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Analisi Armonica Tor Vergata

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Analisi Armonica Tor Vergata

Text of Analisi Armonica Tor Vergata

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    ! $ $

    $ B )$10 T( #%#U B #

    http://www.mat.uniroma2.it/~picard/SMC/didattica/materiali_did/Alg.Lin./ALG_LIN.pdf#algebralineare.succ_decrescenti_hanno_limite

  • -

    *! f g h *$( $ E $ ( T 8 RnU x0 B $ $

    $$( E T 8 $$ EU limxx0 f(x) = limxx0 h(x) 1 g 1 x x0 limxx0 g(x) = limxx0 f(x) = limxx0 h(x)# $ $

    #

    * ! ' $ $ $! $ B $ 6$ $ B #

    * $ $ T$ 2 .#3#% $U! f $*$( x 1# 7

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    )$ *$( ! f : A B g : B C

    $ ( g f : A C B $#

    ( *$( ! A B C R f : A B g : B C ' ( gf : A CB ' (g f)(x) = g(f(x)) f (x)#

    $ *$( ' B $ B #

    http://www.mat.uniroma2.it/~picard/SMC/didattica/materiali_did/Alg.Lin./ALG_LIN.pdf#algebralineare.Teorema_Weierstrass_max_e_min_su_compatti

  • 3

    *$( $! f : [a, b] RB $ $ [a, b] $ * f(a) f(b)#

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  • 4

    ( ! f g ' (fg) = f g + fg# 1 $ *$(

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  • +

    / ' {an > 0}

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  • 0! f : (a, b) R B ' n $$ x0 (a, b) $ $ Pn n

    1 f (x x0)n f(x) = Pn(x) + o ((x x0)n) x x0 6$

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  • %

    7

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    http://www.mat.uniroma2.it/~picard/SMC/didattica/materiali_did/#algebralineare.proprieta_archimedea

  • ,

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  • -

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    http://www.mat.uniroma2.it/~picard/SMC/didattica/materiali_did/Alg.Lin./ALG_LIN.pdf#algebralineare.norme_equivalenti_a_dim_finita

  • 3

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  • 4

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  • +

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  • %

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  • %

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  • %%

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  • %

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  • %,

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  • %.

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  • %-

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  • %3

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    B f(x) B $ x0#

  • %4

    + $

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  • %+

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  • .4

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  • -

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  • P -

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  • 43

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  • 3

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    m(En) } =0}#

    1 p 1 T#3U B $ B $$( 1 Lp 1 (4L4# A $ ( T R% )# - ( %S *

    8 $( $5 $$( 2V 6$ $ 1 $ ' 1

    #

    .1 ) + ' 1 p q ) 1 p q

    1

    p+

    1

    q= 1

    1 = 0)

  • %

    / )' ' 1 p q < 61/p+ 1/q = 17

    a b 0)

    ab 1pap +

    1

    qbq .

    1# ( $ 1 < p q ?> 1 1

    0 < p < 1 1 1 6$ p $$( TK ##U 6$ 5 fp B 8 $ 1= $$( # * * ( p TK #3#U * 6$ * 0 < p < 1 1 # $ 6$ * * 1 $$( # A$ * K ,#-# 1 (

    #

    #

    4" ) 7 5 p W B $# * *$( f $ fp = 0# B *$( $ ( 8 *$( $ $ $ T/#+#,3 #+#%U#

    A 6$ ' * W 1 p $ ( Lp(R) $ ( $ *$('W 6$( *$(#

    2 ( ( 6$( B ( #%#%# ( 6$( $ $ $( +

    .

    .1 ) ' a I) +

    a 5

    b I b a [a]) C

    [a] = {b I : b a} .

  • %,

    .1 ) 6

    '

    +

    .

    f g f = g + + 6

    1 )>)

  • LP %.

    '

    . ( ' $ 2 #3#. T J $; ( #%#U#

    4" ) I = [a, b] B $ 8 $ R $ 2'$ Lp(I)

    5 6$( *$( $' $ [a, b]

    1 b

    a

    |f(x)|p dx < .

