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

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

Coordinate variable transformations*

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

Cylindrical from Cartesian variable transformation

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

Cartesian from cylindrical variable transformation

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

Cartesian from spherical variable transformation

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

Cartesian from parabolic cylindrical variable transformation

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

Spherical from Cartesian variable transformation

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

Spherical from cylindrical variable transformation

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

Cylindrical from spherical variable transformation

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

Cylindrical from parabolic cylindrical variable transformation

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

Unit vectors

Cylindrical from Cartesian unit vectors

$\begin{align} \hat{\boldsymbol\rho} &= \frac{ x \hat{\mathbf x} + y \hat{\mathbf y}}{\sqrt{x^2+y^2}} \\ \hat{\boldsymbol\phi} &= \frac{- y \hat{\mathbf x} + x \hat{\mathbf y}}{\sqrt{x^2+y^2}} \\ \hat{\mathbf z} &= \hat{\mathbf z} \end{align}$ Verified, see page linked in title

Cartesian from cylindrical unit vectors

$\begin{align} \hat{\mathbf x} &= \cos\phi\hat{\boldsymbol\rho} - \sin\phi\hat{\boldsymbol\phi} \\ \hat{\mathbf y} &= \sin\phi\hat{\boldsymbol\rho} + \cos\phi\hat{\boldsymbol\phi} \\ \hat{\mathbf z} &= \hat{\mathbf z} \end{align}$ Verified, see page linked in title

Cartesian from spherical unit vectors

$\begin{align} \hat{\mathbf x} &= \sin\theta\cos\phi\hat{\boldsymbol r} + \cos\theta\cos\phi\hat{\boldsymbol\theta}-\sin\phi\hat{\boldsymbol\phi} \\ \hat{\mathbf y} &= \sin\theta\sin\phi\hat{\boldsymbol r} + \cos\theta\sin\phi\hat{\boldsymbol\theta}+\cos\phi\hat{\boldsymbol\phi} \\ \hat{\mathbf z} &= \cos\theta \hat{\boldsymbol r} - \sin\theta \hat{\boldsymbol\theta} \end{align}$ Verified, see page linked in title

Parabolic cylindrical from Cartesian unit vectors

$\begin{align} \hat{\boldsymbol\sigma} &= \frac{\tau \hat{\mathbf x} - \sigma \hat{\mathbf y}}{\sqrt{\tau^2+\sigma^2}} \\ \hat{\boldsymbol\tau} &= \frac{\sigma \hat{\mathbf x} + \tau \hat{\mathbf y}}{\sqrt{\tau^2+\sigma^2}} \\ \hat{\mathbf z} &= \hat{\mathbf z} \end{align}$

Spherical from Cartesian unit vectors

$\begin{align} \hat{\mathbf r} &= \frac{x \hat{\mathbf x} + y \hat{\mathbf y} + z \hat{\mathbf z}}{\sqrt{x^2+y^2+z^2}} \\ \hat{\boldsymbol\theta} &= \frac{x z \hat{\mathbf x} + y z \hat{\mathbf y} - \left(x^2 + y^2\right) \hat{\mathbf z}}{\sqrt{x^2+y^2} \sqrt{x^2+y^2+z^2}} \\ \hat{\boldsymbol\phi} &= \frac{- y \hat{\mathbf x} + x \hat{\mathbf y}}{\sqrt{x^2+y^2}} \end{align}$ Verified, see page linked in title

Spherical from cylindrical unit vectors

$\begin{align} \hat{\mathbf r} &= \frac{\rho \hat{\boldsymbol\rho} + z \hat{\mathbf z}}{\sqrt{\rho^2 +z^2}} \\ \hat{\boldsymbol\theta} &= \frac{ z \hat{\boldsymbol\rho} - \rho \hat{\mathbf z}}{\sqrt{\rho^2 +z^2}} \\ \hat{\boldsymbol\phi} &= \hat{\boldsymbol\phi} \end{align}$

