Del in cylindrical and spherical coordinates

This is a list of some vector calculus formulae for working with common curvilinear coordinate systems.

Notes

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 <math>\theta \in [0, \pi]</math>: 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 <math>\varphi \in [0, 2\pi]</math>: it is the angle between the x-axis and the projection of the radial vector onto the xy-plane.
The function can be used instead of the mathematical function owing to its domain and image. The classical arctan function has an image of , whereas atan2 is defined to have an image of .

Coordinate conversions

{| class="wikitable"

|+ Conversion between Cartesian, cylindrical, and spherical coordinates

! colspan="2" rowspan="2" |

! colspan="3" | From

! Cartesian

! Cylindrical

! Spherical

! rowspan="3" |To

! Cartesian

| <math>\begin{align}

x&=x\\

y&=y\\

z&=z\\

\end{align}</math>

| <math>\begin{align}

x &= \rho \cos\varphi \\

y &= \rho \sin\varphi \\

z &= z

\end{align}</math>

| <math>\begin{align}

x &= r \sin\theta \cos\varphi \\

y &= r \sin\theta \sin\varphi \\

z &= r \cos\theta \\

\end{align}</math>

! Cylindrical

| <math>\begin{align}

\rho &= \sqrt{x^2 + y^2} \\

\varphi &= \arctan\left(\frac{y}{x}\right) \\

z &= z

\end{align}</math>

| <math>\begin{align}

\rho &=\rho\\

\varphi &=\varphi\\

z&=z\\

\end{align}</math>

| <math>\begin{align}

\rho &= r \sin\theta \\

\varphi &= \varphi \\

z &= r\cos\theta

\end{align}</math>

! Spherical

| <math>\begin{align}

r &= \sqrt{x^2 + y^2 + z^2} \\

\theta &= \arccos\left(\frac{z}{\sqrt{x^2 + y^2 + z^2\right) \\

\varphi &= \arctan\left(\frac{y}{x}\right)

\end{align}</math>

| <math>\begin{align}

r &= \sqrt{\rho^2 + z^2} \\

\theta &= \arctan{\left(\frac{\rho}{z}\right)} \\

\varphi &= \varphi

\end{align}</math>

| <math>\begin{align}

r&=r\\\theta &=\theta \\\varphi &=\varphi

\end{align}</math>

Note that the operation <math>\arctan\left(\frac{A}{B}\right)</math> must be interpreted as the two-argument inverse tangent, atan2.

Unit vector conversions

{| class="wikitable"

|+ Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of destination coordinates

| <math>\nabla^2 A_x \hat{\mathbf x} + \nabla^2 A_y \hat{\mathbf y} + \nabla^2 A_z \hat{\mathbf z} </math>

<math>\begin{align}

\mathopen{}\left(\nabla^2 A_\rho - \frac{A_\rho}{\rho^2} - \frac{2}{\rho^2} \frac{\partial A_\varphi}{\partial \varphi}\right)\mathclose{} &\hat{\boldsymbol \rho} \\

+ \mathopen{}\left(\nabla^2 A_\varphi - \frac{A_\varphi}{\rho^2} + \frac{2}{\rho^2} \frac{\partial A_\rho}{\partial \varphi}\right)\mathclose{} &\hat{\boldsymbol \varphi} \\

{}+ \nabla^2 A_z &\hat{\mathbf z}

\end{align}</math>

<math>\begin{align}

\left(\nabla^2 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_\varphi}{\partial \varphi\right) &\hat{\mathbf r} \\

+ \left(\nabla^2 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_\varphi}{\partial \varphi}\right) &\hat{\boldsymbol \theta} \\

+ \left(\nabla^2 A_\varphi - \frac{A_\varphi}{r^2\sin^2\theta}

+ \frac{2}{r^2\sin\theta} \frac{\partial A_r}{\partial \varphi}

+ \frac{2 \cos\theta}{r^2\sin^2\theta} \frac{\partial A_\theta}{\partial \varphi}\right) &\hat{\boldsymbol \varphi}

