2)

3

+

+

+

dy,

dy,

dy

+. dx

, +

dy,

do, . dyz

dx +

+

do, . dy, t...t dyz

Proceeding to the limit, we have, x, Yı, Yı, Y.,. ...Yn, being the

XY2• ultimate values of x', y's Ya, Ya's. .. .yu's

Do, dv, dv, dy, dv, dy, dv, dy,
dc da

dx
dx
do dy,

.(1).

dx The analogous equations in regard to 0,, vs, V.,. .. . Vn, may be established in the same way. Multiplying the equation (1) by dx, we have

do, 0 = Do, Dv= dy, +

+ot

.dy, dx or, if we express the different partial differentials of v, by suggestive suffixes,

0 = Dv, = 0,9, + dy 0, + dy,9, + dyzu, +. ...+ dyn
=

dų, From the equation (1) together with the (n − 1) analogous equations, making in all n linear equations, we may determine the n differential coefficients

dy dy dy,
,

dyn dx da dx dx in terms of the n(n + 1) partial differential coefficients of V, 02, 03 9. ...0m.

doz. dy.

3

dy,

dyz

3

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(91, 92, Yz,. ...Yn), and that y, y, y. ...Y., are not all of them dependent upon a single variable, but that they are functions of several independent variables. Let u', y', ya', ya's. ...Yn', be corresponding values of u, Y., 42, 439.

1

u – u = f(y', ya', Y;',. ...y," - f(41, 42, Yz,. .. .Yn)
-

= f(ys Y2, Yz,. .. .Yn) - f (yı, Y., Y;, ...y.)
+ f(y', ya', Yz, ....Yn) - f (yi', , 43,. ...Y)
+ f (y;', ya', Ya',. .. .Yn) - flyi', ya', Y3....Yn)

+ f(yi', ya', yz ....Y'n-1, Yn) - f(yi, ya', Ya ....Y',-1) Yu),

or

Y3

Au=e, flyY2Yz...y.)+, f(y's Y2Yzz...yn)+ fly;sY;',Yz,•••yn)

+....+ on f (yi, ya', ya',. .. .Y',-1) Yn). Now when these increments are diminished indefinitely by the continuous approximation of yi, ya', ya',. ...Yn', to the values Y., 42, 43,. .. .Y., and in correspondency with any restrictions

Y.Y 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

Du du + d. u + d. U +.... + d. u.
Ex. Suppose that

u = f(y, Y2, Yz) = , + Ya
y = x, + 2q, yz = x + x,, Y; = x;" + x,";

;
then 8., u = dy,, ,u = dyn, OgU = dyz,

Sugu ду,
d,,u = dy, d, u = dy, d, u = dyz,

Du = dy, + dyz + dyz.
dy, = 8x, +ox,, dy, = x
х, ду X 2 + x," – XP

co

= (x' + x) dx, + (x,' + x) 8x,, dyz = x," – x +

x' + x," - x

(2,"2 + x,'%, + 3,4) 8x, + (x,"2 + x,ʻx, + x,) dx, : hence, by the definition, dy, = dx, + dx,, dy, = 2x, dx, + 2x, dx,, dy, = 3x,* dx, + 3x, dx, ;

; Du = dx, + dx, + 2x, dx, + 2x,dx, + 3x,* dx + 3x, dx,

(1 + 2x, + 3x,?) dx, + (1 + 2%, + 3x,) dxg.

+ Y3

+

= 2

2 1

1

+

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 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., Du du du dz

(1). dc dx da dx In like manner, y being supposed variable and x constant, Du du du dz

(2). dy

dy dz 'dy

dz dz In these equations and are the partial differential coeffidx dy

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

y

and

dx the partial differential coefficients of u with regard to x and

du In the equation (1), represents the value of the ultimate

dz 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

dz 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

+

are

dy

therefore to consider dz as the total differential of 2 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 Du du du d 2

.. (3), dz dx Dz . dx

+

+

du du dz ,

(4). dy dy Dz dy 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 = d u + d,u + (d,2 +0,2)

Dz but, by Art. (24), Dz

D2 = d2 + d z;

dz hence

Du + Du = d u + du + du: but, by Art. (24), we have also

Du = du + d,u + du; hence Du = Du + D,u = d u + d,u + d u.

du

ข

y

y

Partial Differentiation of an explicit Function of n+r Variables,

r independent and n dependent. 26. Let u=f(x,, x,, ,, ..., Yu, Y., 43, ...Y),

•Yn where y., 42, 43,. .. .Yn, are each of them functions of r independent variables X, X, Xg, ...X,.

+.

Then, differentiating successively with regard to x,, X,, Ig,...<,, each of these quantities being taken in turn as the only variable among them, we have, by Art. (21), Du du du dy, du dy, du dyz

du dy, dx, dx, dy, dx, dy, dx,' dy,

dy, da, Du du du dy, du dy,

du dyz dx, dx, dy, da, dy dx dy, dx,

dy dx,
Du du
du dy

du dy, du
,

du dy,
dx, dx, dy, dx, dy, 'dx, dy, dx, dy, 'dx,

ax

dx,

1

du dyn

+

+

+

+.

.+

dy,

dx,

2

2

du dy.

+

Partial Differentiation of an implicit Function of two

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

u = f (x, , 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 a constant, we shall have also

y

Du du du dz
dy dy .dz 'dy

Partial Differentiation of implicit Functions of any number

of independent Variables. 28. Let V, = 0, 0, = 0, 0; 0,...,Vn 0, where ~,, V,, V, ...Vn, are n functions of n + variables Xy, ...X, Y,, Y., Y,...y, then each of the variables y., 42, ,...Y.,