gradient in curvilinear coordinates

We give the formulas for the gradient expressed in various curvilinear coordinate systems.We also show the metric tensors $g_{ij}$ sothat the reader may verify the resultsby working from the basic formulas for the gradient.

1 Cylindrical coordinate system

In thecylindrical system of coordinates $(r,\theta ,z)$ we have

So that

$\nabla f$$={\displaystyle \frac{\partial f}{\partial r}}{𝐞}_r+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial f}{\partial \theta }}{𝐞}_{\theta }+{\displaystyle \frac{\partial f}{\partial z}}𝐤,$

where

${𝐞}_r$$={\displaystyle \frac{\partial }{\partial r}}={\displaystyle \frac{x}{r}}𝐢+{\displaystyle \frac{y}{r}}𝐣$

${𝐞}_{\theta }$$={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial \theta }}=-{\displaystyle \frac{y}{r}}𝐢+{\displaystyle \frac{x}{r}}𝐣$

are the unit vectors in the direction of increase of $r$ and $\theta$. Of course,$𝐢,𝐣,𝐤$ denote the unit vectors along the positive $x,y,z$ axes respectively.

The notations $\partial /\partial r,\partial /\partial \theta$, etc.,denote the tangent vectors corresponding to infinitesimal changes in $r,\theta$, etc. respectively.Concretely, in terms of Cartesian coordinates, $\partial /\partial r$is the vector $𝐢\partial x/\partial r+𝐣\partial y/\partial r+𝐤\partial z/\partial r$.And similarly for the other variables. (There is a deep reason for using the seemingly strange notation:see Leibniz notation for vector fields for details.)

2 Polar coordinate system

This is just the special case of the cylindrical coordinate system where we chop off the $z$ coordinate.Thus

$\nabla f$$={\displaystyle \frac{\partial f}{\partial r}}{𝐞}_r+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial f}{\partial \theta }}{𝐞}_{\theta }.$

3 Spherical coordinate system

To stave off confusion, note that this is the “mathematicians’ ” convention for the spherical coordinatesystem $(\rho ,\varphi ,\theta )$.That is, $\varphi$ is the co-latitude angle, and $\theta$ is the longitudinal angle.

$\nabla f$$={\displaystyle \frac{\partial f}{\partial \rho }}{𝐞}_{\rho }+{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial f}{\partial \varphi }}{𝐞}_{\varphi }+{\displaystyle \frac{1}{\rho \sin \varphi }}{\displaystyle \frac{\partial f}{\partial \theta }}{𝐞}_{\theta },$

where

${𝐞}_{\rho }$$={\displaystyle \frac{\partial }{\partial \rho }}={\displaystyle \frac{x}{\rho }}𝐢+{\displaystyle \frac{y}{\rho }}𝐣+{\displaystyle \frac{z}{\rho }}𝐤$

${𝐞}_{\varphi }$$={\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial }{\partial \varphi }}={\displaystyle \frac{zx}{r\rho }}𝐢+{\displaystyle \frac{zy}{r\rho }}𝐣-{\displaystyle \frac{r}{\rho }}𝐤$

${𝐞}_{\theta }$$={\displaystyle \frac{1}{\rho \sin \theta }}{\displaystyle \frac{\partial }{\partial \theta }}=-{\displaystyle \frac{y}{r}}𝐢+{\displaystyle \frac{x}{r}}𝐣$

are the unit vectors in the direction of increase of $\rho ,\varphi ,\theta$, respectively.