Del in cylindrical and spherical coordinates

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This is a list of some vector calculus formulae for working with common curvilinear coordinate systems.

Notes

• This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and ϕ):
  • The polar angle is denoted by θ: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.
  • The azimuthal angle is denoted by ϕ: it is the angle between the x-axis and the projection of the radial vector onto the xy-plane.
• The function atan2(y, x) can be used instead of the mathematical function arctan(y/x) owing to its domain and image. The classical arctan function has an image of (−π/2, +π/2), whereas atan2 is defined to have an image of (−π, π].

Formulae

Table with the del operator in cylindrical, spherical and parabolic cylindrical coordinates

Operation	Cartesian coordinates (x, y, z)	Cylindrical coordinates (ρ, ϕ, z)	Spherical coordinates (r, θ, ϕ)	Parabolic cylindrical coordinates (σ, τ, z)
Definition of coordinates	${\begin{aligned}\rho &={\sqrt {x^{2}+y^{2}}}\\\phi &=\arctan(y/x)\\z&=z\end{aligned}}$	${\begin{aligned}x&=\rho \cos \phi \\y&=\rho \sin \phi \\z&=z\end{aligned}}$	${\begin{aligned}x&=r\sin \theta \cos \phi \\y&=r\sin \theta \sin \phi \\z&=r\cos \theta \end{aligned}}$	${\begin{aligned}x&=\sigma \tau \\y&={\tfrac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=z\end{aligned}}$
	${\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arccos(z/r)\\\phi &=\arctan(y/x)\end{aligned}}$	${\begin{aligned}r&={\sqrt {\rho ^{2}+z^{2}}}\\\theta &=\arctan {(\rho /z)}\\\phi &=\phi \end{aligned}}$	${\begin{aligned}\rho &=r\sin \theta \\\phi &=\phi \\z&=r\cos \theta \end{aligned}}$	${\begin{aligned}\rho \cos \phi &=\sigma \tau \\\rho \sin \phi &={\tfrac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=z\end{aligned}}$
Definition of unit vectors	${\begin{aligned}{\boldsymbol {\hat {\rho }}}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\{\boldsymbol {\hat {\phi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\\mathbf {\hat {z}} &=\mathbf {\hat {z}} \end{aligned}}$	${\begin{aligned}{\hat {\mathbf {x} }}&=\cos \phi {\boldsymbol {\hat {\rho }}}-\sin \phi {\boldsymbol {\hat {\phi }}}\\{\hat {\mathbf {y} }}&=\sin \phi {\boldsymbol {\hat {\rho }}}+\cos \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=\mathbf {\hat {z}} \end{aligned}}$	${\begin{aligned}{\hat {\mathbf {x} }}&=\sin \theta \cos \phi {\boldsymbol {\hat {r}}}+\cos \theta \cos \phi {\boldsymbol {\hat {\theta }}}-\sin \phi {\boldsymbol {\hat {\phi }}}\\{\hat {\mathbf {y} }}&=\sin \theta \sin \phi {\boldsymbol {\hat {r}}}+\cos \theta \sin \phi {\boldsymbol {\hat {\theta }}}+\cos \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=\cos \theta {\boldsymbol {\hat {r}}}-\sin \theta {\boldsymbol {\hat {\theta }}}\end{aligned}}$	${\begin{aligned}{\boldsymbol {\hat {\sigma }}}&={\frac {\tau {\hat {\mathbf {x} }}-\sigma {\hat {\mathbf {y} }}}{\sqrt {\tau ^{2}+\sigma ^{2}}}}\\{\boldsymbol {\hat {\tau }}}&={\frac {\sigma {\hat {\mathbf {x} }}+\tau {\hat {\mathbf {y} }}}{\sqrt {\tau ^{2}+\sigma ^{2}}}}\\\mathbf {\hat {z}} &=\mathbf {\hat {z}} \end{aligned}}$
	${\begin{aligned}\mathbf {\hat {r}} &={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}+z\mathbf {\hat {z}} }{\sqrt {x^{2}+y^{2}+z^{2}}}}\\{\boldsymbol {\hat {\theta }}}&={\frac {xz{\hat {\mathbf {x} }}+yz{\hat {\mathbf {y} }}-\left(x^{2}+y^{2}\right)\mathbf {\hat {z}} }{{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}}\\{\boldsymbol {\hat {\phi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\end{aligned}}$	${\begin{aligned}\mathbf {\hat {r}} &={\frac {\rho {\boldsymbol {\hat {\rho }}}+z\mathbf {\hat {z}} }{\sqrt {\rho ^{2}+z^{2}}}}\\{\boldsymbol {\hat {\theta }}}&={\frac {z{\boldsymbol {\hat {\rho }}}-\rho \mathbf {\hat {z}} }{\sqrt {\rho ^{2}+z^{2}}}}\\{\boldsymbol {\hat {\phi }}}&={\boldsymbol {\hat {\phi }}}\end{aligned}}$	${\begin{aligned}{\boldsymbol {\hat {\rho }}}&=\sin \theta \mathbf {\hat {r}} +\cos \theta {\boldsymbol {\hat {\theta }}}\\{\boldsymbol {\hat {\phi }}}&={\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=\cos \theta \mathbf {\hat {r}} -\sin \theta {\boldsymbol {\hat {\theta }}}\end{aligned}}$	${\begin{matrix}\end{matrix}}$
