Конспект лекций по математическому анализу. Шерстнев А.Н. - 131 стр.

s U^ETOM 81.3 I 74.2
                            Z1                          Z1
  kf (x + h) , f (x)k = k 0      f 0(x + th)(h) dtk 
                                                   kf 0(x + th)(h)k dt
                        Z1                       0  Z1
                       0 kf 0(x + th)k khk dt = khk 0 kf 0(x + th)k dt:
sKALQRNAQ FUNKCIQ g(t) = kf 0(x + th)k (t 2 [0; 1]) NEPRERYWNA PO t. sLE-
DOWATELXNO
Z1             , PO TEOREME O SREDNEM 50.4 SU]ESTWUET t0 2 [0; 1] TAKOE, ^TO
 0
    k f 0(x + th)k dt = kf 0(x + t0h)k: >
    dLQ FUNKCIJ MNOGIH PEREMENNYH IMEET MESTO TO^NYJ ANALOG FORMU-
LY lAGRANVA.
    3. pUSTX f :       ! R (  Rn) DIFFERENCIRUEMA W I x 2 ; h 2 Rn
TAKOWY, ^TO fx + thj 0  t  1g  . tOGDA SU]ESTWUET t0 2 (0; 1)
TAKOE, ^TO
()                     f (x + h) , f (x) = f 0(x + t0h)(h):
 dOSTATO^NO PRIMENITX OBY^NU@ FORMULU lAGRANVA K SKALQRNOJ FUNK-
CII '(t) = f (x + th) (t 2 [0; 1]): >
    4. z A M E ^ A N I E. fORMULA () UVE NE IMEET MESTA DLQ OTO-
BRAVENIJ f : R2 ! R2. dEJSTWITELXNO, RASSMOTRIM OTOBRAVENIE IZ
P. 77.6. pOLAGAQ W \TOM PRIMERE h = (0; 2) 2 R2, IMEEM f 0(th)(h) =
(,2 sin 2t; 2 cos 2t) =  6  (t 2 [0; 1]). pO\TOMU  = f (h) , f () =6
f (th)(h); 8t 2 [0; 1].
  0

    x83. dLINA PROSTRANSTWENNOJ KRIWOJ
    1. pRIWED    EM TEPERX WYWOD FORMULY 60.4. pUSTX : [a; b] ! R3 |
GLADKAQ WEKTOR-FUNKCIQ, (t) = (x(t); y(t); z(t))(a  t  b), I 
(a = t0 <
t1 < : : : < tn = b) | RAZLOVENIE OTREZKA [a; b]. dLINA `j j -GO ZWENA
LOMANOJ, WPISANNOJ W KRIWU@, QWLQ@]U@SQ OBRAZOM WEKTOR-FUNKCII ,
RAWNA
     `j = [(x(tj ) , x(tj,1))2 + (y(tj ) , y(tj,1))2 + (z(tj ) , z(tj,1))2]1=2
           = k (tj ) , (tj,1)k:

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