Proceeding to the limit, we have, x, Y1, Y1⁄2, Y ̧‚. . . .Yμ, being the ultimate values of x', y', Y, Ya',....Y

dv1 dy2 dv1 dx dy3

dy dx

The analogous equations in regard to v1⁄2, 1⁄2, 1,... .vn, may be established in the same way.

Multiplying the equation (1) by dx, we have

or, if we express the different partial differentials of v, by

suggestive suffixes,

0 = Dv1 = dv1 + d ̧ ̧ ̧ + d ̧ ̧ ̧ + d ̧ ̧21 +...+ dy v1. From the equation (1) together with the (n − 1) analogous equations, making in all n linear equations, we may determine the n differential coefficients

in terms of the n (n + 1) partial differential coefficients of V1 V2 V3.....

Total Differentiation of a Function of Functions of independent

Variables.

24. We have now fully considered the principle of differentiating a function of functions, the subordinate functions being dependent each of them upon one and the same variable. Suppose, however, that

u = f(Y1, Y2, Y3,• • • •Yn),

and that y1, y2, y,....y, are not all of them dependent upon a single variable, but that they are functions of several independent variables. Let u', yi, ya, ya,....y, be corresponding values of u, y1, Y2, Yз,

y: then

n

+ . . . . + d„ ƒ (y1', Y2', Yś'‚· · · ·Y'π-19 Yn). Now when these increments are diminished indefinitely by the continuous approximation of y,', y2, Ys',....yn', to the values Y1, Y2, Y3,....y, 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

12

=

x3

(x,' + x,) dx, + (x2 + x2) dx2,

= (x‚”2 + x,'x ̧ + x ̧2) dx, + (x22 + x2'x2 + x22) 8x2:

hence, by the definition,

1

2

dy1 = dx1 + dx, dy2 = 2x1 dx, + 2x,dx2, dy1 = 3x dx1 + 3x ̧2 dx2; Du = dx ̧ + dx, + 2x ̧dx ̧ + 2x,dx2 + 3x ̧adx + 3x2 dx2

1

Partial Differentiation of an explicit Function of three Variables one of which is a Function of the other two.

25. Suppose that u = ƒ (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 y, we may assume y to remain unchanged while x and therefore z varies: then, the expression

Du
dx

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,

(1).

du

cients of z with regard to x and y respectively; and

du

dx dy

are

the partial differential coefficients of u with regard to x and y.

du
dz

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
dz

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 however to observe that in both cases the actual value of

du 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

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. (dz + dz):

Partial Differentiation of an explicit Function of n+r Variables, r independent and n dependent.

26. Let u= f(x1, X2, X3,....Xr, Y 19 Y2, Y3,... Yn),

where y1, y2 y3" Yn, are each of them functions of r independent variables x,, x, x,....X.

Then, differentiating successively with regard to x1, X2, X3,•••X,, each of these quantities being taken in turn as the only variable among them, we have, by Art. (21),

27. Let z be an implicit function of two independent variables x and y by virtue of an equation

u = ƒ (x, y, z) = 0.

y

Then supposing, as we are evidently at liberty to do, that remains constant while x and consequently z varies, we have, by Art. (22),

Du du du dz

= 0.

= +
dx dx dz dx

Again, supposing y variable and a constant, we shall have also

Partial Differentiation of implicit Functions of any number of independent Variables.

0, v3

where v1, v2, v3,...,, are n functions of n +r variables x1, x, x ̧, ...xr, Y., Y2, Y3,...y: then each of the variables y1, Y2, Yз,...Yn,