Del in cylindrical and spherical coordinates

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

Coordinate conversions
Note that the operation $$\arctan\left(\frac{A}{B}\right)$$ must be interpreted as the two-argument inverse tangent, atan2.

Del formula

 * 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.


 * Defined in Cartesian coordinates as $$\partial_i \mathbf{A} \otimes \mathbf{e}_i$$. An alternative definition is $$\mathbf{e}_i \otimes \partial_i \mathbf{A}$$.


 * 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

 * 1) $$\operatorname{div}  \, \operatorname{grad} f          \equiv \nabla \cdot  \nabla f \equiv \nabla^2 f$$
 * 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) $$\nabla^2 (f g) = f \nabla^2 g + 2 \nabla f \cdot \nabla g + g \nabla^2 f$$
 * 6) $$\nabla^{2}\left(\mathbf{P}\cdot\mathbf{Q}\right)=\mathbf{Q}\cdot\nabla^{2}\mathbf{P}-\mathbf{P}\cdot\nabla^{2}\mathbf{Q}+2\nabla\cdot\left[\left(\mathbf{P}\cdot\nabla\right)\mathbf{Q}+\mathbf{P}\times\nabla\times\mathbf{Q}\right]\quad$$     (From )

Cartesian derivation


$$\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


$$\begin{align} \operatorname{div} \mathbf A &= \lim_{V\to 0} \frac{\iint_{\partial V} \mathbf A \cdot d\mathbf{S}}{\iiint_V dV} \\ &= \frac{A_\rho(\rho+d\rho)(\rho+d\rho)\,d\phi\, dz - A_\rho(\rho)\rho \,d\phi \,dz + A_\phi(\phi+d\phi)\,d\rho\, dz - A_\phi(\phi)\,d\rho\, dz + A_z(z+dz)\,d\rho\, (\rho +d\rho/2)\,d\phi - A_z(z)\,d\rho (\rho +d\rho/2)\, d\phi}{\rho \,d\phi \,d\rho\, dz} \\ &= \frac 1 \rho \frac{\partial (\rho A_\rho)}{\partial \rho} + \frac 1 \rho \frac{\partial A_\phi}{\partial \phi} + \frac{\partial A_z}{\partial z} \end{align}$$

$$\begin{align} (\operatorname{curl} \mathbf A)_\rho &= \lim_{S^{\perp \hat{\boldsymbol \rho}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\boldsymbol{\ell}}{\iint_{S} dS} \\[1ex] &= \frac{A_\phi (z) \left(\rho+d\rho\right)\,d\phi - A_\phi(z+dz) \left(\rho+d\rho\right)\,d\phi + A_z(\phi + d\phi)\,dz - A_z(\phi)\,dz}{\left(\rho+d\rho\right)\,d\phi \,dz} \\[1ex] &= -\frac{\partial A_\phi}{\partial z} + \frac{1}{\rho} \frac{\partial A_z}{\partial \phi} \end{align}$$

$$\begin{align} (\operatorname{curl} \mathbf A)_\phi &= \lim_{S^{\perp \boldsymbol{\hat \phi}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\boldsymbol{\ell}}{\iint_{S} dS} \\ &= \frac{A_z (\rho)\,dz - A_z(\rho + d\rho)\,dz + A_\rho(z+dz)\,d\rho - A_\rho(z)\,d\rho}{d\rho \,dz} \\ &= -\frac{\partial A_z}{\partial \rho} + \frac{\partial A_\rho}{\partial z} \end{align}$$

$$\begin{align} (\operatorname{curl} \mathbf A)_z &= \lim_{S^{\perp \hat{\boldsymbol z}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell}}{\iint_{S} dS} \\[1ex] &= \frac{A_\rho(\phi)\,d\rho - A_\rho(\phi + d\phi)\,d\rho + A_\phi(\rho + d\rho)(\rho + d\rho)\,d\phi - A_\phi(\rho)\rho \,d\phi}{\rho \,d\rho \,d\phi} \\[1ex] &= -\frac{1}{\rho}\frac{\partial A_\rho}{\partial \phi} + \frac{1}{\rho} \frac{\partial (\rho A_\phi)}{\partial \rho} \end{align}$$

$$\begin{align} \operatorname{curl} \mathbf A &= (\operatorname{curl} \mathbf A)_\rho \hat{\boldsymbol \rho} + (\operatorname{curl} \mathbf A)_\phi \hat{\boldsymbol \phi} + (\operatorname{curl} \mathbf A)_z \hat{\boldsymbol z} \\[1ex] &= \left(\frac{1}{\rho} \frac{\partial A_z}{\partial \phi} -\frac{\partial A_\phi}{\partial z} \right) \hat{\boldsymbol \rho} + \left(\frac{\partial A_\rho}{\partial z}-\frac{\partial A_z}{\partial \rho} \right) \hat{\boldsymbol \phi} + \frac{1}{\rho}\left(\frac{\partial (\rho A_\phi)}{\partial \rho} - \frac{\partial A_\rho}{\partial \phi} \right) \hat{\boldsymbol z} \end{align}$$

