Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

8

de cosx=(-1)" sin &. sin cos 2. sin cosé x ...sin cos” – 10. d, log"d=x-1.(log,x)-1. (log? x) -1... (log."–1x) -1.

6

DEVELOPMENT OF FUNCTIONS.

(8.) d. f(x + y)=dy f(x+y).

Taylor's Theorem. Let Ex(u) represent what u becomes when + Dx is substituted for a, then

Dx (D x)? (Dx)3 Ex(u)=u+duu + du +d; u

+ &c.

[ocr errors]

1.2

1.2.3

error

If the n first terms of this series be taken, the limits of

(DX)" are the greatest and least values of d"u

1.2.3...n provided there be no value of x between 8, and + Dw that renders u or any of its differential coefficients infinite.

(L. C. D. 169.) Maclaurin's Theorem. Let (u)x=0, dou represent the values of u, d" u, respectively, when x=0, then

? u=(u).-0 + dou. + du. + decou.

1.2

1.2.3

[ocr errors]

+ &c.

1

E

o de cos” x=(-1)"P, sin cosm–10. d."

du log;"x=Pm (log.m – 1w)-1. Equ=Sm(Dx)m–1.d,m–1:u. Let Exu - U=DzU, then

(−1)n-m+1 +S,Deu1+2.5

2n-m+1. d,u where dz*+1:0.

n

DZU Dx=

[ocr errors][merged small]

m

[blocks in formation]

n-m+1

nem

ag

- 1

je u=S,,20–1.dm-:u.

1

1

+

[ocr errors]

В

+

1.2.3

(9.) Taylor's theorem applied to two variables. Ex,(u)=u+{d, u.Dx+d, u.Dy}

+ {d?u(DX)* + 2d_d,u. Dæ.Dy + d. u (Dy)"} 1.2.3{d}u (Dx)3 + 3 dd,u. (Dx)* Dy

+ 3d,du .(Dy)? + d; u(Dy)} + &c. dm dou=dd u.

(L. C. D. 25—30.) (10). Laplace's Theorem. Let y=y{x+#.$(y)}, in which x is independent of w and y, then f()(*).d.f4} .d,{py().d.f4(x)}

203

.d{p}().d.ft (x)} + &c. Lagrange's Theorem. Let y=x+w.p(y), then

X2 f(y)=f(x) + {p(x).d.f(x)} + d.{P(x)]*.d.f()}

2013

d? {P(x)].d.f(x)} + &c.
This theorem is a particular case of the former, in which

V{x+x.0(y)}=x+8.0(y).
E.,(u)=S»S(Dx)mėn. (Dy)n-1.0,*--:d,»–1:u.

If a third variable z be supposed to receive an increment, then E.,(u)=S»,5,8,(Da)m--. (Dy)**. (Dz)-2. d. *=:2, 97:ds=1:u. Maclaurin's theorem applied to a function of two variables. U=S,2m

in-1.dmo:dd:u, = S, S,2m Y1.

-1.dm-":dn:u. (Jarrett, Camb. Phil. Trans. Vol. III, Pt. 1.) $ f(y)=f(ö) +S dm-!{p4(x)}".d.f4(x)}.

1.2

+

[ocr errors]

1.2.3

[ocr errors]

m-1. S,ya

m

x=0

[ocr errors]

m
m

[merged small][ocr errors]

8

If f(y)=y, then

2013 y=%+ 1.0(ə) + .d.0(:)]° + .dp()+ &c.

(L. C. D. 107–9; Tr. L. Note E; L. D. C. 49–61.)

1.2

1.2.3

IMPLICIT FUNCTIONS. (11.) Let d,u=M, d,u=N, then

M +Ndzy=0, from which equation the value of d y may be found. By successive differentiation, the following equations may be obtained, in which u, un, &c. are functions of w and y: Un + ugd y + uz(dzy) + Nd;y=0, U4 + Uzdzy +48(dzy)” + Uy (dzy)* + (Ug + ugdzy) d’y + Ndiy=0, &c.

&c. from which the values of d; y, dzy, &c. may be obtained.

(L. C. D. 41-8; L. D. C. 1003.)

TRANSFORMATION OF THE INDEPENDENT VARIABLE.

dy

[ocr errors]
[ocr errors]
[ocr errors]

(12.) 0,X =

d; a = dx.

(dzy)3 3(d;y)? dxy.dlly da

&c. = &c. (dzy)" If x and y are considered as functions of a third variable t, then

d, y dey= dex'

(da) 3 diy= (d, *.dog 3d,.dog.dog + 3 (d? a).d,g-d, .doc.

(dx) &c. = '&c.

(L. C. D. 57---69.)

d?y=dw.d?ydịw.dry

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

(13.) Elimination of an arbitrary function.

Let z= f(x,y)=P(t) suppose, then
d,x=dep(t).dt, and d,x=d.p(t).d,t, and hence

d22 d. f(x,y)

d, d,f(x,y) In the same manner, by repeating the differentiation of any given equation, two or more arbitrary functions may be eliminated.

(L. C. D. 77–9.)

PARTICULAR VALUES OF THE VARIABLE. (14.) Let u=0, when x=a, then u must be of the form

v (2 a)",
v being a function of a not containing the factor w a.

deu=v,(w - a)-1
&c. = &c.

dru=vn (x a)"-",
n being the greatest integer $ m.

If m=n, then du. u=vn, which is the first differential coefficient that does not vanish, when w=a.

If m is not an integer, then dr+lu and all succeeding coefficients become infinite, when x=a.

[blocks in formation]

U=

v. (x - a)"

ve (xa) Let n, r be the greatest integers > m, p respectively, then when x=a, if n>r, u=0;

n<r, u, and all its differential coefficients are

infinite;

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

U =

m<p, = . (L. C. D. 143–53.) $(w) If ur

when x=a, the value of u may be thus P(x) found: substitute a +Dx for x in f(x) and $(w), and expand in series of ascending powers of Dx, then

a (DX)". + a,(Dx)", + &c.
6(Dx)": +b, (Dx)". + &c.

a,(Dx)", -". + a,(DX)",=+&c.
If m, >n12U=

6, +b (Dx)"-", + &c. = 0, when Dx=0.

a, +a,(DX)", -m, + &c. If m, <n,u=

b(D x)"; – m. + b (D x)", – m. + &c.

when Dx=0.

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

1

[ocr errors]

U=

If u=f(x).f.(x)=0.0, when x=a, put p(x)=

fi(x) f(x) then

when x=a.
0@)
f(x)
If u=

when x=a, then
P(x)
{p(x)} -1

before:

{f(x)} -1 under which form the value of u may be determined as above.

(L. C. D. 12; L. D. C. 107.) Сс

UE

as

« ΠροηγούμενηΣυνέχεια »