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].

Coordinate conversions[edit]

Conversion between Cartesian, cylindrical, and spherical coordinates
Cartesian Cylindrical Spherical
Cartesian \begin{align}
  x &= x \\
  y &= y \\
  z &= z
\end{align} \begin{align}
  x &= \rho \cos\varphi \\
  y &= \rho \sin\varphi \\
  z &= z
\end{align} \begin{align}
  x &= r \sin\theta \cos\varphi \\
  y &= r \sin\theta \sin\varphi \\
  z &= r \cos\theta
\end{align}
Cylindrical \begin{align}
  \rho &= \sqrt{x^2 + y^2} \\
  \varphi &= \arctan(y / x) \\
  z &= z
\end{align} \begin{align}
  \rho &= \rho \\
  \varphi &= \varphi \\
  z &= z
\end{align} \begin{align}
  \rho &= r \sin\theta \\
  \varphi &= \varphi \\
  z &= r\cos\theta
\end{align}
Spherical \begin{align}
  r &= \sqrt{x^2+y^2+z^2} \\
  \theta &= \arccos(z/r) \\
  \varphi &= \arctan(y/x)
\end{align} \begin{align}
  r &= \sqrt{\rho^2 + z^2} \\
  \theta &= \arctan{(\rho/z)} \\
  \varphi &= \varphi
\end{align} \begin{align}
  r &= r \\
  \theta &= \theta \\
  \varphi &= \varphi
\end{align}

Unit vector conversions[edit]

Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of destination coordinates
Cartesian Cylindrical Spherical
Cartesian \begin{align}
  \hat{\mathbf x} &= \hat{\mathbf x} \\
  \hat{\mathbf y} &= \hat{\mathbf y} \\
  \hat{\mathbf z} &= \hat{\mathbf z}
\end{align} \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} \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}
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} \begin{align}
  \hat{\boldsymbol \rho} &= \hat{\boldsymbol \rho} \\
  \hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi} \\
  \hat{\mathbf z} &= \hat{\mathbf z}
\end{align} \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}
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{(x \hat{\mathbf x} + y \hat{\mathbf y}) z - (x^2 + y^2) \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} \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} \begin{align}
  \hat{\mathbf r} &= \hat{\mathbf r} \\
  \hat{\boldsymbol \theta} &= \hat{\boldsymbol \theta} \\
  \hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi}
\end{align}
Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of source coordinates
Cartesian Cylindrical Spherical
Cartesian \begin{align}
  \hat{\mathbf x} &= \hat{\mathbf x} \\
  \hat{\mathbf y} &= \hat{\mathbf y} \\
  \hat{\mathbf z} &= \hat{\mathbf z}
\end{align} \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} \begin{align}
  \hat{\mathbf x} &= \frac{x (\sqrt{x^2 + y^2} \hat{\mathbf r} + z \hat{\boldsymbol \theta}) - 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 (\sqrt{x^2 + y^2} \hat{\mathbf r} + z \hat{\boldsymbol \theta}) + 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}
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} \begin{align}
  \hat{\boldsymbol \rho} &= \hat{\boldsymbol \rho} \\
  \hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi} \\
  \hat{\mathbf z} &= \hat{\mathbf z}
\end{align} \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}
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} \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} \begin{align}
  \hat{\mathbf r} &= \hat{\mathbf r} \\
  \hat{\boldsymbol \theta} &= \hat{\boldsymbol \theta} \\
  \hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi}
\end{align}

Del formulae[edit]

