Coordinate systems/Derivation of formulas

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The purpose of this resource is to carefully examine the Wikipedia article Del in cylindrical and spherical coordinates for accuracy.

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Transformations between coordinates[edit]

  1. w:Cartesian coordinates (x, y, z)
  2. w:Cylindrical coordinates (ρ, ϕ, z)
  3. w:Spherical coordinates (r, θ, ϕ)
  4. w:Parabolic cylindrical coordinates (σ, τ, z)

Coordinate variable transformations*[edit]

*Asterisk indicates that the title is a link to more discussion

Cylindrical from Cartesian variable transformation[edit]

\rho = \sqrt{x^2+y^2}   ,     
\phi = \arctan(y/x)   ,     
z    = z  verified using mathworld[1]

Cartesian from cylindrical variable transformation[edit]

x = \rho\cos\phi   ,     
y = \rho\sin\phi   ,     
z = z  verified using mathworld[2]

Cartesian from spherical variable transformation[edit]

x = r\sin\theta\cos\phi   ,     
y = r\sin\theta\sin\phi   ,     
z = r\cos\theta  verified using mathworld[3]

Cartesian from parabolic cylindrical variable transformation[edit]

x = \sigma \tau   ,     
y = \tfrac{1}{2} \left( \tau^{2} - \sigma^{2} \right)   ,     
z = z  --no reference

Spherical from Cartesian variable transformation[edit]

r      = \sqrt{x^2+y^2+z^2}   ,     
\theta = \arccos(z/r)   ,     
\phi   = \arctan(y/x)  verified using mathworld[4]

Spherical from cylindrical variable transformation[edit]

r      = \sqrt{\rho^2 + z^2}   ,     
\theta = \arctan{(\rho/z)}   ,     
\phi   = \phi  no reference

Cylindrical from spherical variable transformation[edit]

\rho = r\sin\theta   ,     
\phi = \phi   ,     
z    = r\cos\theta  no reference

Cylindrical from parabolic cylindrical variable transformation[edit]

\rho\cos\phi = \sigma \tau   ,     
\rho\sin\phi = \tfrac{1}{2} \left( \tau^{2} - \sigma^{2} \right)   ,     
z = z  no reference

Unit vectors[edit]

Cylindrical from Cartesian unit vectors[edit]

\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}  Verified, see page linked in title

Cartesian from cylindrical unit vectors[edit]

\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}  Verified, see page linked in title

Cartesian from spherical unit vectors[edit]

\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}  Verified, see page linked in title

Parabolic cylindrical from Cartesian unit vectors[edit]

\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}

Spherical from Cartesian unit vectors[edit]

\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} Verified, see page linked in title

Spherical from cylindrical unit vectors[edit]

\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}

Cylindrical from spherical unit vectors[edit]

\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}

Vector and scalar fields[edit]

\mathbf A is vector field and f is a scalar field. The vector field can be expressed as:

  1. A_x      \hat{\mathbf x}         + A_y      \hat{\mathbf y}         + A_z    \hat{\mathbf z}
  2. A_\rho   \hat{\boldsymbol\rho}   + A_\phi   \hat{\boldsymbol\phi}   + A_z    \hat{\mathbf z}
  3. A_r      \hat{\boldsymbol r}     + A_\theta \hat{\boldsymbol\theta} + A_\phi \hat{\boldsymbol\phi}
  4. A_\sigma \hat{\boldsymbol\sigma} + A_\tau   \hat{\boldsymbol\tau}   + A_\phi \hat{\mathbf z}

Gradient of a scalar field[edit]

\nabla f is the w:gradient of a scaler field.

