Del in cylindrical and spherical coordinates

{{Short description|Mathematical gradient operator in certain coordinate systems}}

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 $\theta \in [0, \pi]$ : 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 $\varphi \in [0, 2\pi]$ : it is the angle between the x-axis and the projection of the radial vector onto the xy-plane.
The function {{nowrap|atan2(y, x)}} can be used instead of the mathematical function {{nowrap|arctan(y/x)}} owing to its domain and image. The classical arctan function has an image of {{nowrap|(−π/2, +π/2)}}, whereas atan2 is defined to have an image of {{nowrap|(−π, π]}}.

Coordinate conversions

class="wikitable" \|+ Conversion between Cartesian, cylindrical, and spherical coordinates{{Cite book\|title=Introduction to Electrodynamics\|last=Griffiths\|first=David J.\|publisher=Pearson\|year=2012\|isbn=978-0-321-85656-2}} ! colspan="2" rowspan="2" \| ! colspan="3" \| From
Cartesian ! Cylindrical ! Spherical
rowspan="3" \|To ! 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\left(\frac{y}{x}\right) \\ 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 &= \arctan\left(\frac{\sqrt{x^2 + y^2}}{z}\right) \\ \varphi &= \arctan\left(\frac{y}{x}\right) \end{align} \| $\begin{align}$ r &= \sqrt{\rho^2 + z^2} \\ \theta &= \arctan{\left(\frac{\rho}{z}\right)} \\ \varphi &= \varphi \end{align} \| $\begin{align}$ r&=r\\\theta &=\theta \\\varphi &=\varphi \end{align}

class="wikitable"

|+ Conversion between Cartesian, cylindrical, and spherical coordinates{{Cite book|title=Introduction to Electrodynamics|last=Griffiths|first=David J.|publisher=Pearson|year=2012|isbn=978-0-321-85656-2}}

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

! colspan="3" | From

Cartesian

! Cylindrical

! Spherical

rowspan="3" |To

! 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\left(\frac{y}{x}\right) \\

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

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

\end{align}

| $\begin{align}$

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

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

\varphi &= \varphi

\end{align}

| $\begin{align}$

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

\end{align}

Note that the operation $\arctan\left(\frac{A}{B}\right)$ 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
! 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{\left(x \hat{\mathbf x} + y \hat{\mathbf y}\right) z - \left(x^2 + y^2\right) \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}

class="wikitable"

|+ 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{\left(x \hat{\mathbf x} + y \hat{\mathbf y}\right) z - \left(x^2 + y^2\right) \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}

class="wikitable" \|+ 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 \left(\sqrt{x^2 + y^2} \hat{\mathbf r} + z \hat{\boldsymbol \theta}\right) - 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 \left(\sqrt{x^2 + y^2} \hat{\mathbf r} + z \hat{\boldsymbol \theta}\right) + 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}

class="wikitable"

|+ 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 \left(\sqrt{x^2 + y^2} \hat{\mathbf r} + z \hat{\boldsymbol \theta}\right) - 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 \left(\sqrt{x^2 + y^2} \hat{\mathbf r} + z \hat{\boldsymbol \theta}\right) + 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 formula

