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[edit]

  • 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[edit]

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{align}
\rho &= \sqrt{x^2+y^2} \\
\phi &= \arctan(y/x) \\
z    &= z \end{align} \begin{align}
x &= \rho\cos\phi \\
y &= \rho\sin\phi \\
z &= z \end{align} \begin{align}
x &= r\sin\theta\cos\phi \\
y &= r\sin\theta\sin\phi \\
z &= r\cos\theta \end{align} \begin{align}
x &= \sigma \tau\\
y &= \tfrac{1}{2} \left( \tau^{2} - \sigma^{2} \right) \\
z &= z \end{align}
\begin{align}
r      &= \sqrt{x^2+y^2+z^2} \\
\theta &= \arccos(z/r)\\
\phi   &= \arctan(y/x) \end{align} \begin{align}
r      &= \sqrt{\rho^2 + z^2} \\
\theta &= \arctan{(\rho/z)}\\
\phi   &= \phi \end{align} \begin{align}
\rho &= r\sin\theta \\
\phi &= \phi\\
z    &= r\cos\theta \end{align} \begin{align}
\rho\cos\phi &= \sigma \tau\\
\rho\sin\phi &= \tfrac{1}{2} \left( \tau^{2} - \sigma^{2} \right) \\
z &= z \end{align}
Definition
of
unit
vectors
\begin{align}
\hat{\boldsymbol\rho} &= \frac{  x \hat{\mathbf x} + y \hat{\mathbf y}}{\sqrt{x^2+y^2}} \\
\hat{\boldsymbol\phi} &= \frac{- y \hat{\mathbf x} + x \hat{\mathbf y}}{\sqrt{x^2+y^2}} \\
\hat{\mathbf z}       &= \hat{\mathbf z}
\end{align} \begin{align}
\hat{\mathbf x} &= \cos\phi\hat{\boldsymbol\rho} - \sin\phi\hat{\boldsymbol\phi} \\
\hat{\mathbf y} &= \sin\phi\hat{\boldsymbol\rho} + \cos\phi\hat{\boldsymbol\phi} \\
\hat{\mathbf z} &= \hat{\mathbf z}
\end{align} \begin{align}
\hat{\mathbf x} &= \sin\theta\cos\phi\hat{\boldsymbol r} + \cos\theta\cos\phi\hat{\boldsymbol\theta}-\sin\phi\hat{\boldsymbol\phi} \\
\hat{\mathbf y} &= \sin\theta\sin\phi\hat{\boldsymbol r} + \cos\theta\sin\phi\hat{\boldsymbol\theta}+\cos\phi\hat{\boldsymbol\phi} \\
\hat{\mathbf z} &= \cos\theta        \hat{\boldsymbol r} - \sin\theta        \hat{\boldsymbol\theta}
\end{align} \begin{align}
\hat{\boldsymbol\sigma} &= \frac{\tau \hat{\mathbf x} - \sigma \hat{\mathbf y}}{\sqrt{\tau^2+\sigma^2}} \\
\hat{\boldsymbol\tau}   &= \frac{\sigma \hat{\mathbf x} + \tau \hat{\mathbf y}}{\sqrt{\tau^2+\sigma^2}} \\
\hat{\mathbf z}         &= \hat{\mathbf z}
\end{align}
\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{x z \hat{\mathbf x} + y z \hat{\mathbf y} - \left(x^2 + y^2\right) \hat{\mathbf z}}{\sqrt{x^2+y^2} \sqrt{x^2+y^2+z^2}} \\
\hat{\boldsymbol\phi}   &= \frac{- y \hat{\mathbf x} + x \hat{\mathbf y}}{\sqrt{x^2+y^2}}
\end{align} \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\phi}   &= \hat{\boldsymbol\phi}
\end{align} \begin{align}
\hat{\boldsymbol\rho} &= \sin\theta \hat{\mathbf r} + \cos\theta \hat{\boldsymbol\theta} \\
\hat{\boldsymbol\phi} &= \hat{\boldsymbol\phi} \\
\hat{\mathbf z}       &= \cos\theta \hat{\mathbf r} - \sin\theta \hat{\boldsymbol\theta}
\end{align} \begin{matrix}
\end{matrix}
A vector field \mathbf A A_x      \hat{\mathbf x}         + A_y      \hat{\mathbf y}         + A_z    \hat{\mathbf z} A_\rho   \hat{\boldsymbol\rho}   + A_\phi   \hat{\boldsymbol\phi}   + A_z    \hat{\mathbf z} A_r      \hat{\boldsymbol r}     + A_\theta \hat{\boldsymbol\theta} + A_\phi \hat{\boldsymbol\phi} A_\sigma \hat{\boldsymbol\sigma} + A_\tau   \hat{\boldsymbol\tau}   + A_\phi \hat{\mathbf z}
Gradient \nabla f {\partial f \over \partial x}\hat{\mathbf x} + {\partial f \over \partial y}\hat{\mathbf y}
+ {\partial f \over \partial z}\hat{\mathbf z} {\partial f \over \partial \rho}\hat{\boldsymbol \rho}
+ {1 \over \rho}{\partial f \over \partial \phi}\hat{\boldsymbol \phi}
+ {\partial f \over \partial z}\hat{\mathbf z} {\partial f \over \partial r}\hat{\boldsymbol r}
+ {1 \over r}{\partial f \over \partial \theta}\hat{\boldsymbol \theta}
+ {1 \over r\sin\theta}{\partial f \over \partial \phi}\hat{\boldsymbol \phi}  \frac{1}{\sqrt{\sigma^{2} + \tau^{2}}} {\partial f \over \partial \sigma}\hat{\boldsymbol \sigma} + \frac{1}{\sqrt{\sigma^{2} + \tau^{2}}} {\partial f \over \partial \tau}\hat{\boldsymbol \tau} + {\partial f \over \partial z}\hat{\mathbf 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{align}
  \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) &\hat{\mathbf z}
\end{align} \begin{align}
  \left(
    \frac{1}{\rho} \frac{\partial A_z}{\partial \phi}
  - \frac{\partial A_\phi}{\partial z}
  \right) &\hat{\boldsymbol \rho} \\
+ \left(
    \frac{\partial A_\rho}{\partial z}
  - \frac{\partial A_z}{\partial \rho}
  \right) &\hat{\boldsymbol \phi} \\
+ \frac{1}{\rho} \left(
    \frac{\partial \left(\rho A_\phi\right)}{\partial \rho}
  - \frac{\partial A_\rho}{\partial \phi}
  \right) &\hat{\mathbf z}
\end{align} \begin{align}
  \frac{1}{r\sin\theta} \left(
    \frac{\partial}{\partial \theta} \left(A_\phi\sin\theta \right)
  - \frac{\partial A_\theta}{\partial \phi}
  \right) &\hat{\boldsymbol r} \\
+ \frac{1}{r} \left(
    \frac{1}{\sin\theta} \frac{\partial A_r}{\partial \phi}
  - \frac{\partial}{\partial r} \left( r A_\phi \right)
  \right) &\hat{\boldsymbol \theta}  \\
+ \frac{1}{r} \left(
    \frac{\partial}{\partial r} \left( r A_\theta \right)
  - \frac{\partial A_r}{\partial \theta}
  \right) &\hat{\boldsymbol \phi}
\end{align} \begin{align}
  \left(
    \frac{1}{\sqrt{\sigma^2 + \tau^2}} \frac{\partial A_z}{\partial \tau}
  - \frac{\partial A_\tau}{\partial z}
  \right) &\hat{\boldsymbol \sigma} \\
- \left(
    \frac{1}{\sqrt{\sigma^2 + \tau^2}} \frac{\partial A_z}{\partial \sigma}
  - \frac{\partial A_\sigma}{\partial z}
  \right) &\hat{\boldsymbol \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) &\hat{\mathbf z}
\end{align}
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 \hat{\mathbf z}
Material derivative[1]