    K p = ( L(I) B 5 6$( *$ ( $' $ I 1 ess supI f(x) < # 1 B '* 5 $ ( 1 B $ (

    $ 2'$ T( #-#% /( #-#U# 25 $ '' ' $ 2'$ 1= ' ! * f(x) $ $ x $ (#

    6 ) 6$( 6$ $6$ f L

    $ *$( $ g ess sup |f | = supx |g(x)|# T

    1 6$ 1 ( 5$ $

    TK #-#3U#

    4" ) +

    @)

  • %-

    $ $# $ $ ' Q [0, 1] ( ((

    ! A = {x0, x1, . . . }# A n N In $

    $1(( 3n 1 xn# E A = n=1In ''

    m(A)

    n=1 3

    n < 1 m(A) > m(I1) > 0 A Q 6$ A B [0, 1]# f *$( A# $ 6$( $ *$( g $ 6$ $6$ 1 5 G $ $ g $ $ '' $ (! * G $ '' 6$1 $ $x0 g T1= $ $ 1 $ U g(x0) = 0 g(x0) = 1# A ( T ( #U'' $ K x K g(x) = 0 g(x) = 1 K#A f = g $ $ K 1= 5 K 1 $ 6$ * 1 f g $$ 6$ $6$# 5 5 Z := {x : g(x) = 0} $ (1= Z '' $ $ $ x0 g $ g(x0) = 0# $ ( '' $ X x0 1 g < 12$ K# A1= A B $ '' $

    J K A# / $ J g < 1

    2 f = 1 1=

    $( f = 1 $ A# P$ $ f g '' $$ 6$

    $6$#7 6$ $ 1 6$( f $*$( g $ 6$ $6$ g = 1 6$ $6$#

    1 =

    10g =

    10f = m(A) < 1 $ (#

    4" ) ) p TK #3#U 1 Lp(R) $ (

    V * C2 T 1 R2U# * V ( Lp(R) $

    $ *$( 1

    f(x) =

    t1 se x [0, 1]t2 se x [1, 2]0 altrimenti

    t1, t2 C#7 t1, t2 C 5 *$( $ $ (

    * C2#

  • LP %3

    V 1 6$ ( B

    * ( p $ K #3#! Lp V p $ (

    2# p = 2 L2 V $ TW 2! K #3#U#

    4" ) ) 2 1 L2(R) $ K #3# * $ L2(R) 6$

    B

    f, g =

    f(x)g(x) dx.

    6 )' 1 $ ( #%#3#

    2 *!

    f, f =

    f(x)f(x) dx =

    |f(x)|2 dx = f22 .

    + )) 2 ' 1 p ) Lp(R) 5 ) Lp(I)

    I R 5 )

    1# p = B ' *! $

    )$10T $*U *$( T ## $U#

    7$ 6$ 1 p < $(( ( $ ' T2 #%#%U#

    fn $

    $ '! (N

    n=1 fnp $ $ 6$1 M > 0# A gn(x) =

    nk=1 |fk|#

    2 $$( >?> T#-#U gnp n

    k=1 fkp M #5 x gn(x) B $ $

    $ TYU 6$ $ +# 2 *$( g B $' TA( #+#,+U ( 2 &$ T#+#.U 1 g|p M # g B 6$ $6$#

  • %4

    A; 6$ x

    fn(x) B $ $ ' $ 6$ $ s(x)# A $ x $ B s(x) = 0# 6$ $6$ *$( s B $$ (sn =

    nk=1 fk(x) 6$ B $ *$( $'# |s(x)| g(x)

    1= |sn(x)| n

    k=1 |fk(x)| = gn(x) g(x)# P$ *$( s Lp |sn(x) s(x)| 2g(x)# E )( T #+#.%U *$( |sn(x) s(x)|p $ 1 sn s Lp 6$

    n fn $ *$(

    Lp#

    4" )$ A 0 < p < 1 ( Lp B $ ( T(ii) 5/( #-#4U# A; ( (C&1 #-#%- 1 6$

    (( Lp $ p > 0 ( d(f, g) =

    |f g|p

    T ( #%#U#

    6 )* 2 ( 1 '' (C&1 #-#%- ' $ ( $ ' T2 #%#%U B $ 5 # $( 8 ' $ ) #-#%+ 1 B #

    ) 1 fn J Lp /

    + +)

    1# p = B * ( # A 0 p < # A1= {fn} B )$10 $ $

    nj 1

    fnj+1 fnjp < 2j . T#,U

    A gk =k

    i=1 |fni+1) fni| g =

    i=1 |fni+1) fni |# 2 $$( >?> T#-#U $$( T#,U 1 gkp < 1 k 6$ $ 2 & $ T #+#.U $ gp < 1# A g(x) < 6$ $6$ 6$

    f(x) := fni +

    ki=1

    (fni+1) fni) T#,,U

  • LP %+

    $ 6$ x T ( ( $ $ ( #U# A 5 $ $ f = 0 1 f $6$# E B 1 1 (

    fni +

    k1i=1

    (fni+1) fni) = fnk

    f(x) T#,,U 6$ x# (( *$( $

    $

    1 $ ( p 1 $ K #3#+!

