Vector Differential Operators by Coordinate System

Vector Calculus in Curvilinear Coordinates

1. Cartesian Coordinates

Coordinates and Basis

Coordinates: $(x,\,y,\,z)$

Basis vectors: $\mathbf{e}_x,\;\mathbf{e}_y,\;\mathbf{e}_z$ (constant vectors)

Line Element and Volume Element

$$d\boldsymbol{\ell} = dx\,\mathbf{e}_x + dy\,\mathbf{e}_y + dz\,\mathbf{e}_z$$ $$dV = dx\,dy\,dz$$

Gradient (grad)

$$\nabla f = \dfrac{\partial f}{\partial x}\,\mathbf{e}_x + \dfrac{\partial f}{\partial y}\,\mathbf{e}_y + \dfrac{\partial f}{\partial z}\,\mathbf{e}_z$$

Divergence (div)

$$\nabla \cdot \mathbf{A} = \dfrac{\partial A_x}{\partial x} + \dfrac{\partial A_y}{\partial y} + \dfrac{\partial A_z}{\partial z}$$

Curl

$$\nabla \times \mathbf{A} = \begin{vmatrix} \mathbf{e}_x & \mathbf{e}_y & \mathbf{e}_z \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ A_x & A_y & A_z \end{vmatrix}$$

that is,

$$\nabla \times \mathbf{A} = \left(\dfrac{\partial A_z}{\partial y} - \dfrac{\partial A_y}{\partial z}\right)\mathbf{e}_x + \left(\dfrac{\partial A_x}{\partial z} - \dfrac{\partial A_z}{\partial x}\right)\mathbf{e}_y + \left(\dfrac{\partial A_y}{\partial x} - \dfrac{\partial A_x}{\partial y}\right)\mathbf{e}_z$$

Laplacian

$$\nabla^2 f = \dfrac{\partial^2 f}{\partial x^2} + \dfrac{\partial^2 f}{\partial y^2} + \dfrac{\partial^2 f}{\partial z^2}$$

2. Cylindrical Coordinates

Coordinates and Transformation

Coordinates: $(\rho,\,\varphi,\,z)$

Transformation: $x = \rho\cos\varphi,\quad y = \rho\sin\varphi,\quad z = z$

Scale factors: $h_\rho = 1,\quad h_\varphi = \rho,\quad h_z = 1$

Derivatives of the Basis Vectors

The basis vectors $\mathbf{e}_\rho,\;\mathbf{e}_\varphi$ in cylindrical coordinates depend on position. The key relations are:

$$\dfrac{\partial \mathbf{e}_\rho}{\partial \varphi} = \mathbf{e}_\varphi, \qquad \dfrac{\partial \mathbf{e}_\varphi}{\partial \varphi} = -\mathbf{e}_\rho$$

All other combinations ($\partial \mathbf{e}_\rho/\partial \rho$, $\partial \mathbf{e}_z/\partial \varphi$, etc.) vanish.

Line Element and Volume Element

$$d\boldsymbol{\ell} = d\rho\,\mathbf{e}_\rho + \rho\,d\varphi\,\mathbf{e}_\varphi + dz\,\mathbf{e}_z$$ $$dV = \rho\,d\rho\,d\varphi\,dz$$

Gradient (grad)

$$\nabla f = \dfrac{\partial f}{\partial \rho}\,\mathbf{e}_\rho + \dfrac{1}{\rho}\dfrac{\partial f}{\partial \varphi}\,\mathbf{e}_\varphi + \dfrac{\partial f}{\partial z}\,\mathbf{e}_z$$

Divergence (div)

$$\nabla \cdot \mathbf{A} = \dfrac{1}{\rho}\dfrac{\partial(\rho A_\rho)}{\partial \rho} + \dfrac{1}{\rho}\dfrac{\partial A_\varphi}{\partial \varphi} + \dfrac{\partial A_z}{\partial z}$$

Curl

$$\nabla \times \mathbf{A} = \left(\dfrac{1}{\rho}\dfrac{\partial A_z}{\partial \varphi} - \dfrac{\partial A_\varphi}{\partial z}\right)\mathbf{e}_\rho + \left(\dfrac{\partial A_\rho}{\partial z} - \dfrac{\partial A_z}{\partial \rho}\right)\mathbf{e}_\varphi + \dfrac{1}{\rho}\left(\dfrac{\partial(\rho A_\varphi)}{\partial \rho} - \dfrac{\partial A_\rho}{\partial \varphi}\right)\mathbf{e}_z$$