Cylindrical from spherical unit vectors

$\begin{align} \hat{\boldsymbol\rho} &= \sin\theta \hat{\mathbf r} + \cos\theta \hat{\boldsymbol\theta} \\ \hat{\boldsymbol\phi} &= \hat{\boldsymbol\phi} \\ \hat{\mathbf z} &= \cos\theta \hat{\mathbf r} - \sin\theta \hat{\boldsymbol\theta} \end{align}$

Vector and scalar fields

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

$A_x \hat{\mathbf x} + A_y \hat{\mathbf y} + A_z \hat{\mathbf z}$
$A_\rho \hat{\boldsymbol\rho} + A_\phi \hat{\boldsymbol\phi} + A_z \hat{\mathbf z}$
$A_r \hat{\boldsymbol r} + A_\theta \hat{\boldsymbol\theta} + A_\phi \hat{\boldsymbol\phi}$
$A_\sigma \hat{\boldsymbol\sigma} + A_\tau \hat{\boldsymbol\tau} + A_\phi \hat{\mathbf z}$

Gradient of a scalar field

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

${\partial f \over \partial x}\hat{\mathbf x} + {\partial f \over \partial y}\hat{\mathbf y} + {\partial f \over \partial z}\hat{\mathbf z}$
${\partial f \over \partial \rho}\hat{\boldsymbol \rho} + {1 \over \rho}{\partial f \over \partial \phi}\hat{\boldsymbol \phi} + {\partial f \over \partial z}\hat{\mathbf z}$
${\partial f \over \partial r}\hat{\boldsymbol r} + {1 \over r}{\partial f \over \partial \theta}\hat{\boldsymbol \theta} + {1 \over r\sin\theta}{\partial f \over \partial \phi}\hat{\boldsymbol \phi}$
$\frac{1}{\sqrt{\sigma^{2} + \tau^{2}}} {\partial f \over \partial \sigma}\hat{\boldsymbol \sigma} + \frac{1}{\sqrt{\sigma^{2} + \tau^{2}}} {\partial f \over \partial \tau}\hat{\boldsymbol \tau} + {\partial f \over \partial z}\hat{\mathbf z}$

Divergence of a vector field*

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

${\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}$
${1 \over \rho}{\partial \left( \rho A_\rho \right) \over \partial \rho} + {1 \over \rho}{\partial A_\phi \over \partial \phi} + {\partial A_z \over \partial z}$
${1 \over r^2}{\partial \left( r^2 A_r \right) \over \partial r} + {1 \over r\sin\theta}{\partial \over \partial \theta} \left( A_\theta\sin\theta \right) + {1 \over r\sin\theta}{\partial A_\phi \over \partial \phi}$
$\frac{1}{\sigma^{2} + \tau^{2}}\left({\partial (\sqrt{\sigma^2+\tau^2} A_\sigma) \over \partial \sigma} + {\partial (\sqrt{\sigma^2+\tau^2} A_\tau) \over \partial \tau}\right) + {\partial A_z \over \partial z}$