\end{align}</math>

! Directional derivative

| <math>\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</math>

|<math>\begin{align}

\left(A_\rho \frac{\partial B_\rho}{\partial \rho}+\frac{A_\varphi}{\rho}\frac{\partial B_\rho}{\partial \varphi}+A_z\frac{\partial B_\rho}{\partial z}-\frac{A_\varphi B_\varphi}{\rho}\right)

&\hat{\boldsymbol \rho} \\

+ \left(A_\rho \frac{\partial B_\varphi}{\partial \rho} + \frac{A_\varphi}{\rho}\frac{\partial B_\varphi}{\partial \varphi} + A_z\frac{\partial B_\varphi}{\partial z} + \frac{A_\varphi B_\rho}{\rho}\right)

&\hat{\boldsymbol \varphi}\\

+ \left(A_\rho \frac{\partial B_z}{\partial \rho}+\frac{A_\varphi}{\rho}\frac{\partial B_z}{\partial \varphi}+A_z\frac{\partial B_z}{\partial z}\right)

&\hat{\mathbf z}

\end{align}</math>

<math>\begin{align}

\left(

A_r \frac{\partial B_r}{\partial r}

+ \frac{A_\theta}{r} \frac{\partial B_r}{\partial \theta}

+ \frac{A_\varphi}{r\sin\theta} \frac{\partial B_r}{\partial \varphi}

- \frac{A_\theta B_\theta + A_\varphi B_\varphi}{r}

\right) &\hat{\mathbf r} \\

+ \left(

A_r \frac{\partial B_\theta}{\partial r}

+ \frac{A_\theta}{r} \frac{\partial B_\theta}{\partial \theta}

+ \frac{A_\varphi}{r\sin\theta} \frac{\partial B_\theta}{\partial \varphi}

+ \frac{A_\theta B_r}{r} - \frac{A_\varphi B_\varphi\cot\theta}{r}

\right) &\hat{\boldsymbol \theta} \\

+ \left(

A_r \frac{\partial B_\varphi}{\partial r}

+ \frac{A_\theta}{r} \frac{\partial B_\varphi}{\partial \theta}

+ \frac{A_\varphi}{r\sin\theta} \frac{\partial B_\varphi}{\partial \varphi}

+ \frac{A_\varphi B_r}{r}

+ \frac{A_\varphi B_\theta \cot\theta}{r}

\right) &\hat{\boldsymbol \varphi}

\end{align}</math>

! Tensor divergence

<math>\begin{align}

\left(\frac{\partial T_{xx{\partial x}+\frac{\partial T_{yx{\partial y}+\frac{\partial T_{zx{\partial z}\right)&\hat{\mathbf x} \\

+\left(\frac{\partial T_{xy{\partial x}+\frac{\partial T_{yy{\partial y}+\frac{\partial T_{zy{\partial z}\right)&\hat{\mathbf y} \\

+\left(\frac{\partial T_{xz{\partial x}+\frac{\partial T_{yz{\partial y}+\frac{\partial T_{zz{\partial z}\right)&\hat{\mathbf z}

\end{align}</math>

<math>\begin{align}

\left[\frac{\partial T_{\rho\rho{\partial\rho}+\frac1\rho\frac{\partial T_{\varphi\rho{\partial\varphi}+\frac{\partial T_{z\rho{\partial z}+\frac1\rho(T_{\rho\rho}-T_{\varphi\varphi})\right]&\hat{\boldsymbol\rho} \\

+\left[\frac{\partial T_{\rho\varphi{\partial\rho}+\frac1\rho\frac{\partial T_{\varphi\varphi{\partial\varphi}+\frac{\partial T_{z\varphi{\partial z}+\frac1\rho(T_{\rho\varphi}+T_{\varphi\rho})\right]&\hat{\boldsymbol\varphi} \\

+\left[\frac{\partial T_{\rho z{\partial\rho}+\frac1\rho\frac{\partial T_{\varphi z{\partial\varphi}+\frac{\partial T_{zz{\partial z}+\frac{T_{\rho z\rho\right]&\hat{\mathbf z}