Scalar Laplacian

$$\nabla^2 f = \dfrac{1}{\rho}\dfrac{\partial}{\partial \rho}\!\left(\rho\,\dfrac{\partial f}{\partial \rho}\right) + \dfrac{1}{\rho^2}\dfrac{\partial^2 f}{\partial \varphi^2} + \dfrac{\partial^2 f}{\partial z^2}$$

Vector Laplacian

Caution: in cylindrical coordinates, $\nabla^2\mathbf{A} \neq (\nabla^2 A_\rho,\;\nabla^2 A_\varphi,\;\nabla^2 A_z)$. Because derivative terms of the basis vectors are added, the components take the following form.

$$(\nabla^2\mathbf{A})_\rho = \nabla^2 A_\rho - \dfrac{A_\rho}{\rho^2} - \dfrac{2}{\rho^2}\dfrac{\partial A_\varphi}{\partial \varphi}$$ $$(\nabla^2\mathbf{A})_\varphi = \nabla^2 A_\varphi - \dfrac{A_\varphi}{\rho^2} + \dfrac{2}{\rho^2}\dfrac{\partial A_\rho}{\partial \varphi}$$ $$(\nabla^2\mathbf{A})_z = \nabla^2 A_z$$

3. Spherical Coordinates

Notation note: this page follows the physics and ISO 80000-2 convention $(r,\,\theta,\,\varphi)$, i.e. $\theta$ is the polar angle (inclination from the $z$-axis, $0 \le \theta \le \pi$) and $\varphi$ is the azimuthal angle (rotation in the $xy$-plane, $0 \le \varphi < 2\pi$). Some mathematics texts swap the roles of $\theta$ and $\varphi$; check the convention when consulting other references.

Coordinates and Transformation

Coordinates: $(r,\,\theta,\,\varphi)$

Transformation: $x = r\sin\theta\cos\varphi,\quad y = r\sin\theta\sin\varphi,\quad z = r\cos\theta$

Scale factors: $h_r = 1,\quad h_\theta = r,\quad h_\varphi = r\sin\theta$

Derivatives of the Basis Vectors

The non-vanishing derivatives of the spherical basis vectors are:

$$\dfrac{\partial \mathbf{e}_r}{\partial \theta} = \mathbf{e}_\theta, \qquad \dfrac{\partial \mathbf{e}_r}{\partial \varphi} = \sin\theta\,\mathbf{e}_\varphi$$ $$\dfrac{\partial \mathbf{e}_\theta}{\partial \theta} = -\mathbf{e}_r, \qquad \dfrac{\partial \mathbf{e}_\theta}{\partial \varphi} = \cos\theta\,\mathbf{e}_\varphi$$ $$\dfrac{\partial \mathbf{e}_\varphi}{\partial \varphi} = -\sin\theta\,\mathbf{e}_r - \cos\theta\,\mathbf{e}_\theta$$

Line Element and Volume Element

$$d\boldsymbol{\ell} = dr\,\mathbf{e}_r + r\,d\theta\,\mathbf{e}_\theta + r\sin\theta\,d\varphi\,\mathbf{e}_\varphi$$ $$dV = r^2\sin\theta\,dr\,d\theta\,d\varphi$$

Gradient (grad)

$$\nabla f = \dfrac{\partial f}{\partial r}\,\mathbf{e}_r + \dfrac{1}{r}\dfrac{\partial f}{\partial \theta}\,\mathbf{e}_\theta + \dfrac{1}{r\sin\theta}\dfrac{\partial f}{\partial \varphi}\,\mathbf{e}_\varphi$$

Divergence (div)

$$\nabla \cdot \mathbf{A} = \dfrac{1}{r^2}\dfrac{\partial(r^2 A_r)}{\partial r} + \dfrac{1}{r\sin\theta}\dfrac{\partial(\sin\theta\,A_\theta)}{\partial \theta} + \dfrac{1}{r\sin\theta}\dfrac{\partial A_\varphi}{\partial \varphi}$$

Curl

$$(\nabla \times \mathbf{A})_r = \dfrac{1}{r\sin\theta}\left(\dfrac{\partial(\sin\theta\,A_\varphi)}{\partial \theta} - \dfrac{\partial A_\theta}{\partial \varphi}\right)$$ $$(\nabla \times \mathbf{A})_\theta = \dfrac{1}{r}\left(\dfrac{1}{\sin\theta}\dfrac{\partial A_r}{\partial \varphi} - \dfrac{\partial(r A_\varphi)}{\partial r}\right)$$ $$(\nabla \times \mathbf{A})_\varphi = \dfrac{1}{r}\left(\dfrac{\partial(r A_\theta)}{\partial r} - \dfrac{\partial A_r}{\partial \theta}\right)$$

