3 22-19 +.... + dmf (y1, Y2', Ys',... ·Y'n-1' Yn). .y', Now when these increments are diminished indefinitely by the continuous approximation of y', Y2, Y.',....Yn', to the values Y1, Y2) Ya... Yμ, and in correspondency with any restrictions to which these variables may be subject, let quantities finite or infinitesimal which are proportional to these vanishing increments be called the differentials of the corresponding incremented quantities: this definition of differentials is merely an extension of the definition for the case of one independent variable to that of any number. Then, by the notation of differentials, we have = d u = бузи = буз, = dy2, du dy,, Du = dy, + dy2+ dyз• = (x‚' + x,) dx, + (x2' + x2) SX2, 13 12 3 12 · (x ̧”2 + x ̧'x ̧ + x ̧2) dx, + (x ̧22 + x2'x2 + x22) Sx2: hence, by the definition, dy, = 2 · dx1 + dx2, dy2 = 2x ̧ d¤ ̧ + 2x ̧dx2, dy = 3x ̧2 dx ̧ + 3x22 dx2; Du = dx1 + dx2 + 2x,dx ̧ + 2x¿dx2 + 3x ̧2 dx + 3x dx2 = 1 2 (1 + 2x, + 3x ̧3) dx ̧ + (1 + 2x2 + 3x2) dx ̧· Partial Differentiation of an explicit Function of three Variables one of which is a Function of the other two. 25. Suppose that u = f(x, y, z), z being some function of two independent variables x and y. Since x and y are supposed to vary independently of each other, the variation of z being dependent upon the variations of x and y, we may assume y to remain unchanged while ≈ and therefore z varies: then, the expression Du being taken to denote the total differential coefficient of u, as far as u is affected, both immediately by the variation of x and indirectly by the variation of z as consequent upon that of x, we have, by Art. (21), Cor., In like manner, y being supposed variable and x constant, du du cients of z with regard to x and y respectively; and are dx dy the partial differential coefficients of u with regard to x and y. du In the equation (1), represents the value of the ultimate ratio of the increment of u to the increment of z, when z receives an increment in consequence of the variation of x: in du the equation (2), represents the value of the ultimate ratio of the increment of u to the increment of z, when z receives an increment in consequence of the variation of y. It is important du however to observe that in both cases the actual value of dz must be the same, the origin of the variation of z evidently not affecting the ultimate ratio in question. We are at liberty therefore to consider dz as the total differential of z in the du denominator of in both equations, the value of du being dz accordingly also the same in both. The equations written in the most expressive form would accordingly be Owing to the complexity of the notation in (3) and (4), it will be desirable to adhere to the form of expression which we have given in (1) and (2). No danger of confusion can arise from the several meanings of Du, du, dz, provided that we remember to regard as monads the expressions Du Du du du du dz dz dx dy dx' dy' dz' dx' dy the denominators of these indissoluble fractions sufficing to suggest the significations of their numerators. Multiplying (3) and (4) by dx and dy respectively, and adding, we have Du + Du = du + du + but, by Art. (24), hence du (dz + dz): Dz D ̧u + D ̧μ = d ̧u + du + du: but, by Art. (24), we have also Du = du + du + du; hence Du = Du + Du = du + du + du. Partial Differentiation of an explicit Function of n+r Variables, r independent and n dependent. 26. Let u= f(x1, x2, xz,....Xr, Y1, Y29 Y35• • • •Y„), , .. where Y1, 1, Y2, Y3, • .yn, are each of them functions of r independent variables x1, X2, X3,. . . . X ̧• ... Then, differentiating successively with regard to ≈1, X2, X3,•••X, 9 each of these quantities being taken in turn as the only variable among them, we have, by Art. (21), Partial Differentiation of an implicit Function of two 27. Let z be an implicit function of two independent variables x and y by virtue of an equation u = f(x, y, z) = 0. Then supposing, as we are evidently at liberty to do, that y remains constant while x and consequently z varies, we have, by Art. (22), Du du du dz = dx dx dz dx Again, supposing y variable and x constant, we shall have also Partial Differentiation of implicit Functions of any number of independent Variables. where v1, v2, v3,...v,, are n functions of n +r variables x1, x, x ̧, •••X+, Y1, Y2, Y ̧‚y: then each of the variables Y1, Y2, Yз,...Yn, may be regarded as a function of r independent variables X1, X2, Xz,....x.. We are therefore at liberty to consider X2, X3,....X,, as all constant, and to regard y1, Y2, Yз....Yn, as functions of a variable x1. We have therefore, by Art. (23), dv, dyn 0 = Dv1 dv1 dv1 dx1 = dx + dy, + dy dx1 Similarly, x2, x,,... 0 dv, dyz dv, dy, dy2 dx1 dy, dx, dyn dx, .x,, successively taking the place of x1, 0 = dx2 1 = == dv, dv, Dv1 + 1 dy + dv dx3 dy2 dy2 dv, dys dx3 dv dv dy dv, dy2 + + dx, dy, dx, dy2 dx, + + 3 dy, dx dyn dx3 There will evidently be also r analogous equations in relation to each of the functions v2, v, ..... We thus have nr differential equations, and may thence determine expressions for the nr partial differential coefficients of the dependent variables y1, y2, y2,.... y1, viz. in terms of the n (n+r) partial differential coefficients of v1, v2, v1, ... v, taken with regard to the variables x,, 29 X33 ... X79. |