\end{align}</math>

<math>\begin{align}

\left[\frac{\partial T_{rr{\partial r}+2\frac{T_{rrr+\frac1r\frac{\partial T_{\theta r{\partial\theta}+\frac{\cot\theta}rT_{\theta r}+\frac1{r\sin\theta}\frac{\partial T_{\varphi r{\partial\varphi}-\frac1r(T_{\theta\theta}+T_{\varphi\varphi})\right]&\hat{\mathbf r} \\

+\left[\frac{\partial T_{r\theta{\partial r}+2\frac{T_{r\thetar+\frac1r\frac{\partial T_{\theta\theta{\partial\theta}+\frac{\cot\theta}rT_{\theta\theta}+\frac1{r\sin\theta}\frac{\partial T_{\varphi\theta{\partial\varphi}+\frac{T_{\theta rr-\frac{\cot\theta}rT_{\varphi\varphi}\right]&\hat{\boldsymbol\theta} \\

+\left[\frac{\partial T_{r\varphi{\partial r}+2\frac{T_{r\varphir+\frac1r\frac{\partial T_{\theta\varphi{\partial\theta}+\frac1{r\sin\theta}\frac{\partial T_{\varphi\varphi{\partial\varphi}+\frac {T_{\varphi r{r}+\frac{\cot\theta}{r} (T_{\theta\varphi}+T_{\varphi\theta})\right]&\hat{\boldsymbol\varphi}

\end{align}</math>

</div>

Differential elements

{| class="wikitable" style="text-align: center;"

! Operation

! Cartesian coordinates

! Cylindrical coordinates

! Spherical coordinates

! Differential displacement )

Cartesian derivation

File:Nabla cartesian.svg

<math display="block">\begin{align}

\operatorname{div} \mathbf A &= \lim_{V\to 0} \frac{\iint_{\partial V} \mathbf A \cdot d\mathbf{S{\iiint_V dV} \\[1ex]

&= \frac{\left[A_x(x{+}dx) - A_x(x)\right] dy\,dz + \left[A_y(y{+}dy) - A_y(y)\right] dx\,dz + \left[A_z(z{+}dz) - A_z(z)\right]dx\,dy}{dx\,dy\,dz} \\

&= \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}

\end{align}</math>

<math display="block">\begin{align}

(\operatorname{curl} \mathbf A)_x &= \lim_{S^{\perp \mathbf{\hat x\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell{\iint_{S} dS} \\[1ex]

&= \frac{\left[A_z(y{+}dy) - A_z(y)\right] dz - \left[A_y(z{+}dz) - A_y(z)\right] dy }{dy\,dz} \\

&= \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}

\end{align}</math>

The expressions for <math>(\operatorname{curl} \mathbf A)_y</math> and <math>(\operatorname{curl} \mathbf A)_z</math> are found in the same way.

Cylindrical derivation

File:Nabla cylindrical2.svg

<math display="block">\begin{align}

\operatorname{div} \mathbf A &= \lim_{V\to 0} \frac{\iint_{\partial V} \mathbf A \cdot d\mathbf{S{\iiint_V dV} \\

&= \frac{\left[A_\rho(\rho{+}d\rho)(\rho{+}d\rho) - A_\rho(\rho)\rho \right] d\phi \,dz + \left[A_\phi(\phi{+}d\phi) - A_\phi(\phi)\right] d\rho\, dz + \left[A_z(z{+}dz) - A_z(z)\right]d\rho (\rho{+}d\rho/2)\, d\phi}{\rho \,d\phi \,d\rho\, dz} \\

&= \frac 1 \rho \frac{\partial (\rho A_\rho)}{\partial \rho} + \frac 1 \rho \frac{\partial A_\phi}{\partial \phi} + \frac{\partial A_z}{\partial z}

\end{align}</math>

<math display="block">\begin{align}

(\operatorname{curl} \mathbf A)_\rho

&= \lim_{S^{\perp \hat{\boldsymbol \rho\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\boldsymbol{\ell{\iint_{S} dS} \\[1ex]

