DIFFERENTIAL CALCULUS

DIFFERENTIAL CALCULUS. The Differential Calculus was embarrassed for many years after its first appearance, by considerations of Motion, Limits, and Infinitesimals. Our countryman, Landen, having made an ineffectual attempt to remove these difficulties, the subject was taken up by Arbogast, who completely new-modelled the Calculus, and founded it upon a pure analytical basis. The genius of La Grange first established the new system, but left the theory obscure and the practical application inconvenient; by attempts to demonstrate matters of definition; by innovations in the notation; and by the universal substitution of the differential coefficients for the differentials themselves. Many of these defects were pointed out by Mr Woodhouse; and the following propositions present a systematic view of the calculus in its latest form, with some further modification of the principles, and some generalization of the formulae.

1. If $f(x+h)$ can be expanded in the form

Then $f\_1(x)$ is called the differential function of $f(x)$, and its relation to $f(x)$ is thus expressed, $f\_1(x)\; =\; Df(x)$.

2. If $(x+h)$ be put under the form $(x)\; +\; D(x)\; \backslash cdot\; h$, it appears that $Dx\; =\; 1$. This, however, is entirely arbitrary, and unity is merely selected for convenience, since by defining the differential function to be the coefficient in the second term of $f(x+a\; \backslash cdot\; h)$ expanded, we should have had $Dx\; =\; a$. Similarly $(c)\; =\; (c)\; +\; D(c)\; \backslash cdot\; h$, and $D(c)\; =\; a$.

3. In different functions of the same principal variable ($x$), we may avoid a perpetual reference to ($x$), and have the advantage of considering such functions as independent variables, by means of another symbol, which shall denote expressions proportional to the differential functions. Thus $d\backslash varphi(x)$, $d\backslash Psi(x)$ are called differentials, and are merely limited to satisfy the equation, $D\backslash varphi(x)\; \backslash times\; d\backslash Psi(x)\; =\; D\backslash Psi(x)\; \backslash times\; d\backslash varphi(x)$.

4. Hence $du$, $dz$, $dv$, &c. are expressions proportional to the differential functions of $u$, $z$, $v$, &c., considered as functions of some variable, the same in all of them.

5. If in the equation $D\backslash varphi(x)\; \backslash cdot\; d\backslash Psi(x)\; =\; D\backslash Psi(x)\; \backslash cdot\; d\backslash varphi(x)$, we take $\backslash varphi(x)\; =\; (x)$, then $D(x)\; \backslash cdot\; d\backslash Psi(x)\; =\; D\backslash Psi(x)\; \backslash cdot\; dx$, or $d\backslash Psi(x)\; =\; D\backslash Psi(x)\; \backslash cdot\; dx$.

6. If there are several characteristics, $D$ only refers to the first, thus $Df\_1\; f\_2(x)\; =\; Df\_1\; \backslash left\backslash \{\; f\_2(x)\; \backslash right\backslash \}$, $Df\_2$ being always regarded as a single symbol.

Differential Calculus. 2. The primary inverse relations are found by determining $z$ from the equation $z^n\; =\; a\; =\; x$. They are

3. The primary reciprocal relations are represented by $a\; \backslash div\; x,\; (a)$ and $(x)$ being interchanged, and are $a\; +\; x,\; a\; \backslash times\; x,\; a^x,\; a^\{\backslash frac\{1\}\{x\}\}$ to $(x)$ terms, &c.

4. The primary inverse reciprocal relations are found by determining $(z)$ from the equation $a\; \backslash div\; z\; =\; x$, thus from $x\; =\; a\; \backslash div\; z\; =\; a^\{\backslash frac\{1\}\{z\}\}$ is deduced $\backslash log(x)$ base $(a)$.

