Del in cylindrical and spherical coordinates

Short description: Mathematical gradient operator in certain coordinate systems

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 $\theta \in [0,\pi ]$: 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 $\phi \in [0,2\pi ]$: 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].

Coordinate conversions

Conversion between Cartesian, cylindrical, and spherical coordinates

		From
		Cartesian	Cylindrical	Spherical
To	Cartesian	$\begin{array}{rl}x& =x\\ y& =y\\ z& =z\\ \end{array}$	$\begin{array}{rl}x& =\rho \cos \phi \\ y& =\rho \sin \phi \\ z& =z\end{array}$	$\begin{array}{rl}x& =r\sin \theta \cos \phi \\ y& =r\sin \theta \sin \phi \\ z& =r\cos \theta \\ \end{array}$
	Cylindrical	$\begin{array}{rl}\rho & =\sqrt{x^2+y^2}\\ \phi & =\arctan \left(\frac{y}{x}\right)\\ z& =z\end{array}$	$\begin{array}{rl}\rho & =\rho \\ \phi & =\phi \\ z& =z\\ \end{array}$	$\begin{array}{rl}\rho & =r\sin \theta \\ \phi & =\phi \\ z& =r\cos \theta \end{array}$
	Spherical	$\begin{array}{rl}r& =\sqrt{x^2+y^2+z^2}\\ \theta & =\arctan \left(\frac{\sqrt{x^2+y^2}}{z}\right)\\ \phi & =\arctan \left(\frac{y}{x}\right)\end{array}$	$\begin{array}{rl}r& =\sqrt{{\rho }^2+z^2}\\ \theta & =\arctan \left(\frac{\rho }{z}\right)\\ \phi & =\phi \end{array}$	$\begin{array}{rl}r& =r\\ \theta & =\theta \\ \phi & =\phi \end{array}$

Note that the operation $\arctan \left(\frac{A}{B}\right)$ must be interpreted as the two-argument inverse tangent, atan2.

Unit vector conversions

Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of destination coordinates

	Cartesian	Cylindrical	Spherical
Cartesian	$\begin{array}{rl}\widehat{\mathbf{𝐱}}& =\widehat{\mathbf{𝐱}}\\ \widehat{\mathbf{𝐲}}& =\widehat{\mathbf{𝐲}}\\ \widehat{\mathbf{𝐳}}& =\widehat{\mathbf{𝐳}}\\ \end{array}$	$\begin{array}{rl}\widehat{\mathbf{𝐱}}& =\cos \phi \hat{\mathit{\rho }}-\sin \phi \hat{\mathit{\phi }}\\ \widehat{\mathbf{𝐲}}& =\sin \phi \hat{\mathit{\rho }}+\cos \phi \hat{\mathit{\phi }}\\ \widehat{\mathbf{𝐳}}& =\widehat{\mathbf{𝐳}}\end{array}$	$\begin{array}{rl}\widehat{\mathbf{𝐱}}& =\sin \theta \cos \phi \widehat{\mathbf{𝐫}}+\cos \theta \cos \phi \hat{\mathit{\theta }}-\sin \phi \hat{\mathit{\phi }}\\ \widehat{\mathbf{𝐲}}& =\sin \theta \sin \phi \widehat{\mathbf{𝐫}}+\cos \theta \sin \phi \hat{\mathit{\theta }}+\cos \phi \hat{\mathit{\phi }}\\ \widehat{\mathbf{𝐳}}& =\cos \theta \widehat{\mathbf{𝐫}}-\sin \theta \hat{\mathit{\theta }}\end{array}$
Cylindrical	$\begin{array}{rl}\hat{\mathit{\rho }}& =\frac{x\widehat{\mathbf{𝐱}}+y\widehat{\mathbf{𝐲}}}{\sqrt{x^2+y^2}}\\ \hat{\mathit{\phi }}& =\frac{-y\widehat{\mathbf{𝐱}}+x\widehat{\mathbf{𝐲}}}{\sqrt{x^2+y^2}}\\ \widehat{\mathbf{𝐳}}& =\widehat{\mathbf{𝐳}}\end{array}$	$\begin{array}{rl}\hat{\mathit{\rho }}& =\hat{\mathit{\rho }}\\ \hat{\mathit{\phi }}& =\hat{\mathit{\phi }}\\ \widehat{\mathbf{𝐳}}& =\widehat{\mathbf{𝐳}}\\ \end{array}$	$\begin{array}{rl}\hat{\mathit{\rho }}& =\sin \theta \widehat{\mathbf{𝐫}}+\cos \theta \hat{\mathit{\theta }}\\ \hat{\mathit{\phi }}& =\hat{\mathit{\phi }}\\ \widehat{\mathbf{𝐳}}& =\cos \theta \widehat{\mathbf{𝐫}}-\sin \theta \hat{\mathit{\theta }}\end{array}$
Spherical	$\begin{array}{rl}\widehat{\mathbf{𝐫}}& =\frac{x\widehat{\mathbf{𝐱}}+y\widehat{\mathbf{𝐲}}+z\widehat{\mathbf{𝐳}}}{\sqrt{x^2+y^2+z^2}}\\ \hat{\mathit{\theta }}& =\frac{(x\widehat{\mathbf{𝐱}}+y\widehat{\mathbf{𝐲}})z-(x^2+y^2)\widehat{\mathbf{𝐳}}}{\sqrt{x^2+y^2+z^2}\sqrt{x^2+y^2}}\\ \hat{\mathit{\phi }}& =\frac{-y\widehat{\mathbf{𝐱}}+x\widehat{\mathbf{𝐲}}}{\sqrt{x^2+y^2}}\end{array}$	$\begin{array}{rl}\widehat{\mathbf{𝐫}}& =\frac{\rho \hat{\mathit{\rho }}+z\widehat{\mathbf{𝐳}}}{\sqrt{{\rho }^2+z^2}}\\ \hat{\mathit{\theta }}& =\frac{z\hat{\mathit{\rho }}-\rho \widehat{\mathbf{𝐳}}}{\sqrt{{\rho }^2+z^2}}\\ \hat{\mathit{\phi }}& =\hat{\mathit{\phi }}\end{array}$	$\begin{array}{rl}\widehat{\mathbf{𝐫}}& =\widehat{\mathbf{𝐫}}\\ \hat{\mathit{\theta }}& =\hat{\mathit{\theta }}\\ \hat{\mathit{\phi }}& =\hat{\mathit{\phi }}\\ \end{array}$

Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of source coordinates

	Cartesian	Cylindrical	Spherical
Cartesian	$\begin{array}{rl}\widehat{\mathbf{𝐱}}& =\widehat{\mathbf{𝐱}}\\ \widehat{\mathbf{𝐲}}& =\widehat{\mathbf{𝐲}}\\ \widehat{\mathbf{𝐳}}& =\widehat{\mathbf{𝐳}}\\ \end{array}$	$\begin{array}{rl}\widehat{\mathbf{𝐱}}& =\frac{x\hat{\mathit{\rho }}-y\hat{\mathit{\phi }}}{\sqrt{x^2+y^2}}\\ \widehat{\mathbf{𝐲}}& =\frac{y\hat{\mathit{\rho }}+x\hat{\mathit{\phi }}}{\sqrt{x^2+y^2}}\\ \widehat{\mathbf{𝐳}}& =\widehat{\mathbf{𝐳}}\end{array}$	$\begin{array}{rl}\widehat{\mathbf{𝐱}}& =\frac{x(\sqrt{x^2+y^2}\widehat{\mathbf{𝐫}}+z\hat{\mathit{\theta }})-y\sqrt{x^2+y^2+z^2}\hat{\mathit{\phi }}}{\sqrt{x^2+y^2}\sqrt{x^2+y^2+z^2}}\\ \widehat{\mathbf{𝐲}}& =\frac{y(\sqrt{x^2+y^2}\widehat{\mathbf{𝐫}}+z\hat{\mathit{\theta }})+x\sqrt{x^2+y^2+z^2}\hat{\mathit{\phi }}}{\sqrt{x^2+y^2}\sqrt{x^2+y^2+z^2}}\\ \widehat{\mathbf{𝐳}}& =\frac{z\widehat{\mathbf{𝐫}}-\sqrt{x^2+y^2}\hat{\mathit{\theta }}}{\sqrt{x^2+y^2+z^2}}\end{array}$
Cylindrical	$\begin{array}{rl}\hat{\mathit{\rho }}& =\cos \phi \widehat{\mathbf{𝐱}}+\sin \phi \widehat{\mathbf{𝐲}}\\ \hat{\mathit{\phi }}& =-\sin \phi \widehat{\mathbf{𝐱}}+\cos \phi \widehat{\mathbf{𝐲}}\\ \widehat{\mathbf{𝐳}}& =\widehat{\mathbf{𝐳}}\end{array}$	$\begin{array}{rl}\hat{\mathit{\rho }}& =\hat{\mathit{\rho }}\\ \hat{\mathit{\phi }}& =\hat{\mathit{\phi }}\\ \widehat{\mathbf{𝐳}}& =\widehat{\mathbf{𝐳}}\\ \end{array}$	$\begin{array}{rl}\hat{\mathit{\rho }}& =\frac{\rho \widehat{\mathbf{𝐫}}+z\hat{\mathit{\theta }}}{\sqrt{{\rho }^2+z^2}}\\ \hat{\mathit{\phi }}& =\hat{\mathit{\phi }}\\ \widehat{\mathbf{𝐳}}& =\frac{z\widehat{\mathbf{𝐫}}-\rho \hat{\mathit{\theta }}}{\sqrt{{\rho }^2+z^2}}\end{array}$
Spherical	$\begin{array}{rl}\widehat{\mathbf{𝐫}}& =\sin \theta (\cos \phi \widehat{\mathbf{𝐱}}+\sin \phi \widehat{\mathbf{𝐲}})+\cos \theta \widehat{\mathbf{𝐳}}\\ \hat{\mathit{\theta }}& =\cos \theta (\cos \phi \widehat{\mathbf{𝐱}}+\sin \phi \widehat{\mathbf{𝐲}})-\sin \theta \widehat{\mathbf{𝐳}}\\ \hat{\mathit{\phi }}& =-\sin \phi \widehat{\mathbf{𝐱}}+\cos \phi \widehat{\mathbf{𝐲}}\end{array}$	$\begin{array}{rl}\widehat{\mathbf{𝐫}}& =\sin \theta \hat{\mathit{\rho }}+\cos \theta \widehat{\mathbf{𝐳}}\\ \hat{\mathit{\theta }}& =\cos \theta \hat{\mathit{\rho }}-\sin \theta \widehat{\mathbf{𝐳}}\\ \hat{\mathit{\phi }}& =\hat{\mathit{\phi }}\end{array}$	$\begin{array}{rl}\widehat{\mathbf{𝐫}}& =\widehat{\mathbf{𝐫}}\\ \hat{\mathit{\theta }}& =\hat{\mathit{\theta }}\\ \hat{\mathit{\phi }}& =\hat{\mathit{\phi }}\\ \end{array}$

