Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR AND TENSOR ANALYSIS
system which takes advantage of the relations of symmetry involved in the particular
problem under consideration. For example, if we are dealing with sphere, we
will ®nd it expedient to describe the position of a point in sphere by the spherical
coordinates (r;;†. Spherical coordinates are a special case of the orthogonal
curvilinear coordinate system. Let us now proceed to discuss these more general
coordinate systems in order to obtain expressions for the gradient, divergence,
curl, and Laplacian. Let the new coordinates u 1 ; u 2 ; u 3 be de®ned by specifying the
Cartesian coordinates (x 1 ; x 2 ; x 3 ) as functions of (u 1 ; u 2 ; u 3 †:
x 1 ˆ f …u 1 ; u 2 ; u 3 †; x 2 ˆ g…u 1 ; u 2 ; u 3 †; x 3 ˆ h…u 1 ; u 2 ; u 3 †; …1:54†
where f, g, h are assumed to be continuous, di€erentiable. A point P (Fig. 1.16) in
space can then be de®ned not only by the rectangular coordinates (x 1 ; x 2 ; x 3 ) but
also by curvilinear coordinates (u 1 ; u 2 ; u 3 ).
If u 2 and u 3 are constant as u 1 varies, P (or its position vector r) describes a curve
which we call the u 1 coordinate curve. Similarly, we can de®ne the u 2 and u 3 coordinate
curves through P. We adopt the convention that the new coordinate system is a
right handed system, like the old one. In the new system dr takes the form:
dr ˆ @r
@u 1
du 1 ‡ @r
@u 2
du 2 ‡ @r
@u 3
du 3 :
The vector @r=@u 1 is tangent to the u 1 coordinate curve at P. If^u 1 is a unit vector
at P in this direction, then ^u 1 ˆ @r=@u 1 =j@r=@u 1 j, so we can write @r=@u 1 ˆ h 1^u 1 ,
where h 1 ˆj@r=@u 1 j. Similarly we can write @r=@u 2 ˆ h 2^u 2 and @r=@u 3 ˆ h 3^u 3
,
where h 2 ˆj@r=@u 2 j and h 3 ˆj@r=@u 3 j, respectively. Then dr can be written
dr ˆ h 1 du 1^u 1 ‡ h 2 du 2^u 2 ‡ h 3 du 3^u 3 :
…1:55†
Figure 1.16.
Curvilinear coordinates.
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