CHAPTER XXVII.

ON DIFFERENTIALS.

365. IN the preceding pages we have given the propositions usually found in treatises on the Differential Calculus, and have used the method of limits in all the demonstrations. We now offer a few remarks on another method of treating the subject.

In the expansion of f(x+h) by Taylor's Theorem, the coefficient of h was shewn to be that function of x which we had called the differential coefficient of f(x) with respect to x. Lagrange proposed to define the differential coefficient of f(x) with respect to x as the coefficient of h in the expansion of f(x+h), and thus to avoid all reference to the theory of limits. Lagrange's views were propounded towards the close of the last century and were generally adopted by elementary writers.

One objection to this method is its use of infinite series without ascertaining that those series are convergent, and the proof that f(x+h) can always be expanded in a series of ascending powers of h, which is made the foundation of the Differential Calculus, labours under serious defects. Another objection is that it is impossible to avoid introducing the notion of a limit in the applications of the subject to geometry and mechanics; the definition of the tangent line to a curve may be given as an example.

366. Nearly all the recent treatises on the Differential Calculus have followed the method of limits, and the only point of importance in which a difference exists among them is with respect to the use of differentials. In the present dy

work has been defined as one symbol, thus-if y = (x)

dx

the limit of

(x + h) - $ (x) when h is indefinitely diminished

dy

is denoted by dx

h

Some writers add the following words—the quantities dx and dy are called the differentials of x and y respectively; their absolute values are indeterminate, and they may be either finite or indefinitely small provided their relative magnitudes be such that is equal to the limit above mentioned.

dy

dx

With this meaning attached to dy and dx such equations

where '(x) is the differential coefficient of (x) or y.

Equations expressed by means of differentials are in general capable of immediate translation into the language of differential coefficients. For example, if x and y be coordinates of a point on a curve and be functions of a third variable t, and if s denote the corresponding arc of the curve beginning at some fixed point, we have, by Art. 307,

A writer who uses differentials will express these results thus,

dx2 + dy2 = ds2,

dx d'x + dy d'y = ds d's.

The student may look upon the latter as merely abbreviated methods of writing the previous equations, and may take dx, dy, d'x, ... as standing for

367. Let u be a function of any number of variables, for example three, so that u(x, y, z). If we suppose

x, y, z, all functions of a variable t, and for shortness put

In works on the Differential Calculus, which use differentials,

we find an equation similar to the above occurring at an

early period, namely,

The introduction and use of this equation form the principal difference between such works and one which, like the present, uses only differential coefficients. To establish (2) the following method is adopted.

Let

and

therefore

u = $(x, y, z),

u + Au = $(x + Ax, y + Ay, z + Az),

Au = (x + Ax, y + ▲y, z + ▲z) – † (x, y, z)

( +A,+A,+Az) - (x, y+A,z+Az)

Δη

=

Ax

+

Ду

Ay

+

Az

(3).

Az

(x, y, z + Az) — 4 (x, y, z)

If Ax, Ay, and Az, diminish without limit, the quantity

$ (x+ Ax, y + Ay, z + Az) − $ (x, y + Ay, z + Az)

+a, we know that a diminishes without limit when

dx

Ax, Ay, and Az, do so. In this manner we may deduce from (3) the equation

where a, ẞ, y, all diminish without limit when Ax, Ay, Az, do so. If then du, dx, dy, and dz, denote quantities whose absolute magnitudes are undetermined, but whose relative magnitudes are those to which Au, Ax, Ay, and Az, respectively approach as their limits when they are all indefinitely diminished, we have

Having thus proved (2), we give an example of its application. Since in establishing (2) we had no occasion to consider whether x, y, and z, were independent or not, the result is universally true, whatever relation be given or supposed between the variables. If, for example, p(x, y, z) is always 0, we have

Now if (x, y, z) =0 is the only equation connecting x, y, and z, we may if we please vary a and z without changing y. Hence in the preceding investigation Ay=0 throughout, and therefore in (5) dy = 0; thus we have

dz where is the differential coefficient of z, supposing x to dx

vary and y to be constant. See Art. 188.

368. It would occupy too much space if we were to proceed further with the subject of differentials. Differential

coefficients have been used exclusively in the present work, from the conviction that the subject is thus presented in the clearest form, and that if some of the operations are thus rendered a little longer than they would otherwise be, there is at the same time far less liability to error. The equation (2) is certainly of great use in applications of the Differential Calculus, particularly in the higher parts of the Geometry of Three Dimensions: after the remarks already made, the student will probably find little difficulty in those applications. Perhaps he may be further assisted by referring to the theorem for the expansion of a function of three variables. If u = p(x, y, z), we have

$ (x+h, y +k, z + 1) − 4 (x, y, z) or Au

where R involves squares and products of h, k, l. Hence the smaller h, k, l, are taken, the smaller is the error contained in the assertion

2. Find the maxima and minima values of (sin x) sinx.

3. Find the area of the greatest isosceles triangle that can be inscribed in a given ellipse, the triangle having its vertex coincident with one extremity of the major axis.

4. APP'B is a semicircle whose diameter is AB, and PP' is parallel to AB. Draw AP' and BP, and let them meet in Q; find the position of P and P' so that the triangle PQP' may be a maximum.