Let f(x, y) = c be the implicit function of two variables, which may be put in the form

u = f(x, y) = c.

Then, since u is equal to a constant, Du = 0; and therefore by the last Article,

(du)

dx +

du

(day) dy = 0;

And without going through the intermediate stage of partial differentiation, we may take (40) or (41) of the preceding Article, and differentiate immediately. Thus

on the original ellipse, but lies a little above it or below it, according to the sign of dy; i. e. it is a point of another ellipse, which second ellipse depends on the variation of the part of c corresponding to the variation of y. And now suppose a to vary, y being constant, in consequence of which the other part of c, which has a variation due to the variation of x, changes, but changes in such a manner as to bring back the point to the original ellipse, the variations of the two parts of the constant being such that the sum of them is zero; whereas then the partial variation of the first carried the point off the ellipse, that of the second brought it back again. This case is exactly that of the total variation of a function of two variables, with the condition that the two partial variations are so related as to render the total variation equal to zero.

49.] Differentiation of a function of many variables, all of which are independent of each other.

Let ur (x, y, z,......) be a function of many variables, x, y, z, all of which are independent of each other; then

.*.

▲U = F(X+▲X, Y + AY, Z + AZ, ...) — F(X, Y, Z, ...)

= F(x+▲x, y + ▲y, z + AZ, ...) — F (X, Y + ▲Y, Z + AZ,
+ F(x, y+AY, Z+AZ,...) - F(x, y, z + AZ, ...)
+ F(X,Y,Z+AZ,...)

+

+

- F(X, Y, Z, ...),

...)

by the introduction of quantities which taken through the whole expression cancel each other. Now let the increments be infinitesimal; then the lines in the right-hand member of the equation become severally the x-, the y-, the z-, ... partial differentials of u; and therefore

Du = DF = d ̧u + d„u + d2u + ... ;

(45)

that is, the total differential is equal to the sum of the partial differentials.

And if the result is expressed in terms of partial derived functions, then

Whence, as in Art. 47, (42) and (43), the ratio of the total differential to that of any one of the variables may be found.

Ex. 1.

u = sin (yz+Zx+xY);

du

= (z+y) cos (yz+zx+xy),

dx

.'.

Du = cos (yz+zx+xy) { (y + z) dx + (z+x) dy +(x+y) dz}.

These results might have been deduced immediately in the same way as those of Ex. 4 and 5 in Art. 47.

50.] If the variables are so combined in an equation that no one is expressed explicitly in terms of the others, then we have an implicit function of many variables of the form

F(X, Y, Z, ......) = c;

DF = 0,

and therefore

and

(7)

whence

may be determined the absolute variation of any one of them which is due to the variations of all the others; thus

Similarly may be found the variation of any one of the variables which is due to the variation of any one of the others; thus, for instance, let it be required to find the relation between the corresponding variations of x and y in the equation

when the other variables do not change value; then

Similarly the ratio of the corresponding variations of x and z are

Now these results are useful in questions like the following: Suppose that there is an expression,

where P, Q, R are functions of x, y and z; and that it expresses a property of the explicit function

are the partial derived-functions of f(x, y);

what is the equivalent form of (50), when (51) is put into the

and this is the equivalent expression. The converse operation is performed by a similar process. It will be observed that we have hereby passed from (50) which is an unsymmetrical expression, to (53) which is symmetrical.

51.] Differentiation of a function of many variables, some of which are dependent on the others.

The principles of differentiation of functions of many variables

which have been explained in the five immediately preceding Articles are general, whatever are the values of the variables; and therefore a fortiori they are applicable to the less general case in which the variables are connected with each other by certain given relations: for what is true in the most general case cannot but be true in the particular.

Firstly let us take the most simple case, where u is a function of two variables, and of the form

u = F(x, y);

and let us suppose y to be a function of x, and of the form

the meaning of the several terms of which expression is sufficiently obvious from their symbols.

a result which is manifestly as it ought to be, if we consider that the relation between u and x, after the elimination of y, is

Again, consider the less simple case, u = F(x, y, z), where