Scalar Laplacian

$$\nabla^2 f = \dfrac{1}{r^2}\dfrac{\partial}{\partial r}\!\left(r^2\dfrac{\partial f}{\partial r}\right) + \dfrac{1}{r^2\sin\theta}\dfrac{\partial}{\partial \theta}\!\left(\sin\theta\,\dfrac{\partial f}{\partial \theta}\right) + \dfrac{1}{r^2\sin^2\theta}\dfrac{\partial^2 f}{\partial \varphi^2}$$

4. General Orthogonal Curvilinear Coordinates

Setup

Coordinates: $(q_1,\,q_2,\,q_3)$, scale factors: $(h_1,\,h_2,\,h_3)$

The scale factors are defined by $h_i = \left|\dfrac{\partial \mathbf{r}}{\partial q_i}\right|$. The line element is

$$d\boldsymbol{\ell} = h_1\,dq_1\,\mathbf{e}_1 + h_2\,dq_2\,\mathbf{e}_2 + h_3\,dq_3\,\mathbf{e}_3$$

and the volume element is $dV = h_1 h_2 h_3\,dq_1\,dq_2\,dq_3$.

Gradient (grad)

$$\nabla f = \displaystyle\sum_{i=1}^{3} \dfrac{1}{h_i}\dfrac{\partial f}{\partial q_i}\,\mathbf{e}_i = \dfrac{1}{h_1}\dfrac{\partial f}{\partial q_1}\,\mathbf{e}_1 + \dfrac{1}{h_2}\dfrac{\partial f}{\partial q_2}\,\mathbf{e}_2 + \dfrac{1}{h_3}\dfrac{\partial f}{\partial q_3}\,\mathbf{e}_3$$

Divergence (div)

$$\nabla \cdot \mathbf{A} = \dfrac{1}{h_1 h_2 h_3}\left[\dfrac{\partial(h_2 h_3 A_1)}{\partial q_1} + \dfrac{\partial(h_1 h_3 A_2)}{\partial q_2} + \dfrac{\partial(h_1 h_2 A_3)}{\partial q_3}\right]$$

Curl

$$\nabla \times \mathbf{A} = \dfrac{1}{h_1 h_2 h_3}\begin{vmatrix} h_1\,\mathbf{e}_1 & h_2\,\mathbf{e}_2 & h_3\,\mathbf{e}_3 \\ \dfrac{\partial}{\partial q_1} & \dfrac{\partial}{\partial q_2} & \dfrac{\partial}{\partial q_3} \\ h_1 A_1 & h_2 A_2 & h_3 A_3 \end{vmatrix}$$

Laplacian

$$\nabla^2 f = \dfrac{1}{h_1 h_2 h_3}\left[\dfrac{\partial}{\partial q_1}\!\left(\dfrac{h_2 h_3}{h_1}\dfrac{\partial f}{\partial q_1}\right) + \dfrac{\partial}{\partial q_2}\!\left(\dfrac{h_1 h_3}{h_2}\dfrac{\partial f}{\partial q_2}\right) + \dfrac{\partial}{\partial q_3}\!\left(\dfrac{h_1 h_2}{h_3}\dfrac{\partial f}{\partial q_3}\right)\right]$$

Sanity check: substituting $h_1 = h_2 = h_3 = 1$ for Cartesian coordinates, $(h_1,h_2,h_3) = (1,\rho,1)$ for cylindrical coordinates, and $(h_1,h_2,h_3) = (1,r,r\sin\theta)$ for spherical coordinates recovers the formulas in each system.