  1. {\partial f \over \partial x}\hat{\mathbf x} + {\partial f \over \partial y}\hat{\mathbf y}
+ {\partial f \over \partial z}\hat{\mathbf z}
  2. {\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}
  3. {\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}
  4.  \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 of a vector field*[edit]

\nabla \cdot \mathbf{A} is the w:divergence of a vector field

  1. {\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}
  2. {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}
  3. {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}
  4.  \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 of a vector field[edit]

\nabla \times \mathbf{A} is the w:curl (mathematics) of A

  \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}
    \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}
  \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}
    \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}

Laplacian of a scalar field[edit]

\Delta f \equiv \nabla^2 f is the w:Laplace operator on a scalar field

  1. {\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2}
  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}
  3. {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}
  4.  \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}}

Laplacian of a vector field[edit]

\Delta \mathbf{A} \equiv \nabla^2 \mathbf{A} is the w:Vector Laplacian of \mathbf{A}

  1. \Delta A_x \hat{\mathbf x} + \Delta A_y \hat{\mathbf y} + \Delta A_z \hat{\mathbf z}
  \mathopen{}\left(\Delta A_\rho - \frac{A_\rho}{\rho^2} - \frac{2}{\rho^2} \frac{\partial A_\phi}{\partial \phi}\right)\mathclose{} \hat{\boldsymbol\rho} 
+ \mathopen{}\left(\Delta A_\phi - \frac{A_\phi}{\rho^2} + \frac{2}{\rho^2} \frac{\partial A_\rho}{\partial \phi}\right)\mathclose{} \hat{\boldsymbol\phi} 
+ \Delta A_z \hat{\mathbf z}
  \left(\Delta A_r - \frac{2 A_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) \hat{\boldsymbol 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) \hat{\boldsymbol\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) \hat{\boldsymbol\phi}

Material derivative of a vector field[edit]

(\mathbf{A} \cdot \nabla) \mathbf{B} might be called the "convective derivative of B along A" (appropriate description if A' is a unit vector) [5]

  \left(A_x \frac{\partial B_x}{\partial x} + A_y \frac{\partial B_x}{\partial y} + A_z \frac{\partial B_x}{\partial z}\right) \hat{\mathbf{x}} 
+ \left(A_x \frac{\partial B_y}{\partial x} + A_y \frac{\partial B_y}{\partial y} + A_z \frac{\partial B_y}{\partial z}\right) \hat{\mathbf{y}} 
+ \left(A_x \frac{\partial B_z}{\partial x} + A_y \frac{\partial B_z}{\partial y} + A_z \frac{\partial B_z}{\partial z}\right) \hat{\mathbf{z}}
  \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)
+ \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)
+ \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)
  \hat{\mathbf z}
    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) \hat{\boldsymbol 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) \hat{\boldsymbol\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) \hat{\boldsymbol\phi}

Differential displacement[edit]

  1. d\mathbf{l} = dx \, \hat{\mathbf x} + dy \, \hat{\mathbf y} + dz \, \hat{\mathbf z}
  2. d\mathbf{l} = d\rho \, \hat{\boldsymbol \rho} + \rho \, d\phi \, \hat{\boldsymbol \phi} + dz \, \hat{\mathbf z}
  3. d\mathbf{l} = dr \, \hat{\mathbf r} + r \, d\theta \, \hat{\boldsymbol \theta} + r \, \sin\theta \, d\phi \, \hat{\boldsymbol \phi}
  4. 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 areas[edit]

Differential normal area d \mathbf S

  dy \, dz \hat{\mathbf x} 
+ dx \, dz \hat{\mathbf y} 
+ dx \, dy \hat{\mathbf z}
  \rho \, d\phi \, dz    \hat{\boldsymbol\rho} 
+         d\rho \, dz    \hat{\boldsymbol\phi} 
+ \rho \, d\rho \, d\phi \hat{\mathbf z}
  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}
  \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}

Differential volume[edit]

  1. dV=dx \, dy \, dz verified[6]
  2. dV=\rho \, d\rho \, d\phi \, dz verified[7]
  3. dV=r^2 \sin\theta \, dr \, d\theta \, d\phi verified[8]
  4. dV=\left(\sigma^2 + \tau^2\right) d\sigma \, d\tau \, dz

nabla's on nabla's[edit]

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


  5. Weisstein, Eric W. "Convective Operator". Mathworld. Retrieved 23 March 2011. 
  6. James Stewart, Calculus: Concepts and Contexts, fourth edition, Brooks Cole 2005 pp. 884-5
  7. James Stewart, Calculus: Concepts and Contexts, fourth edition, Brooks Cole 2005 pp. 884-5
  8. James Stewart, Calculus: Concepts and Contexts, fourth edition, Brooks Cole 2005 pp. 884-5

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