We have used a capital in the differentials Dx, Dy, Dz for the purpose of explanation, and for the avoidance of any confusion between the notation for differentials and for differential coefficients; but when once understood there is no necessity for the continuance of the capital letter, and it is usual in the higher branches of mathematics to denote the same quantities by dx, dy, dz. Hence we shall in future adopt this notation. 138: Equation 6 of Art. 136 may now be written when dx, dy, dz become infinitesimally small. This value of dz is termed the total differential of z with regard to x and y. The total differential of z is therefore equal to the sum of the partial differentials formed under the supposition that y and x are alternately constant. 139. It is easy to pass from a form in which differentials are used to the equivalent form in terms of differential coefficients. For instance, the equation where t is some fourth variable in terms of which each of the variables x, y, z may be expressed; for dz do dy dz .dt, dx= dt, dy .dt (Art. 137). dt Similarly the equation dt ds2 = dx2+dy2 may, by the same article, be written in the language of differential coefficients as 140. Total Differential (Analytical). Two independent variables. We may investigate the total differential of the function p(x, y) analytically as follows: and when a becomes x+h and y becomes y+k, let u become u+du, then and when we proceed to the limit in which h and k become indefinitely small we have ди p(x+h, y+k)−p(x, y+k) = Ltx=0 p(x, y+k) = x; h Also du: dx dy = the ultimate ratios of du: h:k, hence 141. Several independent variables. We may readily extend this result to a function of three or of any number of variables. Let u = p(X1, X2, X3), and let the increments of x,,,,,, be respectively h, ha, h„, and let the corresponding increment of u be su; then Su=4(x1+h1, x2+h2 x3+h ̧)−p(X1, X2, X3) 2 h − p(x, +h‚‚ X2+h„, x3+h3) − p (x, x2+h2, X3+h3) p, 2 h1 +Þ(X1, X3+h2 Xz+hz) − p (X3, X3 X3+hz)p 2 h2 whence, on taking the limit and substituting the ratios du: dx1: dx: dxg instead of the ultimate ratios of Su: h1: h2: hз, we have 1 2 2 3 i.e., the total differential of u when 1, 2, 3, all vary is the sum of the partial differentials obtained under the supposition that when each one in turn varies the others are constant. 142. And in exactly the same way u = $(X1, X2, if xn), ди + dxdx, dx2 = dx du= dx, dx1 = dx we obtain dx And similarly, if u = p(x, x2, ..., Xn), where x1, x2, ..., n, are known functions of x, we obtain 144. An Important Case. The case in which u= p(x, y), y being a function of x, is from its frequent occurrence 145. Differentiation of an Implicit Function. If we have p(x, y) = 0, эф, эф dy then and the du of Art. 140 vanishes. Proceeding as in that article we obtain p(x+h, y+k) = 0, 0, эф dy dx This is a very useful formula for the determination of in cases in which the relation between x and y is an implicit one, of which the solution is inconvenient or impossible. Ex. p(x, y) = x2+y3 — 3axy=0; find dy Here and dx 146. Order of Partial Differentiations Commutative. Suppose we have any relation y = p(x, a), where a is a constant, and that by differentiation we it is obvious that the result of differentiating p(x, a') would be F(x, a'); that is, the operation of changing a to a' may be performed either before or after the differentiation, with the same result. We may put this statement into another form, thus: Let Ea be an |