Del formula

Table with the del operator in cartesian, 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
Vector field A	$A_x\widehat{\mathbf{𝐱}}+A_y\widehat{\mathbf{𝐲}}+A_z\widehat{\mathbf{𝐳}}$	$A_{\rho }\hat{\mathit{\rho }}+A_{\phi }\hat{\mathit{\phi }}+A_z\widehat{\mathbf{𝐳}}$	$A_r\widehat{\mathbf{𝐫}}+A_{\theta }\hat{\mathit{\theta }}+A_{\phi }\hat{\mathit{\phi }}$
Gradient ∇f	$\frac{\partial f}{\partial x}\widehat{\mathbf{𝐱}}+\frac{\partial f}{\partial y}\widehat{\mathbf{𝐲}}+\frac{\partial f}{\partial z}\widehat{\mathbf{𝐳}}$	$\frac{\partial f}{\partial \rho }\hat{\mathit{\rho }}+\frac{1}{\rho }\frac{\partial f}{\partial \phi }\hat{\mathit{\phi }}+\frac{\partial f}{\partial z}\widehat{\mathbf{𝐳}}$	$\frac{\partial f}{\partial r}\widehat{\mathbf{𝐫}}+\frac{1}{r}\frac{\partial f}{\partial \theta }\hat{\mathit{\theta }}+\frac{1}{r\sin \theta }\frac{\partial f}{\partial \phi }\hat{\mathit{\phi }}$
Divergence ∇ ⋅ A	$\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}$	$\frac{1}{\rho }\frac{\partial \left(\rho A_{\rho }\right)}{\partial \rho }+\frac{1}{\rho }\frac{\partial A_{\phi }}{\partial \phi }+\frac{\partial A_z}{\partial z}$	$\frac{1}{r^2}\frac{\partial \left(r^2{A}_r\right)}{\partial r}+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }(A_{\theta }\sin \theta )+\frac{1}{r\sin \theta }\frac{\partial A_{\phi }}{\partial \phi }$
Curl ∇ × A	$\begin{array}{rl}(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})& \widehat{\mathbf{𝐱}}\\ +(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})& \widehat{\mathbf{𝐲}}\\ +(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})& \widehat{\mathbf{𝐳}}\end{array}$	$\begin{array}{rl}(\frac{1}{\rho }\frac{\partial A_z}{\partial \phi }-\frac{\partial A_{\phi }}{\partial z})& \hat{\mathit{\rho }}\\ +(\frac{\partial A_{\rho }}{\partial z}-\frac{\partial A_z}{\partial \rho })& \hat{\mathit{\phi }}\\ +\frac{1}{\rho }(\frac{\partial \left(\rho A_{\phi }\right)}{\partial \rho }-\frac{\partial A_{\rho }}{\partial \phi })& \widehat{\mathbf{𝐳}}\end{array}$	$\begin{array}{rl}\frac{1}{r\sin \theta }(\frac{\partial }{\partial \theta }(A_{\phi }\sin \theta )-\frac{\partial A_{\theta }}{\partial \phi })& \widehat{\mathbf{𝐫}}\\ +\frac{1}{r}(\frac{1}{\sin \theta }\frac{\partial A_r}{\partial \phi }-\frac{\partial }{\partial r}\left(r{A}_{\phi }\right))& \hat{\mathit{\theta }}\\ +\frac{1}{r}(\frac{\partial }{\partial r}\left(r{A}_{\theta }\right)-\frac{\partial A_r}{\partial \theta })& \hat{\mathit{\phi }}\end{array}$
Laplace operator ∇2f ≡ ∆f	$\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{\rho }\frac{\partial }{\partial \rho }\left(\rho \frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2}\frac{{\partial }^{2}f}{\partial {\phi }^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial f}{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial f}{\partial \theta })+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^{2}f}{\partial {\phi }^2}$
Vector gradient ∇A	$\begin{array}{rl}& \frac{\partial A_x}{\partial x}\widehat{\mathbf{𝐱}}\otimes \widehat{\mathbf{𝐱}}+\frac{\partial A_x}{\partial y}\widehat{\mathbf{𝐱}}\otimes \widehat{\mathbf{𝐲}}+\frac{\partial A_x}{\partial z}\widehat{\mathbf{𝐱}}\otimes \widehat{\mathbf{𝐳}}\\ +& \frac{\partial A_y}{\partial x}\widehat{\mathbf{𝐲}}\otimes \widehat{\mathbf{𝐱}}+\frac{\partial A_y}{\partial y}\widehat{\mathbf{𝐲}}\otimes \widehat{\mathbf{𝐲}}+\frac{\partial A_y}{\partial z}\widehat{\mathbf{𝐲}}\otimes \widehat{\mathbf{𝐳}}\\ +& \frac{\partial A_z}{\partial x}\widehat{\mathbf{𝐳}}\otimes \widehat{\mathbf{𝐱}}+\frac{\partial A_z}{\partial