Shifting Orthogonal Coordinates
Length, area, and volume are all calculated differently in different orthogonal coordinate systems. The gradient of a function and the divergence and curl of vector fields, likewise, are calculated according to the orthogonal coordinates which the function or vector field is given in.
For example, in polar coordinates the point (r, a) is the point (rcos(a), rsin(a)) in Cartesian coordinates. Hence the distance from the point (r, a) to the point (s, b) given in polar coordinates is equal to (rcos(a) - scos(b)) 2 + (rsin(a) - ssin(b))2)½. But the distance from (w,x) to (y,z) in Cartesian coordinates is ((w-y)2 + (x-z) 2)½.
Let u1, u2, and u3 be functions defining a three-dimensional orthogonal coordinate system. Then at every point (x, y, z) in space, the surfaces defined by the equations u1 = x, u2 = y, and u3 = z are orthogonal. The intersection of any two of these surfaces is called a coordinate curve. Let c1(t) = (t, y, z). So, the image of c is a coordinate curve. Let s1(t) be the length along the curve c1 from c1(0) to c1(t). Similarly define functions s2 and s3 to be the arc length functions of the other coordinate curves. Then define h1 to be the derivative of s1 with respect to u1. Similarly, let h2 and h3 be the derivatives of s2 and s3 with respect to u2 and u3 respectively. These functions are called the scale factors. Then the arc length of a curve S is given by the line integral
((h1 du1)2 + (h2 du2)2 + (h3 du3)2) ½ where the integral is taken over S.
To make sense of this, suppose that S is a small curve which begins at some point P = (0, 0, 0) and ends at a point Q = (Q1, Q2, Q3) where both P and Q are given in ui coordinates. Then the length of S is approximately given by the distance (in Cartesian coordinates) from (0, 0, 0) to (Q1h1(P), Q2h2(P), Q3h3(P)).
Also the volume of an object O is given by the volume integral
(h1h2h3) du1 du2 du3 where the integral is taken over O.
To make sense of this, suppose that O is a small parallelepiped with vertices (in ui coordinates) given by (0,0,0), (A,0,0), (0,B,0), (0,0,C), (A,B,0), (0,B,C), (A,0,C). Then its volume is approximately equal to the volume of the Euclidean parallelepiped with dimensions h1(P)xA, h2(P)xB, and h3(P)xC.
The gradient of a function f given in ui coordinates is
Grad f = (1/h1 (∂f/∂u1), 1/h2 (∂f/∂u2), 1/h3 (∂f/∂u3))
(with respect to ui coordinates). The divergence and curl of a vector field F = (F1, F2, F3) (given in ui coordinates) are
Div F = 1/h1h2h3 [∂/∂u1(F1h2h3) + ∂/∂u2(F2h1h3) + ∂/∂u3(F3h1h2) ].
Curl F = (1/h2h3 [∂/∂u2(F3h3) - ∂/∂u3(F2h2)], 1/h1h3 [∂/∂u3(F1h1) - ∂/∂u1(F3h3)], 1/h1h2 [∂/∂u1(F2h2) - ∂/∂u2(F1h1)]). (with respect to ui coordinates).
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