00083___1e603906a8c2bd2b9379ff5e4d114b31 - r 2 dc,P...

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62 TENSOR ANALYSIS Chap. 2 (b) In rectangular Cartesian coordinates gap = Jap, the scalar gap$Iap reduces to the form (writing z' = x, x2 = y, z3 = z) a2* d2* d2* 8x2 dy2 dz2 -+-+-. (c) Hence, the Laplace equation in curvilinear coordinates with the scalar field $(x1,z2,x3) is given by 2.19. Let y', y2, y3 (or x, y, z) be rectangular Cartesian coordinates and x1, x2, x3 (or r, 9, 0) be the spherical polar coordinates. Show that the Laplace equation in spherical polar coordinates, when the unknown function is a scalar field $(r, 4, e), is 1 d2* 2d* COtdd* + -- +-- =o. -+--+--
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Unformatted text preview: r 2 dc,P r2sin2#J 802 r dr r 2 ad a2* 1 a2* 2.20. Let Ei(yl, y2, y3) be the components of an unknown vector in rectan- gular Cartesian coordinates. Let each component satisfy Laplace's equation in rectangular coordinates, Show that a generalization of this equation is that Ei(x' , x 2 , x3) in curvilinear coordinates z', z2, x3 will satisfy the system of three differential equations gaPEilap = 0 I where c l a p is the second covariant derivative of Ei(x', x 2 , x3). Hint: Since g = giaGia (i not summed), G"3 is the cofactor of the element g,j in g = (g,j(, show that and that Hence....
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