in
DIFFERENTIAL
INTEGRAL
CALCULUS
C-t», —
£= V^+(af)
S-7 «
*',=sinh (■j+c*)
y’ = f(x.y) (1')
y'=f(x, y)
y |*=*, = I/o
d£=f(x, y) (H
C> (3)
i=fA*)h(y) (O
(3)
(1)
/(*. y)=f(i>i)
<1
(i)
dXy OyXy + byÿy
(6)
(7)
x'djrrr=ï;
C/(*-2)* + (y-l)»=e
y=v(x)§T$dx+c°(x) (6)
^~FfT^/=(JC+1),
HU-Fnt,)+t’3r=(*+,>*
u{ê-2xv)+vTx=-2*
(2)
(4)
|J= liüdx+v' (yy^N (x' y)
<£(*. y, C) = o (1)
*-*+i-[*-*+4r=°
y=x%+^{%) (h
[x+Ÿ (p)]d£~o
/■+(=)
p=v(p)+[*?' (p)+%
F{x> y* ï)=0
(2)
«M*. y> C) = 0
(4)
(5)
ta"»-T!rlbr <6>
ÈL
£_i
0(n) = /(■*. y, y’, ..., 0*"-»)
p_il±Æ*
y'=îhî(l-x)dxs=ij(lx-t)
%=p- Then S=i-
ÿ=-L VT+p*
in{p+/T+7) =-+Ct
p=sinh('zr+c‘)
(3)
= ±Kcl-sr,/»
■f
Y
sin -nrsmJLw-
FT=T- Yï*‘
y" = f(x, y, y')
(3)
y[ y2
W (ÿi. yt) =
y[ y\
W'x (ylt y2) = (y^—ylyA' = y^l—yly*
W'+a.W^O (6)
r0=c
(r),=,.-c=o
ÿx=p = 0, y;_ 3-0
=*H:
^=1, 2
k* + q = 0, q> 0
k\ + pk1 + q = Q
y = ÿ+y* (3)
CJ = Ti(*). C; =
. 1 , 3 *
ÿ*==*(— eX
%+ay=b (io)
% + ay = 0
C[y[n~" + +... + =/(*) j
y* = C1yl + Ciy2+...+Cnyn (5)
(1)
(H
Q^r + *-% + ky = — k
*.—7+ /f3?. *. = -!•- YT-i
y = C1e^t + CJe~^* =(C, +(3)
y”+qy=o (4)
ÿ,=7i?=S¥Wsi"(“(+'p')
D ^ pî y a -x2)2+v2*2
X= J/T=!
(5)
(7)
(3)
(4)
f (6)
(8)
}
(H
lf(0-*(0l<« \
(3)
7ïf-=M*. y)
. . | (4)
__cx0 + gyn-x„X2 „x., , x0X1—cx0—gyn^t
lim y (0 = 0
y=ÿ(t, Cu C2) J
x = y(t, x0, ÿ0) 1 y=V(t, *o. I/o) j
4
J*(0l
ar*+y
cp -.f, x2 c
^2 +A, = 0,
£?-'(*•§) <26>
(1)
y„ =* ç (*<>). Vx = «P (*i). • • •. yn = «p (*■)
(2)
y' = f(x,y) . (1)
y"=%+%y' (3)
yi=y»+-çy<,+j^y*+$\yo
yt=ye+jye+xrtyi>+ -âi-y*
a = x„, x = xk+1 = xk + h:
*/*+!= yk + y y'k + y AyLi + ^ A^i-s + j
y* = /i (*., y0, z0)
—* +4 = 1.
'•‘r Y»3.
+ c* sin yi)+* ^2 (Ca cos yj+c*sin ï%) •
*=/(*> y)
Vn,t yntt •••» ••• (2)
b /
//>=$( $<*• + **)«& )dx
'as. = /(/>,) As,
«M*. y)>®Ax, y)
/(*. y) dxdy
H J •'«"-p’prfp U-j [
= e~ptp dp^dQ = — ~ ^ e~pt j d0 = n (1
S (S*"p,pdp
Æ ss
*i =
(4)
(5)
Mt[h, /(!,-, ri,)]
(3)
V l+ffill. r]i) + fy(li, Tl/)
lim S VT+f?(h, nù + fydi, h,) Asf
Y R1—*1 —y2
A//=(g? + »tf) AS,
Jj(«+n» AS,-
(y ^
SS
(1)
2 A5<-
2 T),AS,
SSdxdy SS
S 0)
>> r
/ = $$$F(0, p, z)pdQdpdz (1)
de
= Yo[
Ans. P*pdpd0 = -j-. 21. J J dy dx. Ans. J J pdpd0=-^-.
