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

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		x & = & hocosphi \ y & = & hosinphi \ z & = & z end{matrix} ight].	x & = & rsin hetacosphi \ y & = & rsin hetasinphi \ z & = & rcos heta end{matrix} ight].
		ho & = & sqrt{x^2 + y^2} \ phi & = & rctan{(y/x)}\ z & = & z end{matrix} ight].	r & = & sqrt{x^2 + y^2 + z^2} \ heta & = & rccos(z / r) = rctan{(sqrt{x^2+y^2}/z)}\ phi & = & rctan{(y/x)} end{matrix} ight].
A vector field $mathbf\{A\}$	$A\_xmathbf\{hat\; x\}\; +\; A\_ymathbf\{hat\; y\}\; +\; A\_zmathbf\{hat\; z\}$	$A\_$ hooldsymbol{hat ho} + A_phioldsymbol{hat phi} + A_zoldsymbol{hat z}
Gradient $$ abla f	$\{partial\; f\; over\; partial\; x\}mathbf\{hat\; x\}\; +\; \{partial\; f\; over\; partial\; y\}mathbf\{hat\; y\}$ + {partial f over partial z}mathbf{hat z}	$\{partial\; f\; over\; partial$ ho}oldsymbol{hat ho} + {1 over ho}{partial f over partial phi}oldsymbol{hat phi} + {partial f over partial z}oldsymbol{hat z}	+ {1 over r}{partial f over partial heta}oldsymbol{hat heta} + {1 over rsin heta}{partial f over partial phi}oldsymbol{hat phi}
Divergence $$ abla cdot mathbf{A}	$\{partial\; A\_x\; over\; partial\; x\}\; +\; \{partial\; A\_y\; over\; partial\; y\}\; +\; \{partial\; A\_z\; over\; partial\; z\}$	$\{1\; over$ ho}{partial ( ho A_ ho ) over partial ho} + {1 over ho}{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 rsin heta}{partial over partial heta} ( A_ hetasin heta ) + {1 over rsin heta}{partial A_phi over partial phi}
Curl $$ abla imes mathbf{A}	displaystyle({partial A_z over partial y} - {partial A_y over partial z}) mathbf{hat x} & + \ displaystyle({partial A_x over partial z} - {partial A_z over partial x}) mathbf{hat y} & + \ displaystyle({partial A_y over partial x} - {partial A_x over partial y}) mathbf{hat z} & end{matrix}	displaystyle({1 over ho}{partial A_z over partial phi} - {partial A_phi over partial z}) oldsymbol{hat ho} & + \ displaystyle({partial A_ ho over partial z} - {partial A_z over partial ho}) oldsymbol{hat phi} & + \ displaystyle{1 over ho}({partial ( ho A_phi ) over partial ho} - {partial A_ ho over partial phi}) oldsymbol{hat z} & end{matrix}	displaystyle{1 over rsin heta}({partial over partial heta} ( A_phisin heta ) - {partial A_ heta over partial phi}) oldsymbol{hat r} & + \ displaystyle{1 over r}({1 over sin heta}{partial A_r over partial phi} - {partial over partial r} ( r A_phi ) ) oldsymbol{hat heta} & + \ displaystyle{1 over r}({partial over partial r} ( r A_ heta ) - {partial A_r over partial heta}) oldsymbol{hat phi} & end{matrix}
Laplace operator $Delta\; f\; =$ abla^2 f	$\{partial^2\; f\; over\; partial\; x^2\}\; +\; \{partial^2\; f\; over\; partial\; y^2\}\; +\; \{partial^2\; f\; over\; partial\; z^2\}$	$\{1\; over$ ho}{partial over partial ho}( ho {partial f over partial ho}) + {1 over ho^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^2sin heta}{partial over partial heta}(sin heta {partial f over partial heta}) + {1 over r^2sin^2 heta}{partial^2 f over partial phi^2} or $\{1\; over\; r\}\{partial^2\; over\; partial\; r^2\}(r\; f)$ + {1 over r^2sin heta}{partial over partial heta}(sin heta {partial f over partial heta}) + {1 over r^2sin^2 heta}{partial^2 f over partial phi^2}