class="wikitable" style="text-align: center;" \|+ Table with the del operator in cartesian, cylindrical and spherical coordinates
Operation ! Cartesian coordinates {{math\|(x, y, z)}} ! Cylindrical coordinates {{math\|(ρ, φ, z)}} ! Spherical coordinates {{math\|(r, θ, φ)}}, where {{math\|θ}} is the polar angle and {{math\|φ}} is the azimuthal angle{{ref\|Alpha\|α}}
Vector field {{math\|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 {{math\|∇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 {{math\|∇ ⋅ 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 {{math\|∇ × 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 {{math\|∇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 gradient {{math\|∇A}}{{ref\|Beta\|β}} \| $\begin{align}{}&\frac{\partial A_x}{\partial x} \hat{\mathbf x} \otimes \hat{\mathbf x} + \frac{\partial A_x}{\partial y} \hat{\mathbf x} \otimes \hat{\mathbf y} + \frac{\partial A_x}{\partial z} \hat{\mathbf x} \otimes \hat{\mathbf z} \\ {}+ &\frac{\partial A_y}{\partial x} \hat{\mathbf y} \otimes \hat{\mathbf x} + \frac{\partial A_y}{\partial y} \hat{\mathbf y} \otimes \hat{\mathbf y} + \frac{\partial A_y}{\partial z} \hat{\mathbf y} \otimes \hat{\mathbf z} \\ {}+ &\frac{\partial A_z}{\partial x} \hat{\mathbf z} \otimes \hat{\mathbf x} + \frac{\partial A_z}{\partial y} \hat{\mathbf z} \otimes \hat{\mathbf y} + \frac{\partial A_z}{\partial z} \hat{\mathbf z} \otimes \hat{\mathbf z}\end{align}$ \| $\begin{align}{}&\frac{\partial A_\rho}{\partial \rho} \hat{\boldsymbol \rho} \otimes \hat{\boldsymbol \rho} + \left(\frac{1}{\rho}\frac{\partial A_\rho}{\partial \varphi}-\frac{A_\varphi}{\rho}\right) \hat{\boldsymbol \rho} \otimes \hat{\boldsymbol \varphi} + \frac{\partial A_\rho}{\partial z} \hat{\boldsymbol \rho} \otimes \hat{\mathbf z} \\ {}+ &\frac{\partial A_\varphi}{\partial \rho} \hat{\boldsymbol \varphi} \otimes \hat{\boldsymbol \rho} + \left(\frac{1}{\rho}\frac{\partial A_\varphi}{\partial \varphi}+\frac{A_\rho}{\rho}\right) \hat{\boldsymbol \varphi} \otimes \hat{\boldsymbol \varphi} + \frac{\partial A_\varphi}{\partial z} \hat{\boldsymbol \varphi} \otimes \hat{\mathbf z} \\ {}+ &\frac{\partial A_z}{\partial \rho} \hat{\mathbf z} \otimes \hat{\boldsymbol \rho} + \frac{1}{\rho}\frac{\partial A_z}{\partial \varphi} \hat{\mathbf z} \otimes \hat{\boldsymbol \varphi} + \frac{\partial A_z}{\partial z} \hat{\mathbf z} \otimes \hat{\mathbf z}\end{align}$ \| $\begin{align}{}&\frac{\partial A_r}{\partial r} \hat{\mathbf r} \otimes \hat{\mathbf r} + \left(\frac{1}{r}\frac{\partial A_r}{\partial \theta}-\frac{A_\theta}{r}\right) \hat{\mathbf r} \otimes \hat{\boldsymbol \theta} + \left(\frac{1}{r \sin\theta}\frac{\partial A_r}{\partial \varphi} - \frac{A_\varphi}{r}\right) \hat{\mathbf r} \otimes \hat{\boldsymbol \varphi} \\ {}+ &\frac{\partial A_\theta}{\partial r} \hat{\boldsymbol \theta} \otimes \hat{\mathbf r} + \left(\frac{1}{r}\frac{\partial A_\theta}{\partial \theta}+\frac{A_r}{r}\right) \hat{\boldsymbol \theta} \otimes \hat{\boldsymbol \theta} + \left(\frac{1}{r \sin\theta}\frac{\partial A_\theta}{\partial \varphi} - \cot\theta \frac{A_\varphi}{r}\right) \hat{\boldsymbol \theta} \otimes \hat{\boldsymbol \varphi} \\ {}+ &\frac{\partial A_\varphi}{\partial r} \hat{\boldsymbol \varphi} \otimes \hat{\mathbf r} + \frac{1}{r}\frac{\partial A_\varphi}{\partial \theta} \hat{\boldsymbol \varphi} \otimes \hat{\boldsymbol \theta} + \left(\frac{1}{r \sin\theta}\frac{\partial A_\varphi}{\partial \varphi} + \cot\theta \frac{A_\theta}{r} + \frac{A_r}{r}\right) \hat{\boldsymbol \varphi} \otimes \hat{\boldsymbol \varphi}\end{align}$
Vector Laplacian {{math\|∇²A ≡ ∆A}}{{cite book \|last1=Arfken \|first1=George \|last2=Weber \|first2=Hans \|last3=Harris \|first3=Frank \|title=Mathematical Methods for Physicists \|date=2012 \|publisher=Academic Press \|isbn=9789381269558 \|page=192 \|edition=Seventh \|ref=arfkenweber}} \| $\nabla^2 A_x \hat{\mathbf x} + \nabla^2 A_y \hat{\mathbf y} + \nabla^2 A_z \hat{\mathbf z}$ \| $\begin{align}$ \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} \| $\begin{align}$ \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}
Directional derivative {{math\|(A ⋅ ∇)B}}{{cite web \|url=http://mathworld.wolfram.com/ConvectiveOperator.html\|title=Convective Operator \|author=Weisstein, Eric W. \|work=Mathworld \|access-date=23 March 2011}} \| $\mathbf{A} \cdot \nabla B_x \hat{\mathbf x} + \mathbf{A} \cdot \nabla B_y \hat{\mathbf y} + \mathbf{A} \cdot \nabla B_z \hat{\mathbf{z}}$ \| $\begin{align}$ \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} \| $\begin{align}$ \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}
Tensor divergence {{math\|∇ ⋅ T}}{{ref\|Gamma\|γ}} \| $\begin{align}$ \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} \| $\begin{align}$ \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} \| $\begin{align}$ \left[\frac{\partial T_{rr}}{\partial r}+2\frac{T_{rr}}r+\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\theta}}r+\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 r}}r-\frac{\cot\theta}rT_{\varphi\varphi}\right]&\hat{\boldsymbol\theta} \\ +\left[\frac{\partial T_{r\varphi}}{\partial r}+2\frac{T_{r\varphi}}r+\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}
Differential displacement {{math\|dℓ}} \| $dx \, \hat{\mathbf x} + dy \, \hat{\mathbf y} + dz \, \hat{\mathbf z}$ \| $d\rho \, \hat{\boldsymbol \rho} + \rho \, d\varphi \, \hat{\boldsymbol \varphi} + dz \, \hat{\mathbf z}$ \| $dr \, \hat{\mathbf r} + r \, d\theta \, \hat{\boldsymbol \theta} + r \, \sin\theta \, d\varphi \, \hat{\boldsymbol \varphi}$
Differential normal area {{math\|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 {{math\|dV}} \| $dx \, dy \, dz$ \| $\rho \, d\rho \, d\varphi \, dz$ \| $r^2 \sin\theta \, dr \, d\theta \, d\varphi$