Spherical derivation
$$\begin{align} \operatorname{div} \mathbf A &= \lim_{V\to 0} \frac{\iint_{\partial V} \mathbf A \cdot d\mathbf{S}}{\iiint_V dV} \\ &= \frac{A_r(r+dr)(r+dr)\,d\theta\, (r+dr)\sin\theta \,d\phi - A_r(r)r\,d\theta\, r\sin\theta \,d\phi + A_\theta(\theta+d\theta)\sin(\theta + d\theta)r\, dr\, d\phi - A_\theta(\theta)\sin(\theta)r \,dr \,d\phi + A_\phi(\phi + d\phi)r\,dr\, d\theta - A_\phi(\phi)r\,dr \,d\theta}{dr\,r\,d\theta\,r\sin\theta\, d\phi} \\ &= \frac{1}{r^2}\frac{\partial (r^2A_r)}{\partial r} + \frac{1}{r \sin\theta} \frac{\partial(A_\theta\sin\theta)}{\partial \theta} + \frac{1}{r \sin\theta} \frac{\partial A_\phi}{\partial \phi} \end{align}$$

$$\begin{align} (\operatorname{curl} \mathbf A)_r = \lim_{S^{\perp \boldsymbol{\hat r}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell}}{\iint_{S} dS} &= \frac{A_\theta(\phi)r \,d\theta + A_\phi(\theta + d\theta)r \sin(\theta + d\theta)\, d\phi - A_\theta(\phi + d\phi)r \,d\theta - A_\phi(\theta)r\sin(\theta)\, d\phi}{r\, d\theta\,r\sin\theta \,d\phi} \\ &= \frac{1}{r\sin\theta}\frac{\partial(A_\phi \sin\theta)}{\partial \theta} - \frac{1}{r\sin\theta} \frac{\partial A_\theta}{\partial \phi} \end{align}$$

$$\begin{align} (\operatorname{curl} \mathbf A)_\theta = \lim_{S^{\perp \boldsymbol{\hat \theta}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell}}{\iint_{S} dS} &= \frac{A_\phi(r)r \sin\theta \,d\phi + A_r(\phi + d\phi)\,dr - A_\phi(r+dr)(r+dr)\sin\theta \,d\phi - A_r(\phi)\,dr}{dr \, r \sin \theta \,d\phi} \\ &= \frac{1}{r\sin\theta}\frac{\partial A_r}{\partial \phi} - \frac{1}{r} \frac{\partial (rA_\phi)}{\partial r} \end{align}$$

$$\begin{align} (\operatorname{curl} \mathbf A)_\phi = \lim_{S^{\perp \boldsymbol{\hat \phi}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell}}{\iint_{S} dS} &= \frac{A_r(\theta)\,dr + A_\theta(r+dr)(r+dr)\,d\theta - A_r(\theta+d\theta)\,dr - A_\theta(r) r \,d\theta}{r\,dr\, d\theta} \\ &= \frac{1}{r}\frac{\partial(rA_\theta)}{\partial r} - \frac{1}{r} \frac{\partial A_r}{\partial \theta} \end{align}$$

$$\begin{align} \operatorname{curl} \mathbf A &= (\operatorname{curl} \mathbf A)_r \, \hat{\boldsymbol r} + (\operatorname{curl} \mathbf A)_\theta \, \hat{\boldsymbol \theta} + (\operatorname{curl} \mathbf A)_\phi \, \hat{\boldsymbol \phi} \\[1ex] &= \frac{1}{r\sin\theta} \left(\frac{\partial(A_\phi \sin\theta)}{\partial \theta}-\frac{\partial A_\theta}{\partial \phi} \right) \hat{\boldsymbol r} +\frac{1}{r} \left(\frac{1}{\sin\theta}\frac{\partial A_r}{\partial \phi} - \frac{\partial (rA_\phi)}{\partial r} \right) \hat{\boldsymbol \theta} + \frac{1}{r}\left(\frac{\partial(rA_\theta)}{\partial r} - \frac{\partial A_r}{\partial \theta} \right) \hat{\boldsymbol \phi} \end{align}$$

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