5. By taking alternately the reciprocal and inverse forms of the first three relations, we shall find that they circulate and only introduce known functions. Thus $x^a$ becomes $ax,\; \backslash log(x)$ base $(a),\; \backslash log(a)$ base $(x),\; a^\{\backslash frac\{1\}\{x\}\},\; x^\{\backslash frac\{1\}\{a\}\},\; x^a.$

6. Every finite expression now used, is composed of these primary relations. It cannot therefore be a matter of surprise that analysis should frequently present expressions appearing to pass into each other discontinuously and without a law, when the laws of succession in these fundamental forms have never been stated; and the functions beyond $x\; \backslash pm\; a$ have not even been mentioned.

PROP. IV.—To find the Differentials of Functions under the first relation.

Similarly

PROP. V.—To find the Differentials of Functions under the second relation.

By a process nearly similar,

PROP. VI.—To find the Differentials of Functions under the third relation.

If the expression be $f(x)^a,$

If the expression be $a^\{f(x)\},$

If the expression be $\backslash log\; f(x)$ base $(a)$ or $\backslash frac\{L\; f(x)\}\{L(a)\},$

$L$ denoting the Napierian logarithm,

Also, $D\; \backslash \{\; \backslash log\; (a)\; \backslash text\{\; base\; \}\; f(x)\; \backslash \}\; =\; D\; \backslash frac\{L(a)\}\{Lf(x)\}$ which is known.

The first three propositions will enable us to extend these formulae to every finite function at present used in analysis, but when we attempt to find the differentials in relations of a higher order, the difficulties increase very rapidly. Indeed, the direct calculus, which is generally considered perfect, is only less limited than the inverse. Let the most expert analyst attempt, in analogy to Prop. 2, to express the differential of the reciprocal function $f(x,\; a)$, by means of the direct function $f(x,\; a)$; let him endeavour to develop $(x+h)^\{\backslash frac\{a\}\{a\}\}$ in analogy to Sir Isaac Newton's expansion of $(x+h)^\{\backslash frac\{a\}\{a\}\}$; or let him search for the differential of $a^\{\backslash frac\{1\}\{x\}\}\; =\; a^\{\backslash frac\{a\}\{ax\}\}$ to $(x)$ terms, and he will be forced to admit the great imperfections of the instrument.

To find Successive Differentials.

If we take the differential of $Df(x)$ considered as a new function, we obtain the second differential function $D\; \backslash \{\; Df(x)\; \backslash \}$, which is represented by $D^2f(x)$, and, in the same manner, we may find the value of $D^n\; f(x)\; =\; D\; \backslash \{\; D\; \backslash \{\; \backslash dots\; Df(x)\; \backslash \}$. The laws observed by the successive differentials of the functions previously examined are very complex, and can only be expressed by separating the differential characteristics, and operating upon them as independent symbols.

PROP. VIII.—To find the Coefficients of the Differential Expansion.

Since $f\; \backslash \{\; x\; +\; (i+h)\; \backslash \}\; =\; f\; \backslash \{\; (x+h)\; +\; i\; \backslash \}$, we develop these expressions separately, and equate their homologous terms. The first becomes

which is equal to

+ ...

The second becomes

which is equal to

+ ...

And now, by equating the same powers of $i$ and $h$, in the second terms of each development, $f(x)\; =$

Hence $f(x+h)$ becomes

Cor. 2. $D^\{-1\}f(x)$ is the inverse differential function, and is of such a nature that $D\; \backslash \{\; D^\{-1\}f(x)\; \backslash \}\; =\; f(x)$.

where $D^\{-1\}f(a)$ is the value of $D^\{-1\}f(x)$, when $x\; =\; a$, and it is called the Correction.

From these theorems the inverse calculus may be deduced by direct processes.