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

|+ Table with the del operator in cartesian, cylindrical and spherical coordinates

Operation

! Cartesian coordinates {{math|(x, y, z)}}

! Cylindrical coordinates {{math|(ρ, φ, z)}}

! Spherical coordinates {{math|(r, θ, φ)}},
where {{math|θ}} is the polar angle and {{math|φ}} is the azimuthal angle{{ref|Alpha|α}}

Vector field {{math|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 {{math|∇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 {{math|∇ ⋅ 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 {{math|∇ × 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 {{math|∇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 gradient {{math|∇A}}{{ref|Beta|β}}

| $\begin{align}{}&\frac{\partial A_x}{\partial x} \hat{\mathbf x} \otimes \hat{\mathbf x} + \frac{\partial A_x}{\partial y} \hat{\mathbf x} \otimes \hat{\mathbf y} + \frac{\partial A_x}{\partial z} \hat{\mathbf x} \otimes \hat{\mathbf z} \\ {}+ &\frac{\partial A_y}{\partial x} \hat{\mathbf y} \otimes \hat{\mathbf x} + \frac{\partial A_y}{\partial y} \hat{\mathbf y} \otimes \hat{\mathbf y} + \frac{\partial A_y}{\partial z} \hat{\mathbf y} \otimes \hat{\mathbf z} \\ {}+ &\frac{\partial A_z}{\partial x} \hat{\mathbf z} \otimes \hat{\mathbf x} + \frac{\partial A_z}{\partial y} \hat{\mathbf z} \otimes \hat{\mathbf y} + \frac{\partial A_z}{\partial z} \hat{\mathbf z} \otimes \hat{\mathbf z}\end{align}$

| $\begin{align}{}&\frac{\partial A_\rho}{\partial \rho} \hat{\boldsymbol \rho} \otimes \hat{\boldsymbol \rho} + \left(\frac{1}{\rho}\frac{\partial A_\rho}{\partial \varphi}-\frac{A_\varphi}{\rho}\right) \hat{\boldsymbol \rho} \otimes \hat{\boldsymbol \varphi} + \frac{\partial A_\rho}{\partial z} \hat{\boldsymbol \rho} \otimes \hat{\mathbf z} \\ {}+ &\frac{\partial A_\varphi}{\partial \rho} \hat{\boldsymbol \varphi} \otimes \hat{\boldsymbol \rho} + \left(\frac{1}{\rho}\frac{\partial A_\varphi}{\partial \varphi}+\frac{A_\rho}{\rho}\right) \hat{\boldsymbol \varphi} \otimes \hat{\boldsymbol \varphi} + \frac{\partial A_\varphi}{\partial z} \hat{\boldsymbol \varphi} \otimes \hat{\mathbf z} \\ {}+ &\frac{\partial A_z}{\partial \rho} \hat{\mathbf z} \otimes \hat{\boldsymbol \rho} + \frac{1}{\rho}\frac{\partial A_z}{\partial \varphi} \hat{\mathbf z} \otimes \hat{\boldsymbol \varphi} + \frac{\partial A_z}{\partial z} \hat{\mathbf z} \otimes \hat{\mathbf z}\end{align}$

