Del in cylindrical and spherical coordinates
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].
Coordinate conversions
Conversion between Cartesian, cylindrical, and spherical coordinates
|
Cartesian |
Cylindrical |
Spherical |
Cartesian |
N/A |
![\begin{align}
x &= \rho \cos\varphi \\
y &= \rho \sin\varphi \\
z &= z
\end{align}](../I/m/2e3041671956747a26652fc9ea74ab30.png) |
![\begin{align}
x &= r \sin\theta \cos\varphi \\
y &= r \sin\theta \sin\varphi \\
z &= r \cos\theta
\end{align}](../I/m/59a4c1a197c519433a8af4365ee52fb0.png) |
Cylindrical |
![\begin{align}
\rho &= \sqrt{x^2 + y^2} \\
\varphi &= \arctan\left(\frac{y}{x}\right) \\
z &= z
\end{align}](../I/m/dd81a3fd9277e818945eb69308b11833.png) |
N/A |
![\begin{align}
\rho &= r \sin\theta \\
\varphi &= \varphi \\
z &= r\cos\theta
\end{align}](../I/m/d1d535442292ff9342fca5c87e8af76c.png) |
Spherical |
![\begin{align}
r &= \sqrt{x^2 + y^2 + z^2} \\
\theta &= \arccos\left(\frac{z}{r}\right) \\
\varphi &= \arctan\left(\frac{y}{x}\right)
\end{align}](../I/m/f9e701c06740b1ecb757eeccfeb3dd07.png) |
![\begin{align}
r &= \sqrt{\rho^2 + z^2} \\
\theta &= \arctan{\left(\frac{\rho}{z}\right)} \\
\varphi &= \varphi
\end{align}](../I/m/23993a5cdd4f8525cca7595c005983b3.png) |
N/A |
Unit vector conversions
Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of destination coordinates
|
Cartesian |
Cylindrical |
Spherical |
Cartesian |
N/A |
![\begin{align}
\hat{\mathbf x} &= \cos\varphi \hat{\boldsymbol \rho} - \sin\varphi \hat{\boldsymbol \varphi} \\
\hat{\mathbf y} &= \sin\varphi \hat{\boldsymbol \rho} + \cos\varphi \hat{\boldsymbol \varphi} \\
\hat{\mathbf z} &= \hat{\mathbf z}
\end{align}](../I/m/d58af0fe467e61c4a2a71e5ce94ed279.png) |
![\begin{align}
\hat{\mathbf x} &= \sin\theta \cos\varphi \hat{\mathbf r} + \cos\theta \cos\varphi \hat{\boldsymbol \theta} - \sin\varphi \hat{\boldsymbol \varphi} \\
\hat{\mathbf y} &= \sin\theta \sin\varphi \hat{\mathbf r} + \cos\theta \sin\varphi \hat{\boldsymbol \theta} + \cos\varphi \hat{\boldsymbol \varphi} \\
\hat{\mathbf z} &= \cos\theta \hat{\mathbf r} - \sin\theta \hat{\boldsymbol \theta}
\end{align}](../I/m/4146b739343c8de1b5d5b3d3bae38512.png) |
Cylindrical |
![\begin{align}
\hat{\boldsymbol \rho} &= \frac{x \hat{\mathbf x} + y \hat{\mathbf y}}{\sqrt{x^2 + y^2}} \\
\hat{\boldsymbol \varphi} &= \frac{-y \hat{\mathbf x} + x \hat{\mathbf y}}{\sqrt{x^2 + y^2}} \\
\hat{\mathbf z} &= \hat{\mathbf z}
\end{align}](../I/m/b6a2579e169adf0ec0f5df19f66b2921.png) |
N/A |
![\begin{align}
\hat{\boldsymbol \rho} &= \sin\theta \hat{\mathbf r} + \cos\theta \hat{\boldsymbol \theta} \\
\hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi} \\
\hat{\mathbf z} &= \cos\theta \hat{\mathbf r} - \sin\theta \hat{\boldsymbol \theta}
\end{align}](../I/m/cb0dfeb0f194113b11d6057f2b90f582.png) |
Spherical |
![\begin{align}
\hat{\mathbf r} &= \frac{x \hat{\mathbf x} + y \hat{\mathbf y} + z \hat{\mathbf z}}{\sqrt{x^2 + y^2 + z^2}} \\
\hat{\boldsymbol \theta} &= \frac{\left(x \hat{\mathbf x} + y \hat{\mathbf y}\right) z - \left(x^2 + y^2\right) \hat{\mathbf z}}{\sqrt{x^2 + y^2 + z^2} \sqrt{x^2 + y^2}} \\
\hat{\boldsymbol \varphi} &= \frac{-y \hat{\mathbf x} + x \hat{\mathbf y}}{\sqrt{x^2 + y^2}}
\end{align}](../I/m/41bcba0daadba89f13589702e1ebbc0f.png) |
![\begin{align}
\hat{\mathbf r} &= \frac{\rho \hat{\boldsymbol \rho} + z \hat{\mathbf z}}{\sqrt{\rho^2 + z^2}} \\
