Page images
PDF
EPUB

in the equation (2), or their equivalents

du du
dy,' dy'

in the equation (1), are called the partial differential coefficients

Du
dx

of u with regard to y,, y2, respectively. Finally, is called the total differential coefficient of u.

The equation (3) shews that the total differential of u is equal to the sum of its partial differentials.

Differentiation of a Function of any number of Functions of a single Variable.

29

21. Let u = f(y1, Y2, y1), a function of three variables y1, Y2, Yз, each of which is a function of x. Then if u', y, y2', y', be simultaneous values of u, y1, Y2, Y,, we have

u = ƒ (Y1, Y2, Y3), u' = ƒ (y1, Y2, Y3'),

[merged small][ocr errors][merged small][ocr errors][ocr errors][merged small][ocr errors][merged small][ocr errors][subsumed][merged small][merged small][merged small][subsumed][ocr errors][ocr errors][subsumed][merged small][subsumed][merged small][ocr errors][subsumed][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

Du

coefficients of u with regard to y1, y2, Yз, respectively; and dx the total differential coefficient of u with regard to x. If we adopt the suffix notation, the equation (1) may be written

Du du dy du dy2 du dy

+

2

=

+

dx dy, dx

2

dy, dx dy, dx

.(2).

[ocr errors]

Multiplying the equation (2) by dx, and putting

[merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][ocr errors]

we have

[ocr errors]

The same theorem may evidently be extended to any number

[merged small][ocr errors][subsumed]

Y1, Y2 Y3,....Y,, being any n functions of x, then

Du

=

n

[blocks in formation]

du dy+ dx dy dx dy, dx dy, dx

or, replacing du in the expressions

du

,

du

du

dy, dy' dys

[merged small][merged small][merged small][merged small][ocr errors]

by du, du, du,.... du respectively, and clearing the equation of fractional forms,"

[merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small]

It may therefore be stated as a general proposition, that the total differential of a function of any number of functions of a variable is equal to the sum of its partial differentials taken on the hypothesis of the separate variation of each of the several subordinate functions of the variable.

[ocr errors]

COR. If any one of the quantities y1, Y, Y,,.... y y1 for instance, be equal to x, which is the most simple form of functionality, then from the above demonstration it is plain that we may replace the corresponding term

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors]

Differentiation of an implicit Function of a single Variable.

[blocks in formation]

y being therefore an implicit function of x: then v', ', y', being corresponding values of v, x, y,

[merged small][merged small][ocr errors][merged small][merged small][ocr errors]

v = p(x, y) = 0, v' = 4 (x', y') = 0,

v' – v = $(x', y') -- p (x, y) = 0,

p(x', y) − p (x, y) ̧ $ (x', y') − p (x', y) y' - y

[ocr errors]

+

x'

[ocr errors][merged small]

y' - y

= 0..(1).

[merged small][merged small][ocr errors]

Now in the limit, when ' approaches indefinitely near to x,

[merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

and, first replacing x' by x, and then y' by y,

Þ (x', y') — $ (x', y) _ p (x, y') − p (x, y) _ dp (x, y) _ dv

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]

dy

(2).

This result gives us the expression for in terms of the

dx

partial differential coefficients of v with regard to x and y taken successively as separately varying. If we replace the symbol dv in the numerators of the fractions

dv dv

dx' dy

by the expressive forms dv, dy, we have, transforming the equation (2) from differential coefficients to differentials,

Dv=d_v + d2 = 0.

This result shews that if any function of x and y be always

C

zero, its total differential or the sum of its partial differentials is always zero.

[blocks in formation]

23. If v1 where v12, V3

32

[ocr errors]
[blocks in formation]

the variables y1, Y2, Y.,... .Y,, being therefore implicit functions

of x, then

[ocr errors]
[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small]

V2, V3, V4•••Vn'

together with n - 1 additional equations involving v2, 3, ... precisely as v, is involved in this.

Let x', y', y', Y.,....y, v, be corresponding values of х, Уг, X, Y, Y2, Y3,....y, v,; then

[ocr errors]
[ocr errors][merged small][subsumed][ocr errors][subsumed][merged small]
[ocr errors]

+ f(x', Y1, Y2, Y3, ....Yn) - f(x', Y1, Y2) Y... Yn)

[ocr errors]
[ocr errors][merged small][merged small][ocr errors][ocr errors][ocr errors]

whence, 8, 1, 2,....,, denoting the partial increments of the functions to which they are prefixed with regard to x, y1, Y2, ....y, respectively, and ▲ the total increment of v,, we have

[subsumed][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small]

Proceeding to the limit, we have, x, y1, Y2, Yз,. . . . Y., being the

ultimate values of x', y', Y, Ya',....Y

[ocr errors]
[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small]

be

The analogous equations in regard to v2, v, ... .vn, may established in the same way.

Multiplying the equation (1) by dx, we have

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]

or, if we express the different partial differentials of v by

[merged small][ocr errors][subsumed]

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

[merged small][ocr errors][merged small]

in terms of the n (n + 1) partial differential coefficients of V1, V2, V39.. .v.

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(Y2, Y2, Y3,• • • •Yn),

and that y1, y2, Y.,....y, are not all of them dependent upon a single variable, but that they are functions of several independent variables. Let u', yi, ya, ya,....y, be corresponding values of u, y1, Y2, Y3,

. Yn :

then

« PreviousContinue »