| $\begin{align}{}&\frac{\partial A_r}{\partial r} \hat{\mathbf r} \otimes \hat{\mathbf r} + \left(\frac{1}{r}\frac{\partial A_r}{\partial \theta}-\frac{A_\theta}{r}\right) \hat{\mathbf r} \otimes \hat{\boldsymbol \theta} + \left(\frac{1}{r \sin\theta}\frac{\partial A_r}{\partial \varphi} - \frac{A_\varphi}{r}\right) \hat{\mathbf r} \otimes \hat{\boldsymbol \varphi} \\ {}+ &\frac{\partial A_\theta}{\partial r} \hat{\boldsymbol \theta} \otimes \hat{\mathbf r} + \left(\frac{1}{r}\frac{\partial A_\theta}{\partial \theta}+\frac{A_r}{r}\right) \hat{\boldsymbol \theta} \otimes \hat{\boldsymbol \theta} + \left(\frac{1}{r \sin\theta}\frac{\partial A_\theta}{\partial \varphi} - \cot\theta \frac{A_\varphi}{r}\right) \hat{\boldsymbol \theta} \otimes \hat{\boldsymbol \varphi} \\ {}+ &\frac{\partial A_\varphi}{\partial r} \hat{\boldsymbol \varphi} \otimes \hat{\mathbf r} + \frac{1}{r}\frac{\partial A_\varphi}{\partial \theta} \hat{\boldsymbol \varphi} \otimes \hat{\boldsymbol \theta} + \left(\frac{1}{r \sin\theta}\frac{\partial A_\varphi}{\partial \varphi} + \cot\theta \frac{A_\theta}{r} + \frac{A_r}{r}\right) \hat{\boldsymbol \varphi} \otimes \hat{\boldsymbol \varphi}\end{align}$

Vector Laplacian {{math|∇²A ≡ ∆A}}{{cite book |last1=Arfken |first1=George |last2=Weber |first2=Hans |last3=Harris |first3=Frank |title=Mathematical Methods for Physicists |date=2012 |publisher=Academic Press |isbn=9789381269558 |page=192 |edition=Seventh |ref=arfkenweber}}

| $\nabla^2 A_x \hat{\mathbf x} + \nabla^2 A_y \hat{\mathbf y} + \nabla^2 A_z \hat{\mathbf z}$

$\begin{align}$

\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}

$\begin{align}$

\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}

Directional derivative {{math|(A ⋅ ∇)B}}{{cite web |url=http://mathworld.wolfram.com/ConvectiveOperator.html|title=Convective Operator |author=Weisstein, Eric W. |work=Mathworld |access-date=23 March 2011}}

| $\mathbf{A} \cdot \nabla B_x \hat{\mathbf x} + \mathbf{A} \cdot \nabla B_y \hat{\mathbf y} + \mathbf{A} \cdot \nabla B_z \hat{\mathbf{z}}$

| $\begin{align}$

\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}

$\begin{align}$

\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}

Tensor divergence {{math|∇ ⋅ T}}{{ref|Gamma|γ}}

$\begin{align}$

\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}

$\begin{align}$

\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}

$\begin{align}$

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

+\left[\frac{\partial T_{r\varphi}}{\partial r}+2\frac{T_{r\varphi}}r+\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}

Differential displacement {{math|dℓ}}

| $dx \, \hat{\mathbf x} + dy \, \hat{\mathbf y} + dz \, \hat{\mathbf z}$

| $d\rho \, \hat{\boldsymbol \rho} + \rho \, d\varphi \, \hat{\boldsymbol \varphi} + dz \, \hat{\mathbf z}$

| $dr \, \hat{\mathbf r} + r \, d\theta \, \hat{\boldsymbol \theta} + r \, \sin\theta \, d\varphi \, \hat{\boldsymbol \varphi}$

Differential normal area {{math|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 {{math|dV}}

| $dx \, dy \, dz$

| $\rho \, d\rho \, d\varphi \, dz$

| $r^2 \sin\theta \, dr \, d\theta \, d\varphi$

:{{note|Alpha|α}} This page uses $\theta$ for the polar angle and $\varphi$ for the azimuthal angle, which is common notation in physics. The source that is used for these formulae uses $\theta$ for the azimuthal angle and $\varphi$ for the polar angle, which is common mathematical notation. In order to get the mathematics formulae, switch $\theta$ and $\varphi$ in the formulae shown in the table above.

