...Since a?уж = a2, ж is a function of ж and у ; therefore differentiating the equation xyz = a1, first with respect to x, and then with respect to y, we get уz + 'Гуаxz = 0, x% + xydyZ = 0. And dxu = 2y + 2ж + 2xdxz dyu = 2a? Making dxu = 0, dyu =...

...result may be written a = <?(x, y, z), and this substituted in (1) gives F[*,y,s,9(*,y,*)] = 0, ---- (3), which will be the equation of the locus. Now...the case of consecutive curves, that the values of -r- and -7- are the same dx dy •whether derived from (1) or (3). Hence the two surfaces (1) and (3)...

...result may be written a = <f(x,y,z), and this substituted in (1) gives F[x,y,z,<f(x,y,z)] = 0, . . . . (3), which will be the equation of the locus. Now...differentiating both (1) and (3) first with respect to .r, and then with respect to y, we readily prove, precisely as in the case of consecutive curves, that...

...is defined by the relation or фх} = . dx dxv • dx By applying this to the function и =f(x, y), first with respect to x and then with respect to y, we have /du\ _ f(x + dx,y) ~f(x,y)^ \dx/ dx (**\/J& W dy and by again applying the same principle to these...

...respect to a, the result may be written and this substituted in (1) gives F[x,y,z,<p(x,y,z)] = 0, . . . . (3), which will be the equation of the locus. Now...the case of consecutive curves, that the values of — and -=- are the same dx dy whether derived from (1) or (3). Hence the two surfaces (1) and (3)...

...hyperbola referred to the same asymptotes. Discussion of the Equation. 78. If we solve equation [55] first with respect to x, and then with respect to y, we have, m "y (i) M x (2) From equation (1) we see that as y increases, x diminishes ; and when y = oo,...

...406. From the theorem just established, and considering the corresponding repeated integrals taken first with respect to x and then with respect to y, we have the following theorem : — Iff(x,y)be any limited summable function, defined in the rectangle...

...A = [a,x] x [b,y] so that (3.2.2) now becomes J ds J Pf(s, " t ) dt = f(s, t)dsdt. Differentiating first with respect to x and then with respect to y, we have immediately that Analogous formulas can be derived in the case of XC Rd. In the general case,...

...we obtain the identity Fx(x, y, x, y)x + Fa(x, y, x, y)y = F(x, y, x, y). (2) Hence, differentiating first with respect to x and then with respect to y, we obtain two equalities: from which it follows that FF F.. xx, _ __ x;; ~_ yy yx " xy x, ' We denote...

...solution which can be found by the method of successive approximations. 2.66 Differentiating the equation first with respect to x and then with respect to y we get QJf S. Existence and Uniqueness Theorems 225 If 1) / 6 <72(n), 2) K € tf 2(fi x fi), 3) K(x,...