial A_x over partial z} - {partial A_z over partial x}) mathbf{hat y} & + ({partial A_y over partial x} - {partial A_x over partial y}) mathbf{hat z} & end{matrix}$	$begin{matrix} ({1 over rho}{partial A_z over partial phi} - {partial A_phi over partial z}) boldsymbol{hat rho} & + ({partial A_rho over partial z} - {partial A_z over partial rho}) boldsymbol{hat phi} & + {1 over rho}({partial ( rho A_phi ) over partial rho} - {partial A_rho over partial phi}) boldsymbol{hat z} & end{matrix}$	$begin{matrix} {1 over rsintheta}({partial over partial theta} ( A_phisintheta ) - {partial A_theta over partial phi}) boldsymbol{hat r} & + {1 over r}({1 over sintheta}{partial A_r over partial phi} - {partial over partial r} ( r A_phi ) ) boldsymbol{hat theta} & + {1 over r}({partial over partial r} ( r A_theta ) - {partial A_r over partial theta}) boldsymbol{hat phi} & end{matrix}$
$Delta f = nabla^2 f$	${partial^2 f over partial x^2} + {partial^2 f over partial y^2} + {partial^2 f over partial z^2}$	${1 over rho}{partial over partial rho}(rho {partial f over partial rho}) + {1 over rho^2}{partial^2 f over partial phi^2} + {partial^2 f over partial z^2}$	${1 over r^2}{partial over partial r}(r^2 {partial f over partial r}) + {1 over r^2sintheta}{partial over partial theta}(sintheta {partial f over partial theta}) + {1 over r^2sin^2theta}{partial^2 f over partial phi^2}$
$Delta mathbf{A} = nabla^2 mathbf{A}$	$Delta A_x mathbf{hat x} + Delta A_y mathbf{hat y} + Delta A_z mathbf{hat z}$	$begin{matrix} (Delta A_rho - {A_rho over rho^2} - {2 over rho^2}{partial A_phi over partial phi}) boldsymbol{hatrho} & + (Delta A_phi - {A_phi over rho^2} + {2 over rho^2}{partial A_rho over partial phi}) boldsymbol{hatphi} & + (Delta A_z ) boldsymbol{hat z} & end{matrix}$	$begin{matrix} (Delta A_r - {2 A_r over r^2} - {2 A_thetacostheta over r^2sintheta} - {2 over r^2}{partial A_theta over partial theta} - {2 over r^2sintheta}{partial A_phi over partial phi}) boldsymbol{hat r} & + (Delta A_theta - {A_theta over r^2sin^2theta} + {2 over r^2}{partial A_r over partial theta} - {2 costheta over r^2sin^2theta}{partial A_phi over partial phi}) boldsymbol{hattheta} & + (Delta A_phi - {A_phi over r^2sin^2theta} + {2 over r^2sin^2theta}{partial A_r over partial phi} + {2 costheta over r^2sin^2theta}{partial A_theta over partial phi}) boldsymbol{hatphi} & end{matrix}$
Differential displacement	$dmathbf{l} = dxmathbf{hat x} + dymathbf{hat y} + dzmathbf{hat z}$	$dmathbf{l} = drhoboldsymbol{hat rho} + rho dphiboldsymbol{hat phi} + dzboldsymbol{hat z}$	$dmathbf{l} = drmathbf{hat r} + rdthetaboldsymbol{hat theta} + rsintheta dphiboldsymbol{hat phi}$
Differential normal area	$begin{matrix}dmathbf{S} = &dydzmathbf{hat x} + &dxdzmathbf{hat y} + &dxdymathbf{hat z}end{matrix}$	$begin{matrix} dmathbf{S} = & rho dphi dzboldsymbol{hat rho} + & drho dzboldsymbol{hat phi} + & rho drho dphi mathbf{hat z} end{matrix}$	$begin{matrix} dmathbf{S} = & r^2 sintheta dtheta dphi mathbf{hat r} + & rsintheta drdphi boldsymbol{hat theta} + & rdrdthetaboldsymbol{hat phi} end{matrix}$
Differential volume	$dv = dxdydz ,$	$dv = rho drho dphi dz,$	$dv = r^2sintheta drdtheta dphi,$
Non-trivial calculation rules: $operatorname{div grad } f = nabla cdot (nabla f) = nabla^2 f = Delta f$ (Laplacian) $operatorname{curl grad } f = nabla times (nabla f) = 0$ $operatorname{div curl } mathbf{A} = nabla cdot (nabla times mathbf{A}) = 0$ $operatorname{curl curl } mathbf{A} = nabla times (nabla times mathbf{A}) = nabla (nabla cdot mathbf{A}) - nabla^2 mathbf{A}$ $Delta f g = f Delta g + 2 nabla f cdot nabla g + g Delta f$ Lagrange's formula for the cross product: $mathbf{A} times (mathbf{B} times mathbf{C}) = mathbf{B} (mathbf{A} cdot mathbf{C}) - mathbf{C} (mathbf{A} cdot mathbf{B})$

Note: This page uses standard physics notation; some (American mathematics) sources define θ as the angle with the $x y$ -plane instead of $φ$ .
Note: 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].

In vector calculus, del is a vector differential operator represented by the nabla symbol, âˆ‡. In the three-dimensional Cartesian coordinate system R3 with coordinates (x, y, z), del can be defined as or alternatively, where is the standard basis in R3. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ... In mathematics, the cross product is a binary operation on vectors in a three-dimensional Euclidean space. ... Atan2 is a two-parameter function for computing the arctangent in the C programming language. ...

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