y}\widehat{\mathbf{𝐳}}\otimes \widehat{\mathbf{𝐲}}+\frac{\partial A_z}{\partial z}\widehat{\mathbf{𝐳}}\otimes \widehat{\mathbf{𝐳}}\end{array}$	$\begin{array}{rl}& \frac{\partial A_{\rho }}{\partial \rho }\hat{\mathit{\rho }}\otimes \hat{\mathit{\rho }}+(\frac{1}{\rho }\frac{\partial A_{\rho }}{\partial \phi }-\frac{A_{\phi }}{\rho })\hat{\mathit{\rho }}\otimes \hat{\mathit{\phi }}+\frac{\partial A_{\rho }}{\partial z}\hat{\mathit{\rho }}\otimes \widehat{\mathbf{𝐳}}\\ +& \frac{\partial A_{\phi }}{\partial \rho }\hat{\mathit{\phi }}\otimes \hat{\mathit{\rho }}+(\frac{1}{\rho }\frac{\partial A_{\phi }}{\partial \phi }+\frac{A_{\rho }}{\rho })\hat{\mathit{\phi }}\otimes \hat{\mathit{\phi }}+\frac{\partial A_{\phi }}{\partial z}\hat{\mathit{\phi }}\otimes \widehat{\mathbf{𝐳}}\\ +& \frac{\partial A_z}{\partial \rho }\widehat{\mathbf{𝐳}}\otimes \hat{\mathit{\rho }}+\frac{1}{\rho }\frac{\partial A_z}{\partial \phi }\widehat{\mathbf{𝐳}}\otimes \hat{\mathit{\phi }}+\frac{\partial A_z}{\partial z}\widehat{\mathbf{𝐳}}\otimes \widehat{\mathbf{𝐳}}\end{array}$	$\begin{array}{rl}& \frac{\partial A_r}{\partial r}\widehat{\mathbf{𝐫}}\otimes \widehat{\mathbf{𝐫}}+(\frac{1}{r}\frac{\partial A_r}{\partial \theta }-\frac{A_{\theta }}{r})\widehat{\mathbf{𝐫}}\otimes \hat{\mathit{\theta }}+(\frac{1}{r\sin \theta }\frac{\partial A_r}{\partial \phi }-\frac{A_{\phi }}{r})\widehat{\mathbf{𝐫}}\otimes \hat{\mathit{\phi }}\\ +& \frac{\partial A_{\theta }}{\partial r}\hat{\mathit{\theta }}\otimes \widehat{\mathbf{𝐫}}+(\frac{1}{r}\frac{\partial A_{\theta }}{\partial \theta }+\frac{A_r}{r})\hat{\mathit{\theta }}\otimes \hat{\mathit{\theta }}+(\frac{1}{r\sin \theta }\frac{\partial A_{\theta }}{\partial \phi }-\cot \theta \frac{A_{\phi }}{r})\hat{\mathit{\theta }}\otimes \hat{\mathit{\phi }}\\ +& \frac{\partial A_{\phi }}{\partial r}\hat{\mathit{\phi }}\otimes \widehat{\mathbf{𝐫}}+\frac{1}{r}\frac{\partial A_{\phi }}{\partial \theta }\hat{\mathit{\phi }}\otimes \hat{\mathit{\theta }}+(\frac{1}{r\sin \theta }\frac{\partial A_{\phi }}{\partial \phi }+\cot \theta \frac{A_{\theta }}{r}+\frac{A_r}{r})\hat{\mathit{\phi }}\otimes \hat{\mathit{\phi }}\end{array}$
Vector Laplacian ∇2A ≡ ∆A	${\mathrm{\nabla }}^2{A}_x\widehat{\mathbf{𝐱}}+{\mathrm{\nabla }}^2{A}_y\widehat{\mathbf{𝐲}}+{\mathrm{\nabla }}^2{A}_z\widehat{\mathbf{𝐳}}$	$\begin{array}{rl}({\mathrm{\nabla }}^2{A}_{\rho }-\frac{A_{\rho }}{{\rho }^2}-\frac{2}{{\rho }^2}\frac{\partial A_{\phi }}{\partial \phi })& \hat{\mathit{\rho }}\\ +({\mathrm{\nabla }}^2{A}_{\phi }-\frac{A_{\phi }}{{\rho }^2}+\frac{2}{{\rho }^2}\frac{\partial A_{\rho }}{\partial \phi })& \hat{\mathit{\phi }}\\ +{\mathrm{\nabla }}^2{A}_z& \widehat{\mathbf{𝐳}}\end{array}$	$\begin{array}{rl}({\mathrm{\nabla }}^2{A}_r-\frac{2A_r}{r^2}-\frac{2}{r^2\sin \theta }\frac{\partial (A_{\theta }\sin \theta )}{\partial \theta }-\frac{2}{r^2\sin \theta }\frac{\partial A_{\phi }}{\partial \phi })& \widehat{\mathbf{𝐫}}\\ +({\mathrm{\nabla }}^2{A}_{\theta }-\frac{A_{\theta }}{r^2{\sin }^2\theta }+\frac{2}{r^2}\frac{\partial A_r}{\partial \theta }-\frac{2\cos \theta }{r^2{\sin }^2\theta }\frac{\partial A_{\phi }}{\partial \phi })& \hat{\mathit{\theta }}\\ +({\mathrm{\nabla }}^2{A}_{\phi }-\frac{A_{\phi }}{r^2{\sin }^2\theta }+\frac{2}{r^2\sin \theta }\frac{\partial A_r}{\partial \phi }+\frac{2\cos \theta }{r^2{\sin }^2\theta }\frac{\partial A_{\theta }}{\partial \phi })& \hat{\mathit{\phi }}\end{array}$
