Advanced calculus; . cations of § 60 are that the equations (22) maythen be solved for x, y in terms of u, v at anypoint of the region and that there is a pair ofthe curves through each point. It is then proper to consider (u, v) asthe coordinates of the points in the region. To any point there corre-spond not only the rectangular coordinates (x, y) but also the curvi-linear coordinates (u, r). The equations connecting the rectangular and curvilinear coordinatesmay be taken in either of the two forms Y l\ 0 X u= (x, y), v = $ (x, y) or x = f(u, v), y = g (u, v), (22) each of which are the solu
Advanced calculus; . cations of § 60 are that the equations (22) maythen be solved for x, y in terms of u, v at anypoint of the region and that there is a pair ofthe curves through each point. It is then proper to consider (u, v) asthe coordinates of the points in the region. To any point there corre-spond not only the rectangular coordinates (x, y) but also the curvi-linear coordinates (u, r). The equations connecting the rectangular and curvilinear coordinatesmay be taken in either of the two forms Y l\ 0 X u= (x, y), v = $ (x, y) or x = f(u, v), y = g (u, v), (22) each of which are the solutions of the other. The Jacobians 2/N \x, y) \u, v are reciprocal each to each; and this rela-tion may be regarded as the analogy ofthe relation (4) of § 2 for the case ofthe function y = (x) and the solutionx = fiy) = <£-1(#) in the case of a singlevariable. The differential of are is = 1Y (27) (x+dvx, y+dvy)(u, v+dv) (x+dx,y+dy) (u + du, v+dv)v + dv ds2 = dx2 -f- dif = Edir -f 2 Fdudr + Gdi. \cu) \cu F ex ex Cll cc cy cy cu cr fdxV s)+ &)?? The differential of area included between two neighboring ^-curves andtwo neighboring r-curves may be written in the form dA = J[ -1-JL- I dudv = dudv -r- J u, vj \x, y These statements will now be proved in detail. (29) 132 DIFFEKENTIAL CALCULUS To prove (27) write out the Jacobians at length and reduce the result. \x, y! \u, »/ du dvdx dx dxdu dydu du dv dy dy dx dv dycv du dx dv dxdx du dx dv du dy dv dydx du dx dv 1 0 dudxdy du dv dxdy dv du dy dv dydy du dy dv 0 1 where the rule for multiplying determinants has been applied and the reductionhas been made by (15), Ex. 9 above, and similar formulas. If the rule for multi-plying determinants is unfamiliar, the Jacobians may be written and multipliedwithout that notation and the reduction may be made by the same formulas asbefore. To establish the formula for the differential of arc it is only necessary to writethe total differentials of dx and dy, to square and add, and then
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