5. Comparison Table

Operator Cartesian $(x,y,z)$ Cylindrical $(\rho,\varphi,z)$ Spherical $(r,\theta,\varphi)$
$\nabla f$ $\dfrac{\partial f}{\partial x}\,\mathbf{e}_x + \dfrac{\partial f}{\partial y}\,\mathbf{e}_y + \dfrac{\partial f}{\partial z}\,\mathbf{e}_z$ $\dfrac{\partial f}{\partial \rho}\,\mathbf{e}_\rho + \dfrac{1}{\rho}\dfrac{\partial f}{\partial \varphi}\,\mathbf{e}_\varphi + \dfrac{\partial f}{\partial z}\,\mathbf{e}_z$ $\dfrac{\partial f}{\partial r}\,\mathbf{e}_r + \dfrac{1}{r}\dfrac{\partial f}{\partial \theta}\,\mathbf{e}_\theta + \dfrac{1}{r\sin\theta}\dfrac{\partial f}{\partial \varphi}\,\mathbf{e}_\varphi$
$\nabla\cdot\mathbf{A}$ $\dfrac{\partial A_x}{\partial x} + \dfrac{\partial A_y}{\partial y} + \dfrac{\partial A_z}{\partial z}$ $\dfrac{1}{\rho}\dfrac{\partial(\rho A_\rho)}{\partial \rho} + \dfrac{1}{\rho}\dfrac{\partial A_\varphi}{\partial \varphi} + \dfrac{\partial A_z}{\partial z}$ $\dfrac{1}{r^2}\dfrac{\partial(r^2 A_r)}{\partial r} + \dfrac{1}{r\sin\theta}\dfrac{\partial(\sin\theta\,A_\theta)}{\partial \theta} + \dfrac{1}{r\sin\theta}\dfrac{\partial A_\varphi}{\partial \varphi}$
$\nabla\times\mathbf{A}$ $\begin{pmatrix} \dfrac{\partial A_z}{\partial y} - \dfrac{\partial A_y}{\partial z} \\[0.3em] \dfrac{\partial A_x}{\partial z} - \dfrac{\partial A_z}{\partial x} \\[0.3em] \dfrac{\partial A_y}{\partial x} - \dfrac{\partial A_x}{\partial y} \end{pmatrix}$ $\begin{pmatrix} \dfrac{1}{\rho}\dfrac{\partial A_z}{\partial \varphi} - \dfrac{\partial A_\varphi}{\partial z} \\[0.3em] \dfrac{\partial A_\rho}{\partial z} - \dfrac{\partial A_z}{\partial \rho} \\[0.3em] \dfrac{1}{\rho}\!\left(\dfrac{\partial(\rho A_\varphi)}{\partial \rho} - \dfrac{\partial A_\rho}{\partial \varphi}\right) \end{pmatrix}$ $\begin{pmatrix} \dfrac{1}{r\sin\theta}\!\left(\dfrac{\partial(\sin\theta\,A_\varphi)}{\partial \theta} - \dfrac{\partial A_\theta}{\partial \varphi}\right) \\[0.3em] \dfrac{1}{r}\!\left(\dfrac{1}{\sin\theta}\dfrac{\partial A_r}{\partial \varphi} - \dfrac{\partial(r A_\varphi)}{\partial r}\right) \\[0.3em] \dfrac{1}{r}\!\left(\dfrac{\partial(r A_\theta)}{\partial r} - \dfrac{\partial A_r}{\partial \theta}\right) \end{pmatrix}$
$\nabla^2 f$ $\dfrac{\partial^2 f}{\partial x^2} + \dfrac{\partial^2 f}{\partial y^2} + \dfrac{\partial^2 f}{\partial z^2}$ $\dfrac{1}{\rho}\dfrac{\partial}{\partial \rho}\!\left(\rho\dfrac{\partial f}{\partial \rho}\right) + \dfrac{1}{\rho^2}\dfrac{\partial^2 f}{\partial \varphi^2} + \dfrac{\partial^2 f}{\partial z^2}$ $\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\!\left(r^2\dfrac{\partial f}{\partial r}\right) + \dfrac{1}{r^2\sin\theta}\dfrac{\partial}{\partial \theta}\!\left(\sin\theta\dfrac{\partial f}{\partial \theta}\right) + \dfrac{1}{r^2\sin^2\theta}\dfrac{\partial^2 f}{\partial \varphi^2}$

Note on the vector Laplacian: in curvilinear coordinate systems, $\nabla^2\mathbf{A}$ cannot be obtained by simply applying the scalar Laplacian to each component. In Cartesian coordinates the basis vectors are constant, so $\nabla^2\mathbf{A} = (\nabla^2 A_x,\;\nabla^2 A_y,\;\nabla^2 A_z)$ holds. In cylindrical and spherical coordinates, however, the basis vectors depend on position, so additional derivative terms involving the basis vectors appear. For the explicit correction terms, see the vector Laplacian in cylindrical coordinates.