(\mathbf{A} \cdot \nabla) \mathbf{B}

Differential displacement d\mathbf{l} = dx \, \hat{\mathbf x} + dy \, \hat{\mathbf y} + dz \, \hat{\mathbf z} d\mathbf{l} = d\rho \, \hat{\boldsymbol \rho} + \rho \, d\phi \, \hat{\boldsymbol \phi} + dz \, \hat{\mathbf z} d\mathbf{l} = dr \, \hat{\mathbf r} + r \, d\theta \, \hat{\boldsymbol \theta} + r \, \sin\theta \, d\phi \, \hat{\boldsymbol \phi} d\mathbf{l} = \sqrt{\sigma^2 + \tau^2} \,  d\sigma \, \hat{\boldsymbol \sigma} + \sqrt{\sigma^2 + \tau^2} \, d\tau \, \hat{\boldsymbol \tau} + dz \, \hat{\mathbf z}
Differential normal area d \mathbf S \begin{align}
  dy \, dz &\hat{\mathbf x} \\
+ dx \, dz &\hat{\mathbf y} \\
+ dx \, dy &\hat{\mathbf z}
\end{align} \begin{align}
  \rho \, d\phi \, dz    &\hat{\boldsymbol\rho} \\
+         d\rho \, dz    &\hat{\boldsymbol\phi} \\
+ \rho \, d\rho \, d\phi &\hat{\mathbf z}
\end{align} \begin{align}
  r^2 \sin\theta \, d\theta \, d\phi   &\hat{\mathbf r} \\
+ r   \sin\theta \, dr      \, d\phi   &\hat{\boldsymbol\theta} \\
+ r              \, dr      \, d\theta &\hat{\boldsymbol\phi}
\end{align} \begin{align}
  \sqrt{\sigma^2 + \tau^2}       \, d\tau   \, dz    &\hat{\boldsymbol\sigma} \\
+ \sqrt{\sigma^2 + \tau^2}       \, d\sigma \, dz    &\hat{\boldsymbol\tau} \\
+ \left(\sigma^2 + \tau^2\right) \, d\sigma \, d\tau &\hat{\mathbf z}
\end{align}
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:
  1. \operatorname{div}  \, \operatorname{grad} f          \equiv \nabla \cdot  \nabla f = \nabla^2 f \equiv \Delta f
  2. \operatorname{curl} \, \operatorname{grad} f          \equiv \nabla \times \nabla f = \mathbf 0
  3. \operatorname{div}  \, \operatorname{curl} \mathbf{A} \equiv \nabla \cdot  (\nabla \times \mathbf{A}) = 0
  4. \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)
  5. \Delta (f g) = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f

See also[edit]

References[edit]

  1. ^ Weisstein, Eric W. "Convective Operator". Mathworld. Retrieved 23 March 2011. 

External links[edit]