Directional derivative (A ⋅ ∇)B	$\mathbf{𝐀}\cdot \mathrm{\nabla }{B}_x\widehat{\mathbf{𝐱}}+\mathbf{𝐀}\cdot \mathrm{\nabla }{B}_y\widehat{\mathbf{𝐲}}+\mathbf{𝐀}\cdot \mathrm{\nabla }{B}_z\widehat{\mathbf{𝐳}}$	$\begin{array}{rl}(A_{\rho }\frac{\partial B_{\rho }}{\partial \rho }+\frac{A_{\phi }}{\rho }\frac{\partial B_{\rho }}{\partial \phi }+A_z\frac{\partial B_{\rho }}{\partial z}-\frac{A_{\phi }{B}_{\phi }}{\rho })& \hat{\mathit{\rho }}\\ +(A_{\rho }\frac{\partial B_{\phi }}{\partial \rho }+\frac{A_{\phi }}{\rho }\frac{\partial B_{\phi }}{\partial \phi }+A_z\frac{\partial B_{\phi }}{\partial z}+\frac{A_{\phi }{B}_{\rho }}{\rho })& \hat{\mathit{\phi }}\\ +(A_{\rho }\frac{\partial B_z}{\partial \rho }+\frac{A_{\phi }}{\rho }\frac{\partial B_z}{\partial \phi }+A_z\frac{\partial B_z}{\partial z})& \widehat{\mathbf{𝐳}}\end{array}$	$\begin{array}{rl}(A_r\frac{\partial B_r}{\partial r}+\frac{A_{\theta }}{r}\frac{\partial B_r}{\partial \theta }+\frac{A_{\phi }}{r\sin \theta }\frac{\partial B_r}{\partial \phi }-\frac{A_{\theta }{B}_{\theta }+A_{\phi }{B}_{\phi }}{r})& \widehat{\mathbf{𝐫}}\\ +(A_r\frac{\partial B_{\theta }}{\partial r}+\frac{A_{\theta }}{r}\frac{\partial B_{\theta }}{\partial \theta }+\frac{A_{\phi }}{r\sin \theta }\frac{\partial B_{\theta }}{\partial \phi }+\frac{A_{\theta }{B}_r}{r}-\frac{A_{\phi }{B}_{\phi }\cot \theta }{r})& \hat{\mathit{\theta }}\\ +(A_r\frac{\partial B_{\phi }}{\partial r}+\frac{A_{\theta }}{r}\frac{\partial B_{\phi }}{\partial \theta }+\frac{A_{\phi }}{r\sin \theta }\frac{\partial B_{\phi }}{\partial \phi }+\frac{A_{\phi }{B}_r}{r}+\frac{A_{\phi }{B}_{\theta }\cot \theta }{r})& \hat{\mathit{\phi }}\end{array}$
Tensor divergence ∇ ⋅ T	$\begin{array}{rl}(\frac{\partial T_{xx}}{\partial x}+\frac{\partial T_{yx}}{\partial y}+\frac{\partial T_{zx}}{\partial z})& \widehat{\mathbf{𝐱}}\\ +(\frac{\partial T_{xy}}{\partial x}+\frac{\partial T_{yy}}{\partial y}+\frac{\partial T_{zy}}{\partial z})& \widehat{\mathbf{𝐲}}\\ +(\frac{\partial T_{xz}}{\partial x}+\frac{\partial T_{yz}}{\partial y}+\frac{\partial T_{zz}}{\partial z})& \widehat{\mathbf{𝐳}}\end{array}$	$\begin{array}{rl}[\frac{\partial T_{\rho \rho }}{\partial \rho }+\frac{1}{\rho }\frac{\partial T_{\phi \rho }}{\partial \phi }+\frac{\partial T_{z\rho }}{\partial z}+\frac{1}{\rho }(T_{\rho \rho }-T_{\phi \phi })]& \hat{\mathit{\rho }}\\ +[\frac{\partial T_{\rho \phi }}{\partial \rho }+\frac{1}{\rho }\frac{\partial T_{\phi \phi }}{\partial \phi }+\frac{\partial T_{z\phi }}{\partial z}+\frac{1}{\rho }(T_{\rho \phi }+T_{\phi \rho })]& \hat{\mathit{\phi }}\\ +[\frac{\partial T_{\rho z}}{\partial \rho }+\frac{1}{\rho }\frac{\partial T_{\phi z}}{\partial \phi }+\frac{\partial T_{zz}}{\partial z}+\frac{T_{\rho z}}{\rho }]& \widehat{\mathbf{𝐳}}\end{array}$	$\begin{array}{rl}[\frac{\partial T_{rr}}{\partial r}+2\frac{T_{rr}}{r}+\frac{1}{r}\frac{\partial T_{\theta r}}{\partial \theta }+\frac{\cot \theta }{r}{T}_{\theta r}+\frac{1}{r\sin \theta }\frac{\partial T_{\phi r}}{\partial \phi }-\frac{1}{r}(T_{\theta \theta }+T_{\phi \phi })]& \widehat{\mathbf{𝐫}}\\ +[\frac{\partial T_{r\theta }}{\partial r}+2\frac{T_{r\theta }}{r}+\frac{1}{r}\frac{\partial T_{\theta \theta }}{\partial \theta }+\frac{\cot \theta }{r}{T}_{\theta \theta }+\frac{1}{r\sin \theta }\frac{\partial T_{\phi \theta }}{\partial \phi }+\frac{T_{\theta r}}{r}-\frac{\cot \theta }{r}{T}_{\phi \phi }]& \hat{\mathit{\theta }}\\ +[\frac{\partial T_{r\phi }}{\partial r}+2\frac{T_{r\phi }}{r}+\frac{1}{r}\frac{\partial T_{\theta \phi }}{\partial \theta }+\frac{1}{r\sin \theta }\frac{\partial T_{\phi \phi }}{\partial \phi }+\frac{T_{\phi r}}{r}+\frac{\cot \theta }{r}(T_{\theta \phi }+T_{\phi \theta })]& \hat{\mathit{\phi }}\end{array}$

