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dr h1 du1 e1 h2 du2 e2 h3 du3 e3
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The quantities h1 ; h2 ; h3 are sometimes caleld scale factors. If e1 ; e2 ; e3 are mutually perpendicular at any point P, the curvilinear coordinates are called orthogonal. Since the basis elements are unit vectors as well as orthogonal this is an orthonormal basis. In such case the element of arc length ds is given by ds2 dr dr h2 du2 h2 du2 h2 du2 1 1 2 2 3 3 19
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and corresponds to the square of the length of the diagonal in the above parallelepiped. Also, in the case of othogonal coordinates, referred to the orthonormal basis e1 ; e2 ; e3 , the volume of the parallelepiped is given by dV jgjk jdu1 du2 du3 j h1 du1 e1 h2 du2 e2 h3 du3 e3 j h1 h2 h3 du1 du2 du3 which can be written as       @r @r  @r  du1 du2 du3  @ x; y; z du1 du2 du3 dV   @u @u @ u ; u ; u  @u3 1 2 1 2 3 20
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where @ x; y; z =@ u1 ; u2 ; u3 is the Jacobian of the transformation. It is clear that when the Jacobian vanishes there is no parallelepiped and explains geometrically the signi cance of the vanishing of a Jacobian as treated in 6. Note: The further signi cance of the Jacobian vanishing is that the transformation degenerates at the point.
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GRADIENT DIVERGENCE, CURL, AND LAPLACIAN IN ORTHOGONAL CURVILINEAR COORDINATES If is a scalar function and A A1 e1 A2 e2 A3 e3 a vector function of orthogonal curvilinear coordinates u1 ; u2 ; u3 , we have the following results. 1: r grad 2: r A div A 1 @ 1 @ 1 @ e e e h1 @u1 1 h2 @u2 2 h3 @u3 3 ! 1 @ @ @ h2 ; h3 A1 h3 h1 A2 h1 h2 A3 h1 h2 h3 @u1 @u2 @u3 h2 e 2 @ @u2 h2 A 2  h3 e 3   @   @u3   h3 A 3 
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  h1 e 1  1  @  3: r A curl A h1 h2 h3  @u1  h A
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     ! 1 @ h2 h3 @ @ h3 h1 @ @ h1 h2 @ 4: r Laplacian of h1 h2 h3 @u1 h1 @u1 @u2 h2 @u2 @u3 h3 @u3 These reduce to the usual expressions in rectangular coordinates if we replace u1 ; u2 ; u3 by x; y; z , in which case e1 ; e2 ; and e3 are replaced by i, j, and k and h1 h2 h3 1.
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SPECIAL CURVILINEAR COORDINATES 1. Cylindrical Coordinates (; ; z). Transformation equations: See Fig. 7-13.
x  cos ; y  sin ; z z
VECTORS
[CHAP. 7
Fig. 7-13
Fig. 7-14
where  A 0; 0 @  < 2; 1 < z < 1. Scale factors: h1 1; h2 ; h3 1 Element of arc length: ds2 d2 2 d2 dz2 Jacobian : @ x; y; z  @ ; ; z dV  d d dz
Element of volume: Laplacian:
  1 @ @U 1 @2 U @2 U @2 U 1 @U 1 @2 U @2 U r U  2 2 2 2 2  @ @  @  @2  @2 @z @ @z
Note that corresponding results can be obtained for polar coordinates in the plane by omitting z dependence. In such case for example, ds2 d2 2 d2 , while the element of volume is replaced by the element of area, dA  d d. 2. Spherical Coordinates (r; ;  . Transformation equations: See Fig. 7-14.
x r sin  cos ; y r sin  sin ; z r cos  where r A 0; 0 @  @ ; 0 @  < 2. Scale factors: h1 1; h2 r; h3 r sin  Element of arc length: ds2 dr2 r2 d2 r2 sin2  d2 Jacobian : @ x; y; z r2 sin  @ r; ;  dV r2 sin  dr d d     1 @ 2 @U 1 @ @U 1 @2 U r sin  2 2 2 @r @ r2 @r r sin  @ r sin  @2
Element of volume: Laplacian:
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