:{{note|Beta|β}} Defined in Cartesian coordinates as $\partial_i \mathbf{A} \otimes \mathbf{e}_i$ . An alternative definition is $\mathbf{e}_i \otimes \partial_i \mathbf{A}$ .

:{{note|Gamma|γ}} Defined in Cartesian coordinates as $\mathbf{e}_i \cdot \partial_i \mathbf{T}$ . An alternative definition is $\partial_i \mathbf{T} \cdot \mathbf{e}_i$ .

= Calculation rules =

$\operatorname{div} \, \operatorname{grad} f \equiv \nabla \cdot \nabla f \equiv \nabla^2 f$
$\operatorname{curl} \, \operatorname{grad} f \equiv \nabla \times \nabla f = \mathbf 0$
$\operatorname{div} \, \operatorname{curl} \mathbf{A} \equiv \nabla \cdot (\nabla \times \mathbf{A}) = 0$
$\operatorname{curl} \, \operatorname{curl} \mathbf{A} \equiv \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$ (Lagrange's formula for del)
$\nabla^2 (f g) = f \nabla^2 g + 2 \nabla f \cdot \nabla g + g \nabla^2 f$
$\nabla^{2}\left(\mathbf{P}\cdot\mathbf{Q}\right)=\mathbf{Q}\cdot\nabla^{2}\mathbf{P}-\mathbf{P}\cdot\nabla^{2}\mathbf{Q}+2\nabla\cdot\left[\left(\mathbf{P}\cdot\nabla\right)\mathbf{Q}+\mathbf{P}\times\nabla\times\mathbf{Q}\right]\quad$ (From {{cite journal | last=Fernández-Guasti | first=M. | title=Green's Second Identity for Vector Fields | journal=ISRN Mathematical Physics | publisher=Hindawi Limited | volume=2012 | year=2012 | issn=2090-4681 | doi=10.5402/2012/973968 | pages=1–7| doi-access=free }} )

Cartesian derivation

File:Nabla cartesian.svg

$\begin{align}$

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

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

$\begin{align}$

(\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}

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

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

\end{align}

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

Cylindrical derivation

File:Nabla cylindrical2.svg

$\begin{align}$

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

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

$\begin{align}$

(\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}

$\begin{align}$

(\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}

$\begin{align}$

(\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}

$\begin{align}$

\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}

Spherical derivation

File:Nabla spherical2.svg

$\begin{align}$

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

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

$\begin{align}$

(\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}

&= \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}

$\begin{align}$

(\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}

&= \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}

$\begin{align}$

(\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}

&= \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}

$\begin{align}$

\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}

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 $\mathbf r$ to change in $\mathbf u$ direction.

Therefore,

$\frac{\partial {\mathbf r}}{\partial u} = \frac{\partial{s}}{\partial u} \mathbf u$

where {{mvar|s}} is the arc length parameter.

For two sets of coordinate systems $u_i$ and $v_j$ , according to chain rule,

$d\mathbf r = \sum_{i} \frac{\partial \mathbf r}{\partial u_i} \, du_i = \sum_{i} \frac{\partial s}{\partial u_i} \hat{\mathbf u}_i du_i = \sum_{j} \frac{\partial s}{\partial v_j} \hat{\mathbf v}_j \, dv_j = \sum_{j}\frac{\partial s}{\partial v_j} \hat{\mathbf v}_j \sum_{i} \frac{\partial v_j}{\partial u_i} \, du_i = \sum_{i} \sum_{j} \frac{\partial s}{\partial v_j} \frac{\partial v_j}{\partial u_i} \hat{\mathbf v}_j \, du_i.$

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

$\frac{\partial s}{\partial u_i} \hat{\mathbf u}_i = \sum_{j} \frac{\partial s}{\partial v_j} \frac{\partial v_j}{\partial u_i} \hat{\mathbf v}_j.$

References

External links

[http://www.csulb.edu/~woollett/ Maxima Computer Algebra system scripts] to generate some of these operators in cylindrical and spherical coordinates.

Category:Vector calculus

Category:Coordinate systems

Del in cylindrical and spherical coordinates

Notes

Coordinate conversions

Unit vector conversions

Del formula

= Calculation rules =

Cartesian derivation

Cylindrical derivation

Spherical derivation

Unit vector conversion formula

See also

References

External links