derivation of unit vectors in curvilinear coordinates

Let (u,v,w) be any non-Cartesian coordinate systemMathworldPlanetmath such that

x=x(u,v,w),y=y(u,v,w),z=z(u,v,w) (1)

We can combine the above three equations into a single vector equation that gives the position vector r of any point P=(x,y,z) in space as a functionMathworldPlanetmath of the coordinatesMathworldPlanetmathPlanetmath u,v,w:


If we held u fixed s.t. u=u0 then the position vector becomes the parametric equation of the surface (called the coordinate surface) u=u0 where v,w play the role of parameters. Furthermore, if we held both u and v fixed s.t u=u0 and v=v0, then the position vector becomes the parametric equation of the curve (called the coordinate curve) formed by the intersection of the surfaces u=u0 and v=v0, in which w acts as a parameter along the curve.

Now, how do we find the tangent vectorsMathworldPlanetmath? Well, what is the meaning of a tangent vector? A tangent vector is a vector which is tangentPlanetmathPlanetmathPlanetmath to a coordinate curve formed by the intersection of the two coodinate surfaces. In other words, it is a vector which indicates the direction in which one of the coordinates, say u, increases while the other two coordinates (i.e. v and w) are held fixed. Sound familiar? Yes, of course, partial derivativesMathworldPlanetmath! A partial derivative with respect to u would take the derivative of the position vector r along the coordinate curve formed by the intesection of the surfaces v=v0 and w=w0 and hence return you a tangent vector along that curve! Hence, by taking the partial derivative of r one by one with respect to all three coordinates, we would get all the three tangent vectors which are tangent to their respective coordinate curves. Thus, we arrive at the following three tangent vectors:

eu=ru  ev=rv  ew=rw

However, these are only tangent vectors. Most often we are interested in unit tangent vectors (a.k.a. standard basis vectors for 3). So we divide them by their respective lengths. Therefore,

e^u=ru|ru|  e^v=rv|rv|  e^w=rw|rw| (2)

But this is very cumbersome to write, so we instead write them as




and, similarly, hv=|ev|,hw=|ew| where hu,hv,hw are known as scale factors (or metric coefficients).

Title derivation of unit vectors in curvilinear coordinates
Canonical name DerivationOfUnitVectorsInCurvilinearCoordinates
Date of creation 2013-03-22 17:02:05
Last modified on 2013-03-22 17:02:05
Owner swapnizzle (13346)
Last modified by swapnizzle (13346)
Numerical id 10
Author swapnizzle (13346)
Entry type Topic
Classification msc 53A45