&= \frac{A_\phi (z) \left(\rho+d\rho\right)\,d\phi - A_\phi(z+dz) \left(\rho+d\rho\right)\,d\phi + A_z(\phi + d\phi)\,dz - A_z(\phi)\,dz}{\left(\rho+d\rho\right)\,d\phi \,dz} \\[1ex]

&= -\frac{\partial A_\phi}{\partial z} + \frac{1}{\rho} \frac{\partial A_z}{\partial \phi}

\end{align}</math>

<math display="block">\begin{align}

(\operatorname{curl} \mathbf A)_\phi &= \lim_{S^{\perp \boldsymbol{\hat \phi\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\boldsymbol{\ell{\iint_{S} dS} \\

&= \frac{A_z (\rho)\,dz - A_z(\rho + d\rho)\,dz + A_\rho(z+dz)\,d\rho - A_\rho(z)\,d\rho}{d\rho \,dz} \\

&= -\frac{\partial A_z}{\partial \rho} + \frac{\partial A_\rho}{\partial z}

\end{align}</math>

<math display="block">\begin{align}

(\operatorname{curl} \mathbf A)_z &= \lim_{S^{\perp \hat{\boldsymbol z\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell{\iint_{S} dS} \\[1ex]

&= \frac{A_\rho(\phi)\,d\rho - A_\rho(\phi + d\phi)\,d\rho + A_\phi(\rho + d\rho)(\rho + d\rho)\,d\phi - A_\phi(\rho)\rho \,d\phi}{\rho \,d\rho \,d\phi} \\[1ex]

&= -\frac{1}{\rho}\frac{\partial A_\rho}{\partial \phi} + \frac{1}{\rho} \frac{\partial (\rho A_\phi)}{\partial \rho}

\end{align}</math>

<math display="block">\begin{align}

\operatorname{curl} \mathbf A &= (\operatorname{curl} \mathbf A)_\rho \hat{\boldsymbol \rho} + (\operatorname{curl} \mathbf A)_\phi \hat{\boldsymbol \phi} + (\operatorname{curl} \mathbf A)_z \hat{\boldsymbol z} \\[1ex]

&= \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 (\rho A_\phi)}{\partial \rho} - \frac{\partial A_\rho}{\partial \phi} \right) \hat{\boldsymbol z}

\end{align}</math>

Spherical derivation

File:Nabla spherical2.svg

<math display="block">\begin{align}

\operatorname{div} \mathbf A &= \lim_{V\to 0} \frac{\iint_{\partial V} \mathbf A \cdot d\mathbf{S{\iiint_V dV} \\

&= \frac{\left[A_r(r{+}dr) (r{+}dr)^2 - A_r(r)r^2\right] \sin\theta \, d\theta\, d\phi + \left[A_\theta(\theta{+}d\theta)\sin(\theta{+}d\theta) - A_\theta(\theta) \sin\theta\right]r \,dr \,d\phi + \left[A_\phi(\phi{+}d\phi) - A_\phi(\phi)\right]r\,dr \,d\theta}{dr\,r\,d\theta\,r\sin\theta\, d\phi} \\

&= \frac{1}{r^2}\frac{\partial (r^2A_r)}{\partial r} + \frac{1}{r \sin\theta} \frac{\partial(A_\theta\sin\theta)}{\partial \theta} + \frac{1}{r \sin\theta} \frac{\partial A_\phi}{\partial \phi}

\end{align}</math>

<math display="block">\begin{align}

(\operatorname{curl} \mathbf A)_r &= \lim_{S^{\perp \boldsymbol{\hat r\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell{\iint_{S} dS} \\[1ex]

&= \frac{A_\theta(\phi)r \,d\theta + A_\phi(\theta + d\theta)r \sin(\theta + d\theta)\, d\phi

- A_\theta(\phi + d\phi)r \,d\theta - A_\phi(\theta)r\sin(\theta)\, d\phi}{r\, d\theta\,r\sin\theta \,d\phi} \\

&= \frac{1}{r\sin\theta}\frac{\partial(A_\phi \sin\theta)}{\partial \theta}

- \frac{1}{r\sin\theta} \frac{\partial A_\theta}{\partial \phi}

\end{align}</math>

<math display="block">\begin{align}

(\operatorname{curl} \mathbf A)_\theta &= \lim_{S^{\perp \boldsymbol{\hat \theta\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell{\iint_{S} dS} \\[1ex]