F+irH
~VT and J ~YT-Ans■ h' V T-
a « 2 FiAs‘ = S (*/» và Axi+Y (xi' üi) Ayî\
=*(*,)•
0 %Y yù*yi = B
(2)
y=y*(x)
[ÿi W<ÿiW].
(M.)
/= f \dJLdJi + dJidl\dt
y, z)ds=lX[v(t), ♦(/), x(t)]V'P'(ty + V(ty + x'(t)tdt
F'
(5)
s
z = f(x, y)
$$zcos(n, z)do = ± J S z(x> y* /(*• y))dxdy
MS
SSk^rnda^wSSrnda
«
j/i+
(ÊL
V
(ÊL
(ÊL
a)
SS^-§dxdy^-SS^cos(n’y)da (7)
+(ïr-SBcos(n’ ^-)cos
dv.
(M.)
T mv2~mvl= j
'“ïï( I
v ==iêc+Jdjj + fo-k
(5)
A L
div (grad «) = !(!)+£(%) + ê{liï)
II (£+0)^=1 [Scos(n- Xî+Tysin{n’ x>]*
III l”to-uàv)dxdyd*=^ (o|-« ^do
III (^^<^^^)d*dyd*=!![Xcos(n’ cos("’ y)+zc°s(n> zïïda
lz)
lz
SSa^da-SSu^da=0
s4= 1 +T+(t+t) = 1 + T + T = 1 + 2'T
S,= 1 +t + (t + t) + (t + T + T + t) = 1 +3’T
t+(t) +(t) +---+(2^n)"+---
f( 1) = «1. / (2) = «2 /(«)-=«» (2)
S / (*) ^
1_±+±_±_L.
rn (x) = m„+i (x) + «„+2 (*)+•••
l*l
(3)
(4)
«1 (*) + «,W+...+H„W+... (5)
+ [«t (*) — “«(«)] + ••• + K (*) — “n (a)] + .. .
KI + K*ol
(5)
fl <"T^
f(x)=J(a) + ^r(a)
+ r(a)+... + {—^r(a) + Rn(x) (1)
R» = i^rnr f(n+" [<*+0 (*—*)]. o < e < î
/«-/«o+^r («)+fe^2f w+... +—r:r”(a)+---
e*= l+* + 2ï + 3, + -- - +^i +
(5)
(6)
ex+iy _ gX (C0S y _|_ (• siu y}
(1)
(2)
/ (0) = 1
MO:
ii
(4)
[(r + *)(r + fc—l) + (r+ *)—/>*]a* + a*_, = 0 or
(4)
...]
(5)
X L* 1 2-3 + 2-4.3-5 2-4-6-3-5-7*‘ • • • J
~Y7 \x~3f+5T-'• "J
, ,. v (~1)V
U\
[-+(T+Je+1)]^+,==S+T + 2T + 3T+Jc + 1
l/(*. 0.) —/(*. 0i)|
S/(*.
\yn—yü\
\y%—yil
\y3—y3\ =
I y—yn\<
t-f^y) (0
(6)
, i , ,i
9.
V*
+ ...
+ 3fsina+4rcosa_--- •
-+*-')4+... •
(x-i-2)n I
73. j-L-, . 4>is. 2 ç£l(l*l < ^2 )• 74
^«s. !+£+£+
+1 ai I +1 I +1Û21 +1 bt | + ... +1 a„ | +1 bn | + ...
(3)
if'**-
(1)
a*=ir f*C0S^^=ir[*!1T-Tn-T f »«"**“*j=
fl#=-sr I f{x)dx=T
a° = iS f
(1)
(2)
(3)
[0, n].
i I rwdx-zg— ;£>*<**+pa)+f+y i>*+p»
t+t i>*+6î>
(i)
(2)
S
(3)
sn(x)—f(x) = ~
J
i 2sinT
2sinT
|^(a)|<[M + M]—V
(2)
(3)
«1 = 7-.
«« = —.
(5)
(7)
(1)
(2)
(3)
(4)
(5)
(6)
s
(2)
A = S A fit and B = 2 Bfit
<*='’2 ») 06)
— y + ... . AnS. y.