Curl of a vector field

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

$\left(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\right) \hat{\mathbf x} + \left(\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\right) \hat{\mathbf y} + \left(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right) \hat{\mathbf z}$
$\left( \frac{1}{\rho} \frac{\partial A_z}{\partial \phi} - \frac{\partial A_\phi}{\partial z} \right) \hat{\boldsymbol \rho} + \left( \frac{\partial A_\rho}{\partial z} - \frac{\partial A_z}{\partial \rho} \right) \hat{\boldsymbol \phi} + \frac{1}{\rho} \left( \frac{\partial \left(\rho A_\phi\right)}{\partial \rho} - \frac{\partial A_\rho}{\partial \phi} \right) \hat{\mathbf z}$
$\frac{1}{r\sin\theta} \left( \frac{\partial}{\partial \theta} \left(A_\phi\sin\theta \right) - \frac{\partial A_\theta}{\partial \phi} \right) \hat{\boldsymbol r} + \frac{1}{r} \left( \frac{1}{\sin\theta} \frac{\partial A_r}{\partial \phi} - \frac{\partial}{\partial r} \left( r A_\phi \right) \right) \hat{\boldsymbol \theta} + \frac{1}{r} \left( \frac{\partial}{\partial r} \left( r A_\theta \right) - \frac{\partial A_r}{\partial \theta} \right) \hat{\boldsymbol \phi}$
$\left( \frac{1}{\sqrt{\sigma^2 + \tau^2}} \frac{\partial A_z}{\partial \tau} - \frac{\partial A_\tau}{\partial z} \right) \hat{\boldsymbol \sigma} - \left( \frac{1}{\sqrt{\sigma^2 + \tau^2}} \frac{\partial A_z}{\partial \sigma} - \frac{\partial A_\sigma}{\partial z} \right) \hat{\boldsymbol \tau}$ $+ \frac{1}{\sqrt{\sigma^2 + \tau^2}} \left( \frac{\partial \left(\sqrt{\sigma^2 + \tau^2} A_\sigma \right)}{\partial \tau} - \frac{\partial \left(\sqrt{\sigma^2 + \tau^2} A_\tau \right)}{\partial \sigma} \right) \hat{\mathbf z}$

Laplacian of a scalar field

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

${\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2}$
${1 \over \rho}{\partial \over \partial \rho}\left(\rho {\partial f \over \partial \rho}\right) + {1 \over \rho^2}{\partial^2 f \over \partial \phi^2} + {\partial^2 f \over \partial z^2}$
${1 \over r^2}{\partial \over \partial r}\!\left(r^2 {\partial f \over \partial r}\right) \!+\!{1 \over r^2\!\sin\theta}{\partial \over \partial \theta}\!\left(\sin\theta {\partial f \over \partial \theta}\right) \!+\!{1 \over r^2\!\sin^2\theta}{\partial^2 f \over \partial \phi^2}$
$\frac{1}{\sigma^{2} + \tau^{2}} \left( \frac{\partial^{2} f}{\partial \sigma^{2}} + \frac{\partial^{2} f}{\partial \tau^{2}} \right) + \frac{\partial^{2} f}{\partial z^{2}}$

Laplacian of a vector field

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

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

Material derivative of a vector field

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

$\left(A_x \frac{\partial B_x}{\partial x} + A_y \frac{\partial B_x}{\partial y} + A_z \frac{\partial B_x}{\partial z}\right) \hat{\mathbf{x}}$ $+ \left(A_x \frac{\partial B_y}{\partial x} + A_y \frac{\partial B_y}{\partial y} + A_z \frac{\partial B_y}{\partial z}\right) \hat{\mathbf{y}}$ $+ \left(A_x \frac{\partial B_z}{\partial x} + A_y \frac{\partial B_z}{\partial y} + A_z \frac{\partial B_z}{\partial z}\right) \hat{\mathbf{z}}$
$\left(A_\rho \frac{\partial B_\rho}{\partial \rho}+\frac{A_\phi}{\rho}\frac{\partial B_\rho}{\partial \phi}+A_z\frac{\partial B_\rho}{\partial z}-\frac{A_\phi B_\phi}{\rho}\right) \hat{\boldsymbol\rho}$ $+ \left(A_\rho \frac{\partial B_\phi}{\partial \rho} + \frac{A_\phi}{\rho}\frac{\partial B_\phi}{\partial \phi} + A_z\frac{\partial B_\phi}{\partial z} + \frac{A_\phi B_\rho}{\rho}\right) \hat{\boldsymbol\phi}$ $+ \left(A_\rho \frac{\partial B_z}{\partial \rho}+\frac{A_\phi}{\rho}\frac{\partial B_z}{\partial \phi}+A_z\frac{\partial B_z}{\partial z}\right) \hat{\mathbf z}$
$\left( A_r \frac{\partial B_r}{\partial r} + \frac{A_\theta}{r} \frac{\partial B_r}{\partial \theta} + \frac{A_\phi}{r\sin\theta} \frac{\partial B_r}{\partial \phi} - \frac{A_\theta B_\theta + A_\phi B_\phi}{r} \right) \hat{\boldsymbol r}$ $+ \left( A_r \frac{\partial B_\theta}{\partial r} + \frac{A_\theta}{r} \frac{\partial B_\theta}{\partial \theta} + \frac{A_\phi}{r\sin\theta} \frac{\partial B_\theta}{\partial \phi} + \frac{A_\theta B_r}{r} - \frac{A_\phi B_\phi\cot\theta}{r} \right) \hat{\boldsymbol\theta}$ $+ \left( A_r \frac{\partial B_\phi}{\partial r} + \frac{A_\theta}{r} \frac{\partial B_\phi}{\partial \theta} + \frac{A_\phi}{r\sin\theta} \frac{\partial B_\phi}{\partial \phi} + \frac{A_\phi B_r}{r} + \frac{A_\phi B_\theta \cot\theta}{r} \right) \hat{\boldsymbol\phi}$

