Advanced Calculus fi..

118 Advanced Calculus, Fifth Editionrr - Tc~+p = 0. one hasancquation inw. fi. I/. 8. Sincc w = ;)T/ilV,ctc.. thc rclationof form (b) is obtained. The others are provcd in ~hc samc way.]A pair of functionscan be regarded as a mapping from thc uv-plane to the .I-!-plane (see Section 2.7).Undcr appropriate conditions this mapping is a one-to-one correspondence betweena domain Dl,, in the uv-plane and a domain D,, in the .ry-plane (Fig. 2.8). Onc canthen consider the ir111er.c.e mapping that takes each point !.r. !) in D,, to the uniquepoint (u, LI) such that (2.77) holds. The inverse mapping is given by functionsHowever, we may have difficulty in solving (2.77) to obtain these functionsexpl~citly.We may, nevertheless, consider Eqs. (2.77), in tho formas implicit equations for the functions (2.78). We can then seek partial derivativesas in the preceding section. With F(.r, y. 11. ti) = ,f(u. 11) - .r and G(.Y. y. 11. 11) =g(rt. v ) - v we have, for example.The Jacobian determinant in the denominator is simply the Jacobian of the mapping(2.77). As earlier, we assume that it is not zero at the points considered. This condition.together with the continuity of the first partial derivatives appearing, ensuresthat the Implicit Function Theorem is applicable. so that the inverse mapping (2.78)is well defined.The fact that the functions (2.78) are solutions of (2.77) means thatFigure 2.8Mapping. inves\c mapping. and curvilinear coordinates.