• ^α This page uses $\theta$ for the polar angle and $\phi$ for the azimuthal angle, which is common notation in physics. The source that is used for these formulae uses $\theta$ for the azimuthal angle and $\phi$ for the polar angle, which is common mathematical notation. In order to get the mathematics formulae, switch $\theta$ and $\phi$ in the formulae shown in the table above.
• ^β Defined in Cartesian coordinates as ${\partial }_i\mathbf{𝐀}\otimes {\mathbf{𝐞}}_i$. An alternative definition is ${\mathbf{𝐞}}_i\otimes {\partial }_i\mathbf{𝐀}$.
• ^γ Defined in Cartesian coordinates as ${\mathbf{𝐞}}_i\cdot {\partial }_i\mathbf{𝐓}$. An alternative definition is ${\partial }_i\mathbf{𝐓}\cdot {\mathbf{𝐞}}_i$.

Differential elements

Operation	Cartesian coordinates (x, y, z)	Cylindrical coordinates (ρ, φ, z)	Spherical coordinates (r, θ, φ)
Differential displacement dℓ	$dx\widehat{\mathbf{𝐱}}+dy\widehat{\mathbf{𝐲}}+dz\widehat{\mathbf{𝐳}}$	$d\rho \hat{\mathit{\rho }}+\rho d\phi \hat{\mathit{\phi }}+dz\widehat{\mathbf{𝐳}}$	$dr\widehat{\mathbf{𝐫}}+rd\theta \hat{\mathit{\theta }}+r\sin \theta d\phi \hat{\mathit{\phi }}$
Differential normal area dS	$\begin{array}{rl}dydz& \widehat{\mathbf{𝐱}}\\ +dxdz& \widehat{\mathbf{𝐲}}\\ +dxdy& \widehat{\mathbf{𝐳}}\end{array}$	$\begin{array}{rl}\rho d\phi dz& \hat{\mathit{\rho }}\\ +d\rho dz& \hat{\mathit{\phi }}\\ +\rho d\rho d\phi & \widehat{\mathbf{𝐳}}\end{array}$	$\begin{array}{rl}{r}^2\sin \theta d\theta d\phi & \widehat{\mathbf{𝐫}}\\ +r\sin \theta drd\phi & \hat{\mathit{\theta }}\\ +rdrd\theta & \hat{\mathit{\phi }}\end{array}$
Differential volume dV	$dxdydz$	$\rho d\rho d\phi dz$	$r^2\sin \theta drd\theta d\phi$

Calculation rules

1. $\mathrm{div}\mathrm{grad}f\equiv \mathrm{\nabla }\cdot \mathrm{\nabla }f\equiv {\mathrm{\nabla }}^{2}f$
2. $\mathrm{curl}\mathrm{grad}f\equiv \mathrm{\nabla }\times \mathrm{\nabla }f=\mathbf{𝟎}$
3. $\mathrm{div}\mathrm{curl}\mathbf{𝐀}\equiv \mathrm{\nabla }\cdot (\mathrm{\nabla }\times \mathbf{𝐀})=0$
4. $\mathrm{curl}\mathrm{curl}\mathbf{𝐀}\equiv \mathrm{\nabla }\times (\mathrm{\nabla }\times \mathbf{𝐀})=\mathrm{\nabla }(\mathrm{\nabla }\cdot \mathbf{𝐀})-{\mathrm{\nabla }}^2\mathbf{𝐀}$ (Lagrange's formula for del)
5. ${\mathrm{\nabla }}^2(fg)=f{\mathrm{\nabla }}^{2}g+2\mathrm{\nabla }f\cdot \mathrm{\nabla }g+g{\mathrm{\nabla }}^{2}f$
6. ${\mathrm{\nabla }}^2(\mathbf{𝐏}\cdot \mathbf{𝐐})=\mathbf{𝐐}\cdot {\mathrm{\nabla }}^2\mathbf{𝐏}-\mathbf{𝐏}\cdot {\mathrm{\nabla }}^2\mathbf{𝐐}+2\mathrm{\nabla }\cdot [(\mathbf{𝐏}\cdot \mathrm{\nabla })\mathbf{𝐐}+\mathbf{𝐏}\times \mathrm{\nabla }\times \mathbf{𝐐}]$ (From )

Cartesian derivation

File:Nabla cartesian.svg

$$\begin{array}{rl}\mathrm{div}\mathbf{𝐀}& =\underset{V\to 0}{\lim }\frac{\underset{\partial V}{\iint }\mathbf{𝐀}\cdot d\mathbf{𝐒}}{\underset{V}{\iiint }dV}\\ & =\frac{[A_x(x+dx)-A_x(x)]dydz+[A_y(y+dy)-A_y(y)]dxdz+[A_z(z+dz)-A_z(z)]dxdy}{dxdydz}\\ & =\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}\end{array}$$

$$\begin{array}{rl}(\mathrm{curl}\mathbf{𝐀}{)}_x& =\underset{S^{\perp \hat{\mathbf{x}}}\to 0}{\lim }\frac{\underset{\partial S}{\int }\mathbf{𝐀}\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}\\ & =\frac{[A_z(y+dy)-A_z(y)]dz-[A_y(z+dz)-A_y(z)]dy}{dydz}\\ & =\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}\end{array}$$

The expressions for $(\mathrm{curl}\mathbf{𝐀}{)}_y$ and $(\mathrm{curl}\mathbf{𝐀}{)}_z$ are found in the same way.