Cor. 4. There is nothing in the nature of analysis which necessarily limits expansions to the differential form. The series may be conceived to proceed by other functions of $x$ and $h$; it even may not consist of a succession of sums, but may ascend by products, or according to any other relation. Thus

we may find the successive terms of $(x\; \backslash times\; i^h)$ when developed in the form

For, let $f(x)$ be called the factorial function of $f(x)$, and let its relation to $f(x)$ be thus expressed, $f(x)\; =\; Pf(x)$; then by expanding

and by equating the like powers of $i$ and $h$, we shall find $f(x\; \backslash times\; s^h)$ become

a theorem entirely analogous to that of Taylor, and presenting a factorial calculus on principles similar to the differential. It is not, however, necessary to deduce the values of $Pf(x)$, by making it the subject of distinct investigations, since Mr Herschel, on seeing the formula, has discovered an expression for $Pf(x)$ by means of $Df(x)$, and observes, that the factorial calculus so harmonizes with the differential, that either may be established, and the other deduced from it.

Differentials depend upon the development of $f\; \backslash left\backslash \{\; x\; \backslash pm\; a\; \backslash frac\{h\}\{2\}\; \backslash right\backslash \}$, factorials upon $f\; \backslash left\backslash \{\; x\; \backslash pm\; a\; \backslash frac\{h\}\{2\}\; \backslash right\backslash \}$, and it is obvious that a wide field is open for $f\; \backslash left\backslash \{\; x\; \backslash pm\; a\; \backslash frac\{h\}\{2\}\; \backslash right\backslash \}$.

Mr Whewell further suggests the idea of an expansion, in which the relation of the successive terms shall be expressed by a functional characteristic.

The object of the calculus is not to show that every function of the form $f(x+h)$ must necessarily proceed by integer powers of $h$. The legitimate design is to exhibit the relation subsisting among the successive terms, and to find the value of $Df(x)$ in complex expressions, by means of the values in the primary functions. It supposes the general form of development given, and leaves the determination of the second terms, in the expansion of the primary functions, to the artifices of the ordinary algebra.

PROP. IX.—To find the Differentials of Continuous Quantities.

Let $(x)$ be a known discrete variable, and let any continuous quantity, which varies with $(x)$, be represented by $\backslash Psi(x)$, an unknown function of $(x)$. If then $x$ becomes $x+h$, $\backslash Psi(x)$ becomes $\backslash Psi(x+h)$, and we obtain the following equal ratios:

Let $Q$ and $Q\text{'}$ be discrete known variables, one always greater, and the other always less than $\backslash Delta\; \backslash Psi(x)$. Also, let the ratios

be such, that both may be made to approach nearer than by any assignable difference, to the ratio

Then, since $Q$ and $Q\text{'}$ are one greater and the other less than $\backslash Delta\; \backslash Psi(x)$, the ratio of

cannot possibly differ from that ratio to which

approaches nearer than by any assignable difference. It may further be proved, that the only ratio which can be true is,

If, therefore, we are investigating any curve, or the motion of any body, where the analytical value is unknown, Newton lays it down as a practical rule for finding the differentials, that they will be in the limiting ratio of the corresponding increments, viz. in the ratio to which the increments approach nearer than by any assignable difference. This practical rule, offered by the great inventor, may be demonstrated ex absurdo, with all the rigour of the ancient geometry. The formula of Taylor not having been distinctly expressed, Newton could not lay down the analytical theory of his calculus, but he left a practical rule, which is the best that has yet been proposed, and he left an illustration, the most beautiful in the circle of the sciences. He compared the differentials to the velocities of a body in motion. Many modern writers still define differentials by means of the limiting ratio; but this is as inconsistent, as it would be to define multiplication by the mechanical rule for finding the product. The method of infinitesimals is now universally exploded, having been originally proposed by Leibnitz, who, perceiving that Newton's results depended solely on the second terms, supposed that his calculus must consist in making the subsequent terms vanish. The results were here happily correct, though a similar attempt to appropriate the theory of gravitation was wholly unsuccessful. The latest English work on this branch of analysis, is the Translation of La Croix's Treatise, where Mr Peacock gives a luminous view of the different theories on the subject. (x. x.)