Differential displacement

$d\mathbf{l} = dx \, \hat{\mathbf x} + dy \, \hat{\mathbf y} + dz \, \hat{\mathbf z}$
$d\mathbf{l} = d\rho \, \hat{\boldsymbol \rho} + \rho \, d\phi \, \hat{\boldsymbol \phi} + dz \, \hat{\mathbf z}$
$d\mathbf{l} = dr \, \hat{\mathbf r} + r \, d\theta \, \hat{\boldsymbol \theta} + r \, \sin\theta \, d\phi \, \hat{\boldsymbol \phi}$
$d\mathbf{l} = \sqrt{\sigma^2 + \tau^2} \, d\sigma \, \hat{\boldsymbol \sigma} + \sqrt{\sigma^2 + \tau^2} \, d\tau \, \hat{\boldsymbol \tau} + dz \, \hat{\mathbf z}$

Differential normal areas

These vector differentials cannot be integrated for curved surfaces. Click the title above to see why.

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

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

nabla's on nabla's

Non-trivial calculation rules:

$\operatorname{div} \, \operatorname{grad} f \equiv \nabla \cdot \nabla f = \nabla^2 f \equiv \Delta f$
$\operatorname{curl} \, \operatorname{grad} f \equiv \nabla \times \nabla f = \mathbf 0$
$\operatorname{div} \, \operatorname{curl} \mathbf{A} \equiv \nabla \cdot (\nabla \times \mathbf{A}) = 0$
$\operatorname{curl} \, \operatorname{curl} \mathbf{A} \equiv \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$ (Lagrange's formula for del)
$\Delta (f g) = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f$

References

↑ http://mathworld.wolfram.com/CylindricalCoordinates.html
↑ http://mathworld.wolfram.com/CylindricalCoordinates.html
↑ http://mathworld.wolfram.com/SphericalCoordinates.html
↑ http://mathworld.wolfram.com/SphericalCoordinates.html
↑
↑ James Stewart, Calculus: Concepts and Contexts, fourth edition, Brooks Cole 2005 pp. 884-5
↑ James Stewart, Calculus: Concepts and Contexts, fourth edition, Brooks Cole 2005 pp. 884-5
↑ James Stewart, Calculus: Concepts and Contexts, fourth edition, Brooks Cole 2005 pp. 884-5

^[1]

^[2]

↑ Weisstein, Eric W. "Convective Operator". Mathworld. Retrieved 23 March 2011.
↑ Huba J.D. (1994). "NRL Plasma Formulary revised". Office of Naval Research. Retrieved 11 June 2014.

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