    ) , 0 p p )

    4" ) A p 1 ( Lp (

    Np(f) = fp :=(

    |f |p) 1

    p

    ,

    $ * 1 B 5 2'$!* ( (( '' $ 2 &$ T #+#.U 1 5 2'$ TB $

    ) #-#%+U# * $ 5 2 &$ ''

    $ ((# 7 $( qk 5 2k/n qk *$( n 5$ 6$ 1 Lp ( n ! 9 n *$(

    ( 1 6$ *$( ( Lp 2'$ B ' # 2 $

    n B $10 Lp

    T 2'$ U Lp #T)$! 0 < p < 1 6$ Np = p B $ T/(

    #-#4U Npp B $ ( 6$ L

    p B ! K#-#%3U#

    6 ) f Lp g L# $$ fg Lp#

  • '

    # 25$ B ' p = 6$ $ p < # V$J $ f 1= f !6$ $ f 0# A > 0 $ *$( u f $ *$( v f 1

    (v u)p 0 k N 1 1k< # In =

    [nk, n+1

    k

    )#

    : In * $ * $ $ $ B $ R#A1= f B $ B $ 5$ $ In# (f) 5 f TB5 (f) = {y : y = f(x) 6$1 x R}U f < M N = kM ! '' (f) Nn=N1In# A 6$ n Jn = f1(In)# 7 {Jn : n = N 1, . . . , N} B $ * $' $ 1 Nn=N1Jn = R 6$

    $' $ ( #+#3#E $ $ *$( 1 f $*# 8 *

    1 ; 6$ 5 ( * B $# A *$(

    Jn n 1 ( Jn 1 ''

    $ ( ## *$( =

    Nn=N1 fnn

    fn = f(xn) 6$1 xn Jn# B $ 8! ( f 5 Jn In

  • C(I) LP (I)

    1 f L 6$ $6$ $ $ $ (# E

    1 5 *

    $ *$( f $ $ A B

    ess infA f = sup{ infxA\O

    f(x) : O A, m(O) = 0} T#,+U

    5 $ (# 7 In = [ess infxAnj f(x) ess supxAnj f(x)) B 5 * * $ ( f Jn $; $ Zn Jn $ $ $ 6$ f

    ! f $ $ $ $ $ 1= $ f $ 6$ ( 2'$ $ $6$ *

    6$ ' xn Jn \Zn# ) 6$ $ *$( B $ *$( 1 f < 6$ $6$! * x Jn f(x) fn = f(xn) ' 5 In $1(( 1

    k< # P$ 5$ 1 $

    5( *$( T $$( T#,3U T#,4UU#) 5( *$( 1 B 8 # A K #+# > 0 n = N 1. . . , N $ On Jn 1m(On \ Jn) < /(2N + 2)# E = Nn=N1On \ Jn# 7 m(E) < # 2 #+# (i) n $ * $' $ Anj j = 1, 2, . . . 1 On = j=1Anj # 71 Anj $ $ n B ' 1 $ An0j01 $ An1j1 n0 n1 1= On (! 6$ $;

    $ On0 On1 1B $ $ T( /(N+1) 1=

    On0 On1 = (On0 On1) \ (Jn0 Jn1) = (On0 \ Jn0) (On1 \ Jn1) ,

    1 Jn $#E $ $ *$( 1 f $* $ $ # ) 6$ 5( * 8 '' $# *$( Anj nj *$( =

    Nn=N1

    j=1 fnjnj fnj = f(xnj)