Cylindrical derivation

File:Nabla cylindrical2.svg

$$\begin{array}{rl}\mathrm{div}\mathbf{𝐀}& =\underset{V\to 0}{\lim }\frac{\underset{\partial V}{\iint }\mathbf{𝐀}\cdot d\mathbf{𝐒}}{\underset{V}{\iiint }dV}\\ & =\frac{[A_{\rho }(\rho +d\rho )(\rho +d\rho )-A_{\rho }(\rho )\rho ]d\varphi dz+[A_{\varphi }(\varphi +d\varphi )-A_{\varphi }(\varphi )]d\rho dz+[A_z(z+dz)-A_z(z)]d\rho (\rho +d\rho /2)d\varphi }{\rho d\varphi d\rho dz}\\ & =\frac{1}{\rho }\frac{\partial (\rho A_{\rho })}{\partial \rho }+\frac{1}{\rho }\frac{\partial A_{\varphi }}{\partial \varphi }+\frac{\partial A_z}{\partial z}\end{array}$$

$$\begin{array}{rl}(\mathrm{curl}\mathbf{𝐀}{)}_{\rho }& =\underset{S^{\perp \hat{\mathit{\rho }}}\to 0}{\lim }\frac{\underset{\partial S}{\int }\mathbf{𝐀}\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}\\ & =\frac{A_{\varphi }(z)(\rho +d\rho )d\varphi -A_{\varphi }(z+dz)(\rho +d\rho )d\varphi +A_z(\varphi +d\varphi )dz-A_z(\varphi )dz}{(\rho +d\rho )d\varphi dz}\\ & =-\frac{\partial A_{\varphi }}{\partial z}+\frac{1}{\rho }\frac{\partial A_z}{\partial \varphi }\end{array}$$

$$\begin{array}{rl}(\mathrm{curl}\mathbf{𝐀}{)}_{\varphi }& =\underset{S^{\perp \hat{\mathit{\varphi }}}\to 0}{\lim }\frac{\underset{\partial S}{\int }\mathbf{𝐀}\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}\\ & =\frac{A_z(\rho )dz-A_z(\rho +d\rho )dz+A_{\rho }(z+dz)d\rho -A_{\rho }(z)d\rho }{d\rho dz}\\ & =-\frac{\partial A_z}{\partial \rho }+\frac{\partial A_{\rho }}{\partial z}\end{array}$$

$$\begin{array}{rl}(\mathrm{curl}\mathbf{𝐀}{)}_z& =\underset{S^{\perp \hat{\mathit{z}}}\to 0}{\lim }\frac{\underset{\partial S}{\int }\mathbf{𝐀}\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}\\ & =\frac{A_{\rho }(\varphi )d\rho -A_{\rho }(\varphi +d\varphi )d\rho +A_{\varphi }(\rho +d\rho )(\rho +d\rho )d\varphi -A_{\varphi }(\rho )\rho d\varphi }{\rho d\rho d\varphi }\\ & =-\frac{1}{\rho }\frac{\partial A_{\rho }}{\partial \varphi }+\frac{1}{\rho }\frac{\partial (\rho A_{\varphi })}{\partial \rho }\end{array}$$

$$\begin{array}{rl}\mathrm{curl}\mathbf{𝐀}& =(\mathrm{curl}\mathbf{𝐀}{)}_{\rho }\hat{\mathit{\rho }}+(\mathrm{curl}\mathbf{𝐀}{)}_{\varphi }\hat{\mathit{\varphi }}+(\mathrm{curl}\mathbf{𝐀}{)}_z\hat{\mathit{z}}\\ & =(\frac{1}{\rho }\frac{\partial A_z}{\partial \varphi }-\frac{\partial A_{\varphi }}{\partial z})\hat{\mathit{\rho }}+(\frac{\partial A_{\rho }}{\partial z}-\frac{\partial A_z}{\partial \rho })\hat{\mathit{\varphi }}+\frac{1}{\rho }(\frac{\partial (\rho A_{\varphi })}{\partial \rho }-\frac{\partial A_{\rho }}{\partial \varphi })\hat{\mathit{z}}\end{array}$$

