Mathematical Methods for Physics and Engineering - Matematica.NET

VECTOR CALCULUS10.10 General curvilinear coordinatesAs indicated earlier, the contents of this section are more formal and technicallycomplicated than hitherto. The section could be omitted until the reader has hadsome experience of using its results.Cylindrical and spherical polars are just two examples of what are calledgeneral curvilinear coordinates. In the general case, the position of a point Phaving Cartesian coordinates x, y, z may be expressed in terms of the threecurvilinear coordinates u 1 ,u 2 ,u 3 ,whereand similarlyx = x(u 1 ,u 2 ,u 3 ), y = y(u 1 ,u 2 ,u 3 ), z = z(u 1 ,u 2 ,u 3 ),u 1 = u 1 (x, y, z), u 2 = u 2 (x, y, z), u 3 = u 3 (x, y, z).We assume that all these functions are continuous, differentiable and have asingle-valued inverse, except perhaps at or on certain isolated points or lines,so that there is a one-to-one correspondence between the x, y, z and u 1 ,u 2 ,u 3systems. The u 1 -, u 2 -andu 3 - coordinate curves of a general curvilinear systemare analogous to the x-, y- andz- axes of Cartesian coordinates. The surfacesu 1 = c 1 , u 2 = c 2 and u 3 = c 3 , where c 1 ,c 2 ,c 3 are constants, are called thecoordinate surfaces and each pair of these surfaces has its intersection in a curvecalled a coordinate curve or line (see figure 10.11).If at each point in space the three coordinate surfaces passing through the pointmeet at right angles then the curvilinear coordinate system is called orthogonal.For example, in spherical polars u 1 = r, u 2 = θ, u 3 = φ and the three coordinatesurfaces passing through the point (R,Θ, Φ) are the sphere r = R, the circularcone θ = Θ and the plane φ = Φ, which intersect at right angles at thatpoint. Therefore spherical polars form an orthogonal coordinate system (as docylindrical polars) .If r(u 1 ,u 2 ,u 3 ) is the position vector of the point P then e 1 = ∂r/∂u 1 is a vectortangent to the u 1 -curve at P (for which u 2 and u 3 are constants) in the directionof increasing u 1 . Similarly, e 2 = ∂r/∂u 2 and e 3 = ∂r/∂u 3 are vectors tangent tothe u 2 -andu 3 - curves at P in the direction of increasing u 2 and u 3 respectively.Denoting the lengths of these vectors by h 1 , h 2 and h 3 ,theunit vectors in each ofthese directions are given byê 1 = 1 ∂r, ê 2 = 1 ∂r, ê 3 = 1 ∂r,h 1 ∂u 1 h 2 ∂u 2 h 3 ∂u 3where h 1 = |∂r/∂u 1 |, h 2 = |∂r/∂u 2 | and h 3 = |∂r/∂u 3 |.The quantities h 1 , h 2 , h 3 are the scale factors of the curvilinear coordinatesystem. The element of distance associated with an infinitesimal change du i inone of the coordinates is h i du i . In the previous section we found that the scale364