gradient in curvilinear coordinates

We give the formulas for the gradient expressed in various curvilinear coordinate systems. We also show the metric tensorsMathworldPlanetmath gij 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,θ,z) we have


So that

f =fr𝐞r+1rfθ𝐞θ+fz𝐤,


𝐞r =r=xr𝐢+yr𝐣
𝐞θ =1rθ=-yr𝐢+xr𝐣

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

The notations /r,/θ, etc., denote the tangent vectorsMathworldPlanetmath corresponding to infinitesimalMathworldPlanetmath changes in r,θ, etc. respectively. Concretely, in terms of Cartesian coordinatesMathworldPlanetmath, /r is the vector 𝐢x/r+𝐣y/r+𝐤z/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

f =fr𝐞r+1rfθ𝐞θ.

3 Spherical coordinate system

To stave off confusion, note that this is the “mathematicians’ ” convention for the spherical coordinate system (ρ,ϕ,θ). That is, ϕ is the co-latitude angle, and θ is the longitudinal angle.

f =fρ𝐞ρ+1ρfϕ𝐞ϕ+1ρsinϕfθ𝐞θ,


𝐞ρ =ρ=xρ𝐢+yρ𝐣+zρ𝐤
𝐞ϕ =1ρϕ=zxrρ𝐢+zyrρ𝐣-rρ𝐤
𝐞θ =1ρsinθθ=-yr𝐢+xr𝐣

are the unit vectors in the direction of increase of ρ,ϕ,θ, 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