# 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

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*

*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:

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

$\nabla f$ 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*

$\nabla \cdot \mathbf{A}$ 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

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

1. $\left(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\right) \hat{\mathbf x} + \left(\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\right) \hat{\mathbf y} + \left(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right) \hat{\mathbf z}$
2. $\left( \frac{1}{\rho} \frac{\partial A_z}{\partial \phi} - \frac{\partial A_\phi}{\partial z} \right) \hat{\boldsymbol \rho} + \left( \frac{\partial A_\rho}{\partial z} - \frac{\partial A_z}{\partial \rho} \right) \hat{\boldsymbol \phi} + \frac{1}{\rho} \left( \frac{\partial \left(\rho A_\phi\right)}{\partial \rho} - \frac{\partial A_\rho}{\partial \phi} \right) \hat{\mathbf z}$
3. $\frac{1}{r\sin\theta} \left( \frac{\partial}{\partial \theta} \left(A_\phi\sin\theta \right) - \frac{\partial A_\theta}{\partial \phi} \right) \hat{\boldsymbol r} + \frac{1}{r} \left( \frac{1}{\sin\theta} \frac{\partial A_r}{\partial \phi} - \frac{\partial}{\partial r} \left( r A_\phi \right) \right) \hat{\boldsymbol \theta} + \frac{1}{r} \left( \frac{\partial}{\partial r} \left( r A_\theta \right) - \frac{\partial A_r}{\partial \theta} \right) \hat{\boldsymbol \phi}$
4. $\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

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

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

1. $\Delta A_x \hat{\mathbf x} + \Delta A_y \hat{\mathbf y} + \Delta A_z \hat{\mathbf z}$
2. $\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}$
3. $\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]

1. $\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}}$
2. $\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}$
3. $\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

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

Differential normal area $d \mathbf S$

1. $dy \, dz \hat{\mathbf x} + dx \, dz \hat{\mathbf y} + dx \, dy \hat{\mathbf z}$
2. $\rho \, d\phi \, dz \hat{\boldsymbol\rho} + d\rho \, dz \hat{\boldsymbol\phi} + \rho \, d\rho \, d\phi \hat{\mathbf z}$
3. $r^2 \sin\theta \, d\theta \, d\phi \hat{\mathbf r} + r \sin\theta \, dr \, d\phi \hat{\boldsymbol\theta} + r \, dr \, d\theta \hat{\boldsymbol\phi}$
4. $\sqrt{\sigma^2 + \tau^2} \, d\tau \, dz \hat{\boldsymbol\sigma} + \sqrt{\sigma^2 + \tau^2} \, d\sigma \, dz \hat{\boldsymbol\tau} + \left(\sigma^2 + \tau^2\right) \, d\sigma \, d\tau \hat{\mathbf z}$

### Differential volume

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

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. $\Delta (f g) = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f$

## References

1. http://mathworld.wolfram.com/CylindricalCoordinates.html
2. http://mathworld.wolfram.com/CylindricalCoordinates.html
3. http://mathworld.wolfram.com/SphericalCoordinates.html
4. http://mathworld.wolfram.com/SphericalCoordinates.html
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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