Spherical derivation

File:Nabla spherical2.svg $$\begin{array}{rl}\mathrm{div}\mathbf{𝐀}& =\underset{V\to 0}{\lim }\frac{\underset{\partial V}{\iint }\mathbf{𝐀}\cdot d\mathbf{𝐒}}{\underset{V}{\iiint }dV}\\ & =\frac{[A_r(r+dr)(r+dr{)}^2-A_r(r)r^2]\sin \theta d\theta d\varphi +[A_{\theta }(\theta +d\theta )\sin (\theta +d\theta )-A_{\theta }(\theta )\sin \theta ]rdrd\varphi +[A_{\varphi }(\varphi +d\varphi )-A_{\varphi }(\varphi )]rdrd\theta }{drrd\theta r\sin \theta d\varphi }\\ & =\frac{1}{r^2}\frac{\partial (r^2{A}_r)}{\partial r}+\frac{1}{r\sin \theta }\frac{\partial (A_{\theta }\sin \theta )}{\partial \theta }+\frac{1}{r\sin \theta }\frac{\partial A_{\varphi }}{\partial \varphi }\end{array}$$

$$\begin{array}{rl}(\mathrm{curl}\mathbf{𝐀}{)}_r& =\underset{S^{\perp \hat{\mathit{r}}}\to 0}{\lim }\frac{\underset{\partial S}{\int }\mathbf{𝐀}\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}\\ & =\frac{A_{\theta }(\varphi )rd\theta +A_{\varphi }(\theta +d\theta )r\sin (\theta +d\theta )d\varphi -A_{\theta }(\varphi +d\varphi )rd\theta -A_{\varphi }(\theta )r\sin (\theta )d\varphi }{rd\theta r\sin \theta d\varphi }\\ & =\frac{1}{r\sin \theta }\frac{\partial (A_{\varphi }\sin \theta )}{\partial \theta }-\frac{1}{r\sin \theta }\frac{\partial A_{\theta }}{\partial \varphi }\end{array}$$

$$\begin{array}{rl}(\mathrm{curl}\mathbf{𝐀}{)}_{\theta }& =\underset{S^{\perp \hat{\mathit{\theta }}}\to 0}{\lim }\frac{\underset{\partial S}{\int }\mathbf{𝐀}\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}\\ & =\frac{A_{\varphi }(r)r\sin \theta d\varphi +A_r(\varphi +d\varphi )dr-A_{\varphi }(r+dr)(r+dr)\sin \theta d\varphi -A_r(\varphi )dr}{drr\sin \theta d\varphi }\\ & =\frac{1}{r\sin \theta }\frac{\partial A_r}{\partial \varphi }-\frac{1}{r}\frac{\partial (r{A}_{\varphi })}{\partial r}\end{array}$$

$$\begin{array}{rl}(\mathrm{curl}\mathbf{𝐀}{)}_{\varphi }& =\underset{S^{\perp \hat{\mathit{\varphi }}}\to 0}{\lim }\frac{\underset{\partial S}{\int }\mathbf{𝐀}\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}\\ & =\frac{A_r(\theta )dr+A_{\theta }(r+dr)(r+dr)d\theta -A_r(\theta +d\theta )dr-A_{\theta }(r)rd\theta }{rdrd\theta }\\ & =\frac{1}{r}\frac{\partial (r{A}_{\theta })}{\partial r}-\frac{1}{r}\frac{\partial A_r}{\partial \theta }\end{array}$$

$$\begin{array}{rl}\mathrm{curl}\mathbf{𝐀}& =(\mathrm{curl}\mathbf{𝐀}{)}_r\hat{\mathit{r}}+(\mathrm{curl}\mathbf{𝐀}{)}_{\theta }\hat{\mathit{\theta }}+(\mathrm{curl}\mathbf{𝐀}{)}_{\varphi }\hat{\mathit{\varphi }}\\ & =\frac{1}{r\sin \theta }(\frac{\partial (A_{\varphi }\sin \theta )}{\partial \theta }-\frac{\partial A_{\theta }}{\partial \varphi })\hat{\mathit{r}}+\frac{1}{r}(\frac{1}{\sin \theta }\frac{\partial A_r}{\partial \varphi }-\frac{\partial (r{A}_{\varphi })}{\partial r})\hat{\mathit{\theta }}+\frac{1}{r}(\frac{\partial (r{A}_{\theta })}{\partial r}-\frac{\partial A_r}{\partial \theta })\hat{\mathit{\varphi }}\end{array}$$

Unit vector conversion formula

The unit vector of a coordinate parameter u is defined in such a way that a small positive change in u causes the position vector $\mathbf{𝐫}$ to change in $\mathbf{𝐮}$ direction.

Therefore, $$\frac{\partial \mathbf{𝐫}}{\partial u}=\frac{\partial s}{\partial u}\mathbf{𝐮}$$ where s is the arc length parameter.

For two sets of coordinate systems $u_i$ and $v_j$, according to chain rule, $$\begin{array}{rl}d\mathbf{𝐫}& =\sum _i\frac{\partial \mathbf{𝐫}}{\partial u_i}d{u}_i=\sum _i\frac{\partial s}{\partial u_i}{\widehat{\mathbf{𝐮}}}_{i}d{u}_i=\sum _j\frac{\partial s}{\partial v_j}{\widehat{\mathbf{𝐯}}}_{j}d{v}_j\\ & =\sum _j\frac{\partial s}{\partial v_j}{\widehat{\mathbf{𝐯}}}_j\sum _i\frac{\partial v_j}{\partial u_i}d{u}_i\\ & =\sum _i\sum _j\frac{\partial s}{\partial v_j}\frac{\partial v_j}{\partial u_i}{\widehat{\mathbf{𝐯}}}_{j}d{u}_i.\end{array}$$

Now, we isolate the $i$^th component. For $i\ne k$, let $\mathrm{d}{u}_k=0$. Then divide on both sides by $\mathrm{d}{u}_i$ to get: $$\frac{\partial s}{\partial u_i}{\widehat{\mathbf{𝐮}}}_i=\sum _j\frac{\partial s}{\partial v_j}\frac{\partial v_j}{\partial u_i}{\widehat{\mathbf{𝐯}}}_j.$$

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.