Vector Analysis and the Theory of Relativity
General Books LLC, 2010 - 52 pages
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1922 Excerpt: ...that (ds) cannot be zero, for real values of the variables a;(r) and dx(r) save in the trivial case when all the dxM = 0, it can easily be shown that the angle defined in this way is always real for real curves. Let us write instead of dx(r) the expression XZ(r) + /xm(r) and thus form the quadratic expression in X and p 9r, (Klir) + /um(r))(XZ(.) + This is not to vanish for real values of X, p save when X = 0, n = 0 (we suppose the quantities Z(r) and m(r) all real and the two directions as distinct). Using grjyW = 1 = 0 m i M we have that X2 + 2Xju cos 6 + fi2 = 0 must have complex roots when regarded as an equation in h: M-Hence 1--cos2 0 0 so that the angle as defined above is always real for real directions under the assumption that (ds) cannot vanish on a real curve. It must be remembered however that this assumption is not always made, e.g., in Relativity Theory. When cos 6=0 the curves are said to be orthogonal or at right angles at the point in question. COS Examples In ordinary space with the x's as rectangular Cartesian coordinates we have the usual expression cos 0 = lmmm + Z(2)m(2) + Z(3)m(3) where (lm, P Z(3)), m(2), m(3)) are the direction cosines of the two curves. If now we use any " curvilinear " coordinates (ym, 2/(2), y(3)) the angle between two curves is In particular if we use orthogonal coordinates (ds)2 = /n( ))2 + /22(fy(2))2 + f33dyTM)2 Thus for a curve in polar coordinates r, 8, j It will now be clear why those coordinates in terms of which (ds)2 has no product terms are said to be orthogonal. For dx1 da; f, -Qim T-tt vrr (from its covariant character) If now all the coordinates y but one, yM say, are kept constant we have a curve whose equations, in the x coordinates, may be conveniently specified by m...
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