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)
A vector field A A_x      \hat{\mathbf x}         + A_y      \hat{\mathbf y}         + A_z    \hat{\mathbf z} A_\rho   \hat{\boldsymbol \rho}   + A_\varphi   \hat{\boldsymbol \varphi}   + A_z    \hat{\mathbf z} A_r      \hat{\mathbf r}     + A_\theta \hat{\boldsymbol \theta} + A_\varphi \hat{\boldsymbol \varphi}
Gradient 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 \varphi}\hat{\boldsymbol \varphi}
+ {\partial f \over \partial z}\hat{\mathbf z} {\partial f \over \partial r}\hat{\mathbf r}
+ {1 \over r}{\partial f \over \partial \theta}\hat{\boldsymbol \theta}
+ {1 \over r\sin\theta}{\partial f \over \partial \varphi}\hat{\boldsymbol \varphi}
Divergence ∇ ⋅ 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_\varphi \over \partial \varphi}
+ {\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_\varphi \over \partial \varphi}
Curl ∇ × 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 \varphi}
  - \frac{\partial A_\varphi}{\partial z}
  \right) &\hat{\boldsymbol \rho} \\
+ \left(
    \frac{\partial A_\rho}{\partial z}
  - \frac{\partial A_z}{\partial \rho}
  \right) &\hat{\boldsymbol \varphi} \\
+ \frac{1}{\rho} \left(
    \frac{\partial \left(\rho A_\varphi\right)}{\partial \rho}
  - \frac{\partial A_\rho}{\partial \varphi}
  \right) &\hat{\mathbf z}
\end{align} \begin{align}
  \frac{1}{r\sin\theta} \left(
    \frac{\partial}{\partial \theta} \left(A_\varphi\sin\theta \right)
  - \frac{\partial A_\theta}{\partial \varphi}
  \right) &\hat{\mathbf r} \\
+ \frac{1}{r} \left(
    \frac{1}{\sin\theta} \frac{\partial A_r}{\partial \varphi}
  - \frac{\partial}{\partial r} \left( r A_\varphi \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 \varphi}
\end{align}
Laplace operator 2f ≡ ∆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 \varphi^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 \varphi^2}
Vector Laplacian 2A ≡ ∆A \nabla^2 A_x \hat{\mathbf x} + \nabla^2 A_y \hat{\mathbf y} + \nabla^2 A_z \hat{\mathbf z}
Material derivative[1] (A ⋅ ∇)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}}
Differential displacement d dx \, \hat{\mathbf x} + dy \, \hat{\mathbf y} + dz \, \hat{\mathbf z} d\rho \, \hat{\boldsymbol \rho} + \rho \, d\varphi \, \hat{\boldsymbol \varphi} + dz \, \hat{\mathbf z} dr \, \hat{\mathbf r} + r \, d\theta \, \hat{\boldsymbol \theta} + r \, \sin\theta \, d\varphi \, \hat{\boldsymbol \varphi}
Differential normal area dS \begin{align}
  dy \, dz &\, \hat{\mathbf x} \\
{} + dx \, dz &\, \hat{\mathbf y} \\
{} + dx \, dy &\, \hat{\mathbf z}
\end{align} \begin{align}
  \rho \, d\varphi \, dz &\, \hat{\boldsymbol \rho} \\
{} + d\rho \, dz &\, \hat{\boldsymbol \varphi} \\
{} + \rho \, d\rho \, d\varphi &\, \hat{\mathbf z}
\end{align} \begin{align}
  r^2 \sin\theta \, d\theta \, d\varphi &\, \hat{\mathbf r} \\
{} + r \sin\theta \, dr \, d\varphi &\, \hat{\boldsymbol \theta} \\
{} + r \, dr \, d\theta &\, \hat{\boldsymbol \varphi}
\end{align}
Differential volume dV dx \, dy \, dz \rho \, d\rho \, d\varphi \, dz r^2 \sin\theta \, dr \, d\theta \, d\varphi

Non-trivial calculation rules[edit]

  1. \operatorname{div}  \, \operatorname{grad} f          \equiv \nabla \cdot  \nabla f = \nabla^2 f \equiv \nabla^2 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. \nabla^2 (f g) = f \nabla^2 g + 2 \nabla f \cdot \nabla g + g \nabla^2 f

See also[edit]

References[edit]

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

External links[edit]