(1)
1 + iR + lJ-O (5)
&+A%-c*ê-CL&~°
(1)
= (1)
(3)
[-*&LH- H
(4)
(5)
(6)
(7)
(8) (9)
u(x, 0) = (p (x) u (0, 0 = fi(0 u(l, t) = ^(t)
(1)
(2)
(3)
; p <8>
(l—;+ a*tJt(«/+i.* + (9)
ir=a-w
+ ( j «P (a)
*'
„ vu
*(P) =
(1)
«|o = *(M) (2)
(8)
IW(-t~div(pv))d£ù=0
An=~^^^cosnt dt
= i+L [(ie,'('")B+(ie'<<<"p>)n]
+
(i)
«Uti —+
Ans-0(x’ /)=5E(àrrpsin
u(0, o = o,
(2)
(3)
-f(f)+/(«o ai
(4)
(5)
O)
(2)
(
(3)
L{r(t)}=le-r‘f'(t)dt (2)
p«F(p)-f-*r(o
ao]e~ptÿzdt + ai]e~ptiï£rdt +
(6)
w+x=l
i£+*w+2x=t
(i)
(p + T)2+(l^a*—t)
yr“.-4-
(B-4)
(2)
VJ
ir+4in'+3!/=0
\h (T) t
S fi(*)/*(t—x)dx
{} fi (T)/* (<—T)dtj = 5
Pi(p)FAp)
]Â~T
+
Y^4
(4)
Y
l/ a
(6)
S- + 2n| + ^ = 0 (1)
O)
(2)
(3)
(4)
(5)
* <*> = (*«-«,*)* + 4^ {(*»-«>«) sin o,/ —2niocoso)/}
11 i ! ! ! 1 M !
(7)
(8)
(9)
J_
O)
60(t-h)
(2)
(3)
S 1 (4)
S = ô(0 (6)
|r = /(0 + ô(0
= 0)
(O
P(i4) = P(fl1)-P(i4/fl1) + P(fl,)-P(i4/flt) + P(fl3)*P(»/Ba)
p^>=ôÆo.8=ë=a675
AA ... AÂ J ... Â
(x = i) = Cj
P«<*,= l-[p(*=y)+p(*=3-)+p(*=|-)]
p(^4)=p(^=4)+,,(^4)+p (*=4)
M [x] = 2 **/>* (1)
= 2 xkPk (O
M [*] = np
M [l] = £ lkpk-
(2)
l{x)~^7üexp
J
2. G(0) = 0.
(4)
(5)
(5')
f(a-,<;<„+,)=£ [®(-pT)—'® (-ïtt);
PH<ï<0 = ®(^-) (7)
I exP [-T^l
,w=ï?iïexp (“■»■)
(2)
r<-£<;<£)= J
(3)
p (a < i < W-4-[é ( t£)-(i. (2=2)
d = Jj1-E»)'dx-7751*'exp (■~£)'ix
-775[“0’exp(-w-)].'“-pH
O)
(3)
(4)
[(6)
2 2p/y=l (U
=_L/'ü_oU---W-
w= Sj(x'v)dv
(8)
2ol J
(6)
yfe:exp (-p!liM "Hh;exp (-p‘|)*
2 pü = i (2)
= ~n (3)
2 aï
^<-<^[H7fm)-H?T7k)} <8>
(7)
(i)
(4)
y=\\yiÿ* ••• 0.11
(6)
(10)
(O
(2)
(8)
(1)
(2)
(3)
(4)
(5)
(6)
(1)
(2)
(3)
(4)
(5)
(2)
(3)
(4)
(5)
(3)
(4)
(5)
(6)
(5)
(6)
(7)
(8)
(9)
(10)
*î = «n*i+'“21*2+ «8.*s ]
(3)
(4)
(5)
(6)
(7)
(8)
(1)
(2)
(3)
y» ~ ^3-^3
(4)
(5)
(6)
O)
(2)
(8) (9)
(10)
(5)
(6)
(7)
(8)
(10)
(2)
(3)
(l)
Il fl (011 = 11 fl// (011 (<=1.2 «; /=!. 2 n) (2)
(3)
j_
(0
(4)
(5)
D||a|| = ||£>û|| (6)
S
(7)
(8)
(9)
d\ ^ (1)
= ®mxl "t" ®n2 X2 “h • • • 4" annxn
(0
(5)
(8)
«„
(9)
(10)
(H)
(1)
fon-.
(2)
(3)
a-Il-<11=Il “•INI •«il (5)
0)
ll*(0 II = Il *0 II + S II a (Z|) Il (|| *,11
+ | Il <*(**) II (il *o II + J II a (z,) || (.. .)d2^jd2^dzt
