Конспект лекций по математическому анализу. Шерстнев А.Н. - 135 стр.

TO ESTX '(x) < 161 
 r2 < '(x), ^TO PROTIWORE^IT TOMU, ^TO ' DOSTIGAET
MINIMUMA W x ). fUNKCIQ ' DIFFERENCIRUEMA W x, I W SILU 84.2
                n                                                                       
   '0(x) = ,2 P (yi , f i(x)) @x     @f i (x); : : :; ,2 Pn (yi , f i (x)) @fni (x)
                     i=1          2 1 11 3                  i=1                @x
                     @f i  6 y , f (x) 7
            = ,2 @xj (x) 
 4            : : : 5 = 0:
                                   yn , f n (x)
                                     @f i 
nO MATRICA qKOBI f (x) = @xj (x) OBRATIMA (SM. ZAME^ANIE POSLE
                           0
FORMULY (2)), TAK ^TO y , f (x) = , ^TO I TREBOWALOSX.
     (W). pUSTX y 2 V PROIZWOLEN, y + k 2 V; x = g(y) I g(y + k) , g(y) = h.
tOGDA k = f (x + h) , f (x). w SILU (3) kkk  21 khk, TAK ^TO k ! 
WLE^ET h ! . oTS@DA SLEDUET NEPRERYWNOSTX OTOBRAVENIQ g. dALEE k =
f 0(x)h + o(h) (h ! ). pOSKOLXKU LINEJNOE OTOBRAVENIE f 0(x) OBRATIMO,
IMEEM
                   g(y + k) , g(y) = f 0(x),1k + o(h) (h ! ):
nAKONEC, klim  ko(h)k = lim ko(h)k 
 khk = 0, OTKUDA
             ! kk k         k! khk        kkk
               g(y + k) , g(y) = f 0(g(y)),1k + o(k) (k ! ): >
     2. p R I M E R. oTOBRAVENIE f (x; y ) = (ex cos y; ex sin y )((x; y ) 2 R2 ) NE-
PRERYWNO DIFFERENCIRUEMO, PRI^EM KASATELXNOE OTOBRAVENIE (SM. 77.6)
OBRATIMO W KAVDOJ TO^KE (x; y) 2 R2, TAK KAK det f 0(x; y) = e2x =                    6 0:
nAJDEM PROIZWODNU@ OBRATNOGO (K f ) OTOBRAVENIQ
                    g(u; v) = (g1(u; v); g2(u; v)) ((u; v) 2 R2):
mY IMEEM
 f  g(u; v) = (expfg1(u; v)g cos g2(u; v); expfg1(u; v)g sin g2(u; v)) = (u; v);
OTKUDA expfg1(u; v)g cos g2(u; v) = u; expfg1(u; v)g sin g2(u; v) = v. pO\TO-
MU
    g0(u; v) = f 0(g1(u; v); g2(u; v)),1
                  fg1(u; v)g cos g2(u; v) , expfg1(u; v)g sin g2(u; v) ,1
             = exp
                  uexpf,gv (u;
                          1 v )g sin g 2 (u; v ) expfg 1 (u; v )g cos g 2(u; v )
                              ,1                        
             = v u               = u2 +1 v2 ,uv uv :

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