\hat{\boldsymbol \theta} &= \frac{z \hat{\boldsymbol \rho} - \rho \hat{\mathbf z}}{\sqrt{\rho^2 + z^2}} \\
\hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi}
\end{align}](../I/m/754b150407c83ac86d6ddacffae5d5e6.png) |
N/A |
Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of source coordinates
|
Cartesian |
Cylindrical |
Spherical |
Cartesian |
N/A |
![\begin{align}
\hat{\mathbf x} &= \frac{x \hat{\boldsymbol \rho} - y \hat{\boldsymbol \varphi}}{\sqrt{x^2 + y^2}} \\
\hat{\mathbf y} &= \frac{y \hat{\boldsymbol \rho} + x \hat{\boldsymbol \varphi}}{\sqrt{x^2 + y^2}} \\
\hat{\mathbf z} &= \hat{\mathbf z}
\end{align}](../I/m/fa90ebe559becc7badcf38238dd2f583.png) |
![\begin{align}
\hat{\mathbf x} &= \frac{x \left(\sqrt{x^2 + y^2} \hat{\mathbf r} + z \hat{\boldsymbol \theta}\right) - y \sqrt{x^2 + y^2 + z^2} \hat{\boldsymbol \varphi}}{\sqrt{x^2 + y^2} \sqrt{x^2 + y^2 + z^2}} \\
\hat{\mathbf y} &= \frac{y \left(\sqrt{x^2 + y^2} \hat{\mathbf r} + z \hat{\boldsymbol \theta}\right) + x \sqrt{x^2 + y^2 + z^2} \hat{\boldsymbol \varphi}}{\sqrt{x^2 + y^2} \sqrt{x^2 + y^2 + z^2}} \\
\hat{\mathbf z} &= \frac{z \hat{\mathbf r} - \sqrt{x^2 + y^2} \hat{\boldsymbol \theta}}{\sqrt{x^2 + y^2 + z^2}}
\end{align}](../I/m/064195bdf435d0bdb62e29173cd0813a.png) |
Cylindrical |
![\begin{align}
\hat{\boldsymbol \rho} &= \cos\varphi \hat{\mathbf x} + \sin\varphi \hat{\mathbf y} \\
\hat{\boldsymbol \varphi} &= -\sin\varphi \hat{\mathbf x} + \cos\varphi \hat{\mathbf y} \\
\hat{\mathbf z} &= \hat{\mathbf z}
\end{align}](../I/m/a90a292a4ec4ec8d1d7e306b27af3f79.png) |
N/A |
![\begin{align}
\hat{\boldsymbol \rho} &= \frac{\rho \hat{\mathbf r} + z \hat{\boldsymbol \theta}}{\sqrt{\rho^2 + z^2}} \\
\hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi} \\
\hat{\mathbf z} &= \frac{z \hat{\mathbf r} - \rho \hat{\boldsymbol \theta}}{\sqrt{\rho^2 + z^2}}
\end{align}](../I/m/907c94b4754d397613691620779c028c.png) |
Spherical |
![\begin{align}
\hat{\mathbf r} &= \sin\theta \left(\cos\varphi \hat{\mathbf x} + \sin\varphi \hat{\mathbf y}\right) + \cos\theta \hat{\mathbf z} \\
\hat{\boldsymbol \theta} &= \cos\theta \left(\cos\varphi \hat{\mathbf x} + \sin\varphi \hat{\mathbf y}\right) - \sin\theta \hat{\mathbf z} \\
\hat{\boldsymbol \varphi} &= -\sin\varphi \hat{\mathbf x} + \cos\varphi \hat{\mathbf y}
\end{align}](../I/m/b07942c883717c4ce59cc84a5636122b.png) |
![\begin{align}
\hat{\mathbf r} &= \sin\theta \hat{\boldsymbol \rho} + \cos\theta \hat{\mathbf z} \\
\hat{\boldsymbol \theta} &= \cos\theta \hat{\boldsymbol \rho} - \sin\theta \hat{\mathbf z} \\
\hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi}
\end{align}](../I/m/0a2691fe14eaacb59693ac81b30e8bfa.png) |
N/A |
Del formulae
Non-trivial calculation rules
-
![\operatorname{div} \, \operatorname{grad} f \equiv \nabla \cdot \nabla f \equiv \nabla^2 f](../I/m/88e3736bedd6253794c6d83f8336beec.png)
-
![\operatorname{curl} \, \operatorname{grad} f \equiv \nabla \times \nabla f = \mathbf 0](../I/m/7e694b3647546e286748acf745953209.png)
-
![\operatorname{div} \, \operatorname{curl} \mathbf{A} \equiv \nabla \cdot (\nabla \times \mathbf{A}) = 0](../I/m/9404ca9f21280b4004dad9abcf34e47e.png)
-
(Lagrange's formula for del)
-
![\nabla^2 (f g) = f \nabla^2 g + 2 \nabla f \cdot \nabla g + g \nabla^2 f](../I/m/5cd99f77038a802859ca4bd7e7feee7e.png)
See also
References
External links