A vector field $\mathbf {A}$	$A_{x}{\hat {\mathbf {x} }}+A_{y}{\hat {\mathbf {y} }}+A_{z}\mathbf {\hat {z}}$	$A_{\rho }{\boldsymbol {\hat {\rho }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}+A_{z}\mathbf {\hat {z}}$	$A_{r}{\boldsymbol {\hat {r}}}+A_{\theta }{\boldsymbol {\hat {\theta }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}$	$A_{\sigma }{\boldsymbol {\hat {\sigma }}}+A_{\tau }{\boldsymbol {\hat {\tau }}}+A_{\phi }\mathbf {\hat {z}}$
$f$ scalar field Gradient $\nabla f$	${\partial f \over \partial x}{\hat {\mathbf {x} }}+{\partial f \over \partial y}{\hat {\mathbf {y} }}+{\partial f \over \partial z}\mathbf {\hat {z}}$	${\partial f \over \partial \rho }{\boldsymbol {\hat {\rho }}}+{1 \over \rho }{\partial f \over \partial \phi }{\boldsymbol {\hat {\phi }}}+{\partial f \over \partial z}\mathbf {\hat {z}}$	${\partial f \over \partial r}{\boldsymbol {\hat {r}}}+{1 \over r}{\partial f \over \partial \theta }{\boldsymbol {\hat {\theta }}}+{1 \over r\sin \theta }{\partial f \over \partial \phi }{\boldsymbol {\hat {\phi }}}$	${\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial f \over \partial \sigma }{\boldsymbol {\hat {\sigma }}}+{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial f \over \partial \tau }{\boldsymbol {\hat {\tau }}}+{\partial f \over \partial z}\mathbf {\hat {z}}$
Divergence $\nabla \cdot \mathbf {A}$	${\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}$	${1 \over \rho }{\partial \left(\rho A_{\rho }\right) \over \partial \rho }+{1 \over \rho }{\partial A_{\phi } \over \partial \phi }+{\partial A_{z} \over \partial z}$	${1 \over r^{2}}{\partial \left(r^{2}A_{r}\right) \over \partial r}+{1 \over r\sin \theta }{\partial \over \partial \theta }\left(A_{\theta }\sin \theta \right)+{1 \over r\sin \theta }{\partial A_{\phi } \over \partial \phi }$	${\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\partial ({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\sigma }) \over \partial \sigma }+{\partial ({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\tau }) \over \partial \tau }\right)+{\partial A_{z} \over \partial z}$
Curl $\nabla \times \mathbf {A}$	${\begin{aligned}\left({\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\right)&{\hat {\mathbf {x} }}+\\+\left({\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}}\right)&{\hat {\mathbf {y} }}+\\+\left({\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}\right)&\mathbf {\hat {z}} \end{aligned}}$	${\begin{aligned}\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}-{\frac {\partial A_{\phi }}{\partial z}}\right)&{\boldsymbol {\hat {\rho }}}\\+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right)&{\boldsymbol {\hat {\phi }}}\\+{\frac {1}{\rho }}\left({\frac {\partial \left(\rho A_{\phi }\right)}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \phi }}\right)&\mathbf {\hat {z}} \end{aligned}}$	${\begin{aligned}{\frac {1}{r\sin \theta }}\left({\frac {\partial }{\partial \theta }}\left(A_{\phi }\sin \theta \right)-{\frac {\partial A_{\theta }}{\partial \phi }}\right)&{\boldsymbol {\hat {r}}}\\+{\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {\partial }{\partial r}}\left(rA_{\phi }\right)\right)&{\boldsymbol {\hat {\theta }}}\\+{\frac {1}{r}}\left({\frac {\partial }{\partial r}}\left(rA_{\theta }\right)-{\frac {\partial A_{r}}{\partial \theta }}\right)&{\boldsymbol {\hat {\phi }}}\end{aligned}}$	${\begin{aligned}\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial A_{z}}{\partial \tau }}-{\frac {\partial A_{\tau }}{\partial z}}\right)&{\boldsymbol {\hat {\sigma }}}\\-\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial A_{z}}{\partial \sigma }}-{\frac {\partial A_{\sigma }}{\partial z}}\right)&{\boldsymbol {\hat {\tau }}}\\+{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}\left({\frac {\partial \left({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\sigma }\right)}{\partial \tau }}-{\frac {\partial \left({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\tau }\right)}{\partial \sigma }}\right)&\mathbf {\hat {z}} \end{aligned}}$
Laplace operator $\Delta f\equiv \nabla ^{2}f$	${\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}$	${1 \over \rho }{\partial \over \partial \rho }\left(\rho {\partial f \over \partial \rho }\right)+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \phi ^{2}}+{\partial ^{2}f \over \partial z^{2}}$	${1 \over r^{2}}{\partial \over \partial r}\!\left(r^{2}{\partial f \over \partial r}\right)\!+\!{1 \over r^{2}\!\sin \theta }{\partial \over \partial \theta }\!\left(\sin \theta {\partial f \over \partial \theta }\right)\!+\!{1 \over r^{2}\!\sin ^{2}\theta }{\partial ^{2}f \over \partial \phi ^{2}}$	${\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\frac {\partial ^{2}f}{\partial \sigma ^{2}}}+{\frac {\partial ^{2}f}{\partial \tau ^{2}}}\right)+{\frac {\partial ^{2}f}{\partial z^{2}}}$
Vector Laplacian $\Delta \mathbf {A} \equiv \nabla ^{2}\mathbf {A}$	$\Delta A_{x}{\hat {\mathbf {x} }}+\Delta A_{y}{\hat {\mathbf {y} }}+\Delta A_{z}\mathbf {\hat {z}}$	Template:Mdash View by clicking show Template:Mdash ${\begin{aligned}{\mathopen {}}\left(\Delta A_{\rho }-{\frac {A_{\rho }}{\rho ^{2}}}-{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\phi }}{\partial \phi }}\right){\mathclose {}}&{\boldsymbol {\hat {\rho }}}\\+{\mathopen {}}\left(\Delta A_{\phi }-{\frac {A_{\phi }}{\rho ^{2}}}+{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\rho }}{\partial \phi }}\right){\mathclose {}}&{\boldsymbol {\hat {\phi }}}\\+\Delta A_{z}&\mathbf {\hat {z}} \end{aligned}}$	Template:Mdash View by clicking show Template:Mdash ${\begin{aligned}\left(\Delta A_{r}-{\frac {2A_{r}}{r^{2}}}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial \left(A_{\theta }\sin \theta \right)}{\partial \theta }}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{\phi }}{\partial \phi }}\right)&{\boldsymbol {\hat {r}}}\\+\left(\Delta A_{\theta }-{\frac {A_{\theta }}{r^{2}\sin ^{2}\theta }}+{\frac {2}{r^{2}}}{\frac {\partial A_{r}}{\partial \theta }}-{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial A_{\phi }}{\partial \phi }}\right)&{\boldsymbol {\hat {\theta }}}\\+\left(\Delta A_{\phi }-{\frac {A_{\phi }}{r^{2}\sin ^{2}\theta }}+{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}+{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial A_{\theta }}{\partial \phi }}\right)&{\boldsymbol {\hat {\phi }}}\end{aligned}}$
Material derivative $(\mathbf {A} \cdot \nabla )\mathbf {B}$	$\mathbf {A} \cdot \nabla B_{x}{\hat {\mathbf {x} }}+\mathbf {A} \cdot \nabla B_{y}{\hat {\mathbf {y} }}+\mathbf {A} \cdot \nabla B_{z}{\hat {\mathbf {z} }}$	Template:Mdash View by clicking show Template:Mdash ${\begin{aligned}\left(A_{\rho }{\frac {\partial B_{\rho }}{\partial \rho }}+{\frac {A_{\phi }}{\rho }}{\frac {\partial B_{\rho }}{\partial \phi }}+A_{z}{\frac {\partial B_{\rho }}{\partial z}}-{\frac {A_{\phi }B_{\phi }}{\rho }}\right)&{\boldsymbol {\hat {\rho }}}\\+\left(A_{\rho }{\frac {\partial B_{\phi }}{\partial \rho }}+{\frac {A_{\phi }}{\rho }}{\frac {\partial B_{\phi }}{\partial \phi }}+A_{z}{\frac {\partial B_{\phi }}{\partial z}}+{\frac {A_{\phi }B_{\rho }}{\rho }}\right)&{\boldsymbol {\hat {\phi }}}\\+\left(A_{\rho }{\frac {\partial B_{z}}{\partial \rho }}+{\frac {A_{\phi }}{\rho }}{\frac {\partial B_{z}}{\partial \phi }}+A_{z}{\frac {\partial B_{z}}{\partial z}}\right)&\mathbf {\hat {z}} \end{aligned}}$	Template:Mdash View by clicking show Template:Mdash ${\begin{aligned}\left(A_{r}{\frac {\partial B_{r}}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{r}}{\partial \theta }}+{\frac {A_{\phi }}{r\sin \theta }}{\frac {\partial B_{r}}{\partial \phi }}-{\frac {A_{\theta }B_{\theta }+A_{\phi }B_{\phi }}{r}}\right)&{\boldsymbol {\hat {r}}}\\+\left(A_{r}{\frac {\partial B_{\theta }}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{\theta }}{\partial \theta }}+{\frac {A_{\phi }}{r\sin \theta }}{\frac {\partial B_{\theta }}{\partial \phi }}+{\frac {A_{\theta }B_{r}}{r}}-{\frac {A_{\phi }B_{\phi }\cot \theta }{r}}\right)&{\boldsymbol {\hat {\theta }}}\\+\left(A_{r}{\frac {\partial B_{\phi }}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{\phi }}{\partial \theta }}+{\frac {A_{\phi }}{r\sin \theta }}{\frac {\partial B_{\phi }}{\partial \phi }}+{\frac {A_{\phi }B_{r}}{r}}+{\frac {A_{\phi }B_{\theta }\cot \theta }{r}}\right)&{\boldsymbol {\hat {\phi }}}\end{aligned}}$
Differential displacement	$d\mathbf {l} =dx\,{\hat {\mathbf {x} }}+dy\,{\hat {\mathbf {y} }}+dz\,\mathbf {\hat {z}}$	$d\mathbf {l} =d\rho \,{\boldsymbol {\hat {\rho }}}+\rho \,d\phi \,{\boldsymbol {\hat {\phi }}}+dz\,\mathbf {\hat {z}}$	$d\mathbf {l} =dr\,\mathbf {\hat {r}} +r\,d\theta \,{\boldsymbol {\hat {\theta }}}+r\,\sin \theta \,d\phi \,{\boldsymbol {\hat {\phi }}}$	$d\mathbf {l} ={\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\sigma \,{\boldsymbol {\hat {\sigma }}}+{\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\tau \,{\boldsymbol {\hat {\tau }}}+dz\,\mathbf {\hat {z}}$
Differential normal area $d\mathbf {S}$	${\begin{aligned}dy\,dz&{\hat {\mathbf {x} }}\\+dx\,dz&{\hat {\mathbf {y} }}\\+dx\,dy&\mathbf {\hat {z}} \end{aligned}}$	${\begin{aligned}\rho \,d\phi \,dz&{\boldsymbol {\hat {\rho }}}\\+d\rho \,dz&{\boldsymbol {\hat {\phi }}}\\+\rho \,d\rho \,d\phi &\mathbf {\hat {z}} \end{aligned}}$	${\begin{aligned}r^{2}\sin \theta \,d\theta \,d\phi &\mathbf {\hat {r}} \\+r\sin \theta \,dr\,d\phi &{\boldsymbol {\hat {\theta }}}\\+r\,dr\,d\theta &{\boldsymbol {\hat {\phi }}}\end{aligned}}$	${\begin{aligned}{\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\tau \,dz&{\boldsymbol {\hat {\sigma }}}\\+{\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\sigma \,dz&{\boldsymbol {\hat {\tau }}}\\+\left(\sigma ^{2}+\tau ^{2}\right)\,d\sigma \,d\tau &\mathbf {\hat {z}} \end{aligned}}$
Differential volume $dV$	$dx\,dy\,dz$	$\rho \,d\rho \,d\phi \,dz$	$r^{2}\sin \theta \,dr\,d\theta \,d\phi$	$\left(\sigma ^{2}+\tau ^{2}\right)d\sigma \,d\tau \,dz$
Non-trivial calculation rules: $\operatorname {div} \,\operatorname {grad} f\equiv \nabla \cdot \nabla f=\nabla ^{2}f\equiv \Delta f$ $\operatorname {curl} \,\operatorname {grad} f\equiv \nabla \times \nabla f=\mathbf {0}$ $\operatorname {div} \,\operatorname {curl} \mathbf {A} \equiv \nabla \cdot (\nabla \times \mathbf {A} )=0$ $\operatorname {curl} \,\operatorname {curl} \mathbf {A} \equiv \nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A}$ (Lagrange's formula for del) $\Delta (fg)=f\Delta g+2\nabla f\cdot \nabla g+g\Delta f$

See also

• Del
• Orthogonal coordinates
• Curvilinear coordinates
• Vector fields in cylindrical and spherical coordinates

External links

• Maxima Computer Algebra system scripts to generate some of these operators in cylindrical and spherical coordinates.