gradient in curvilinear coordinates
We give the formulas for the gradient expressed in various curvilinear coordinate systems. We also show the metric tensors 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 we have
are the unit vectors in the direction of increase of and . Of course, denote the unit vectors along the positive axes respectively.
The notations , etc., denote the tangent vectors corresponding to infinitesimal changes in , etc. respectively. Concretely, in terms of Cartesian coordinates, is the vector . 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 coordinate. Thus
3 Spherical coordinate system
To stave off confusion, note that this is the “mathematicians’ ” convention for the spherical coordinate system . That is, is the co-latitude angle, and is the longitudinal angle.
are the unit vectors in the direction of increase of , respectively.
|Title||gradient in curvilinear coordinates|
|Date of creation||2013-03-22 15:27:32|
|Last modified on||2013-03-22 15:27:32|
|Last modified by||stevecheng (10074)|