Advanced calculus; a text upon select parts of differential calculus, differential equations, integral calculus, theory of functions with numerous exercises - Tapa blanda

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. 1911 Excerpt: ... be given by mathematical induction. Each of the theorems has been proved in the simplest cases and it remains only to show that the theorems are true for p functions in case they are for p--1. Expand the determinant J. cF?F cF J = Jt--1 + J2--± + V Jp--Ji,.., Jp, minors. dx1 "&x2 dxp For the first theorem J 0 and hence at least one of the minors J1,..., Jp must fail to vanish. Let that one be Jv which is the Jacobian of Ft,.., Fp with respect to x2,, xp. By the assumption that the theorem holds for the casep--1, these p--1 equations may be solved for x2,, xp in terms of the n + 1 variables a:j, 6. If u = u (x, y), = u(x, v), and z = x(f, ij), y = y(t, ij), prove State the extension to any number of variables. How may (27') be used to prove (27)? Again state the extension to any number of variables.--) dudvdw = dudvdw + JI----!--) is the element of u, v, w/ x, y, z) volume in space with curvilinear coordinates u,v,w = consts. 8. In what parts of the plane can u = x2 + y2, v = xy not be used as curvilinear coordinates 1 Express ds2 for these coordinates. 9. Prove that 2 u = x2--y2, v = xy is a conformal transformation. u v 10. Prove that x =, y = is a conformal transformation. u2 + v2 u2 + v 11. Define conformal transformation in space. If the transformation x = au + bv + cw, y = a'u + b'v + c'w, z = a"u + b"v + c"w is conformal, is it orthogonal? See Ex. 10 (f), p. 100. 12. Show that the areas of the triangles whose vertices are (w, B), (u + du, v), (u, v + dv) and (u + du, v + dv), (u + du, »), (u, v + dv) are infinitesimals of the same order, as suggested In § 63. 13. Would the condition F = 0 in (28) mean that the set of curves u = const, were perpendicular to the set v = const.? 14. Express E, F, G in (28) in terms...