gradient in curvilinear coordinates

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

1 Cylindrical coordinate system

In the cylindrical 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 coordinate system $(\rho ,\varphi ,\theta )$. 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.

Title	gradient in curvilinear coordinates
Canonical name	GradientInCurvilinearCoordinates
Date of creation	2013-03-22 15:27:32
Last modified on	2013-03-22 15:27:32
Owner	stevecheng (10074)
Last modified by	stevecheng (10074)
Numerical id	5
Author	stevecheng (10074)
Entry type	Result
Classification	msc 26B12
Classification	msc 26B10
Related topic	gradient
Related topic	Gradient