&= \frac{A_\phi(r)r \sin\theta \,d\phi + A_r(\phi + d\phi)\,dr

- A_\phi(r+dr)(r+dr)\sin\theta \,d\phi - A_r(\phi)\,dr}{dr \, r \sin \theta \,d\phi} \\

&= \frac{1}{r\sin\theta}\frac{\partial A_r}{\partial \phi}

- \frac{1}{r} \frac{\partial (rA_\phi)}{\partial r}

\end{align}</math>

<math display="block">\begin{align}

(\operatorname{curl} \mathbf A)_\phi &= \lim_{S^{\perp \boldsymbol{\hat \phi\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell{\iint_{S} dS} \\[1ex]

&= \frac{A_r(\theta)\,dr + A_\theta(r+dr)(r+dr)\,d\theta

- A_r(\theta+d\theta)\,dr - A_\theta(r) r \,d\theta}{r\,dr\, d\theta} \\

&= \frac{1}{r}\frac{\partial(rA_\theta)}{\partial r}

- \frac{1}{r} \frac{\partial A_r}{\partial \theta}

\end{align}</math>

<math display="block">\begin{align}

\operatorname{curl} \mathbf A

&= (\operatorname{curl} \mathbf A)_r \, \hat{\boldsymbol r} + (\operatorname{curl} \mathbf A)_\theta \, \hat{\boldsymbol \theta} + (\operatorname{curl} \mathbf A)_\phi \, \hat{\boldsymbol \phi} \\[1ex]

&= \frac{1}{r\sin\theta} \left(\frac{\partial(A_\phi \sin\theta)}{\partial \theta}-\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 (rA_\phi)}{\partial r} \right) \hat{\boldsymbol \theta} + \frac{1}{r}\left(\frac{\partial(rA_\theta)}{\partial r} - \frac{\partial A_r}{\partial \theta} \right) \hat{\boldsymbol \phi}

\end{align}</math>

Unit vector conversion formula

The unit vector of a coordinate parameter u is defined in such a way that a small positive change in u causes the position vector <math>\mathbf r</math> to change in <math>\mathbf u</math> direction.

Therefore,

<math display="block">\frac{\partial {\mathbf r{\partial u} = \frac{\partial{s{\partial u} \mathbf u</math>

where is the arc length parameter.

For two sets of coordinate systems <math>u_i</math> and <math>v_j</math>, according to chain rule,

<math display="block">\begin{align}

d\mathbf r &= \sum_{i} \frac{\partial \mathbf r}{\partial u_i} \, du_i

= \sum_{i} \frac{\partial s}{\partial u_i} \hat{\mathbf u}_i du_i

= \sum_{j} \frac{\partial s}{\partial v_j} \hat{\mathbf v}_j \, dv_j \\

&= \sum_{j} \frac{\partial s}{\partial v_j} \hat{\mathbf v}_j \sum_{i} \frac{\partial v_j}{\partial u_i} \, du_i \\

&= \sum_{i} \sum_{j} \frac{\partial s}{\partial v_j} \frac{\partial v_j}{\partial u_i} \hat{\mathbf v}_j \, du_i.

\end{align}</math>

Now, we isolate the <math>i</math>th component. For <math>i{\neq}k</math>, let <math>\mathrm d u_k=0</math>. Then divide on both sides by <math>\mathrm d u_i</math> to get:

<math display="block">\frac{\partial s}{\partial u_i} \hat{\mathbf u}_i = \sum_{j} \frac{\partial s}{\partial v_j} \frac{\partial v_j}{\partial u_i} \hat{\mathbf v}_j.</math>

References

External links

Maxima Computer Algebra system scripts to generate some of these operators in cylindrical and spherical coordinates.

Notes

Coordinate conversions

Unit vector conversions

Differential elements

Cartesian derivation

Cylindrical derivation

Spherical derivation

Unit vector conversion formula

See also

References

External links