$Delta\; mathbf\{A\}\; =$ abla^2 mathbf{A}	$Delta\; A\_x\; mathbf\{hat\; x\}\; +\; Delta\; A\_y\; mathbf\{hat\; y\}\; +\; Delta\; A\_z\; mathbf\{hat\; z\}$	displaystyle(Delta A_ ho - {A_ ho over ho^2} - {2 over ho^2}{partial A_phi over partial phi}) oldsymbol{hat ho} & + \ displaystyle(Delta A_phi - {A_phi over ho^2} + {2 over ho^2}{partial A_ ho over partial phi}) oldsymbol{hatphi} & + \ displaystyle(Delta A_z ) oldsymbol{hat z} & end{matrix}	(Delta A_r - {2 A_r over r^2} - {2 over r^2sin heta}{partial (A_ heta sin heta) over partial heta} - {2 over r^2sin heta}{partial A_phi over partial phi}) oldsymbol{hat r} & + \ (Delta A_ heta - {A_ heta over r^2sin^2 heta} + {2 over r^2}{partial A_r over partial heta} - {2 cos heta over r^2sin^2 heta}{partial A_phi over partial phi}) oldsymbol{hat heta} & + \ (Delta A_phi - {A_phi over r^2sin^2 heta} + {2 over r^2sin^2 heta}{partial A_r over partial phi} + {2 cos heta over r^2sin^2 heta}{partial A_ heta over partial phi}) oldsymbol{hatphi} & end{matrix}
Differential displacement	$dmathbf\{l\}\; =\; dxmathbf\{hat\; x\}\; +\; dymathbf\{hat\; y\}\; +\; dzmathbf\{hat\; z\}$	$dmathbf\{l\}\; =\; d$ hooldsymbol{hat ho} + ho dphioldsymbol{hat phi} + dzoldsymbol{hat z}
Differential normal area	&dxdzmathbf{hat y} + \ &dxdymathbf{hat z}end{matrix}	dmathbf{S} = & ho dphi dzoldsymbol{hat ho} + \ & d ho dzoldsymbol{hat phi} + \ & ho d ho dphi mathbf{hat z} end{matrix}	dmathbf{S} = & r^2 sin heta d heta dphi mathbf{hat r} + \ & rsin heta drdphi oldsymbol{hat heta} + \ & rdrd hetaoldsymbol{hat phi} end{matrix}
Differential volume	$dv\; =\; dxdydz\; ,$	$dv\; =$ ho d ho dphi dz,	$dv\; =\; r^2sin\; heta\; drd\; heta\; dphi,$
Non-trivial calculation rules: $operatorname\{div\; grad\; \}\; f\; =$ abla cdot ( abla f) = abla^2 f = Delta f (Laplacian) $operatorname\{curl\; grad\; \}\; f\; =$ abla imes ( abla f) = 0 $operatorname\{div\; curl\; \}\; mathbf\{A\}\; =$ abla cdot ( abla imes mathbf{A}) = 0 $operatorname\{curl\; curl\; \}\; mathbf\{A\}\; =$ abla imes ( abla imes mathbf{A}) = abla ( abla cdot mathbf{A}) - abla^2 mathbf{A} $Delta\; f\; g\; =\; f\; Delta\; g\; +\; 2$ abla f cdot abla g + g Delta f Lagrange's formula for the cross product: $mathbf\{A\}\; imes\; (mathbf\{B\}\; imes\; mathbf\{C\})$ = mathbf{B} (mathbf{A} cdot mathbf{C}) - mathbf{C} (mathbf{A} cdot mathbf{B})

Contents

Remarks

See also

Remarks

★ This page uses standard physics notation; some (American mathematics) sources define $phi$ as the angle from the z-axis instead of $heta$.

★ 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].

See also

★ Orthogonal coordinates

★ Curvilinear coordinates

★ Vector fields in cylindrical and spherical coordinates