    6$1 xnj Jn T $ 8 xnj Jn \ Zn#U

  • ,

    $ 6$ x AnjJn = Anj\E ' f(x) fnj = f(xnj) 5 In $1(( 1k < ! 6$

    |f(x) fnj | < T#.U

    x Anj \ E# A; *$( T$ $' U $* f *$ 5E $ #

    A 5$ $ ( || < |f | T || < |f |U

    *$( T *$( U 1 Jn Anj \ E 6$ $6$ $ * 6$ f # 7 6$

    B $J cnj = ess infIn |f(x)| T $ T#,+U! cnj $$( T#.U cnj |f(x)| x Anj \ E Jn# A; *$( TU =

    Nn=N

    j=1 cnjnj * $$( 1# 7

    *$( 5$ $ =N

    n=N

    j=1 tnn

    tn = ess infxJn |f(x)# V $( 1 B B f #

    / * L Lp ( Lp p ' > 0 K > 0

    f Lp p < fK |fK | K f fKp < )

    1# A1= f Lp $ E R *$ 6$ 1

    R\E |f |p < p/2# A fK $$ ( *$ E#

    $ 1 f # A n > 0

    fn(x) = f(x) |f(x)| n f(x) = n T nU f(x) > nT f(x) < nU# 7 |fn| n 6$ x 1fn(x) f(x) n # |f fn| |f | 6$ 1 *$( t tp Tt 0U B $ 1 |f fn|p |f |p# A )( T #+#.%U 1

    limn

    f fnp = limn

    |f fn|p = 0 .

  • C(I) LP (I) .

    A 6$1 K ''( E|f fK |p < p/2# 7

    R|f fK |p =

    E

    |f fK |p +R\E

    |f fK |p 0 s p < )1# P$ $ B $( 5$ ( 2 #+# T( $ $' $ U#

    * 1) ! Lp

    7 6$ ( $ ( *$( $ $ ( 1 (#. ,#-# $ ( ' ( #%#

    V B $ ( 1 T B $ 1

    limvv0

    T (v) = T (v0)

    T Tv 1 T (v)U#

  • LP +

    ' V,W T : V W 5 T 5 C > 0 v V

    TvW CvV .

    1! ( 6$ J $; ( *$ $ ' 8 A( ,#-# $#

    .1 ' V,W T : V W 5 T

    T = inf {C > 0 : TvW CvV v V }

    .1 ' X )

    X ( $ X

    X )

    .1 2

    #

    T Lp T (f) R f Lp R T (f) 0 f Lp 6

    5 7)

    / ' (i) M T Lp T = ReT + i ImT ReT ImT ) ' T 5 ReT ImT )

    (ii) M T Lp

    T = T+T T+ T ) ' T 5 T+

    T)

    1# A *$( T (ReT )(f) = ReT (f) $ *$( f # V 1 ReT B $ *$( |(ReT )(f)| |T (f)| Tfp $ $ ReT H ( ImT # f B T (f) =T (Re f)+iT (Im f) = (ReT )f+i(ImT )f ?; T ReT+i ImT$ *$( H 1= f Lp B '( $*$( f = Re f + Im f 1 T = ReT + i ImT # P$ (i)#

    A *$( T f T+(f) =sup {T (h) : 0 h f}# T+ B $ *$( ! * f 0 1

  • ,

    T+(f) T+(0) = 0# E 1 1 f 0 T (f) = T+(f)# f f 9( $ f = f+ f f+ = max f, 0 f = min f, 0# E0 f+ |f | f# P$ |T+(f)| |T+(f+)|+|T+(f)| =|T (f+)| + |T (f)| 2|T (f)| T+ 2T# f T (f) = T (f+) T (f) = T+(f) T(f) $ T = T+ T#P$ (ii)#

    B $ 6$ (# / 1 $ Lp Lq 1 p < q B 5

    $ p# K p = 2 1 $ ( F'L2 B ( L2 $ * 1 $ ( F' ( ( ,#-#,#

    + ) % # Lp Lq ' I R 1 p < f Lp(I)) ' q 1/p + 1/q = 1 61 )

  • LP ,

    )( ; #+#.# A1= T B $ $ 1 (A) =n (An) 6$ B $ $ T( #+#U# )

    $ 2'$ mH 1 m(A) =0 A B *$( $ Lp 6$ (A) = 0# A 6$ K>0 #+#%3! $ *$( g L1(I) g 0 1

    T (A) =

    g(x)A(x) dx

    $' A# h L(I) $ *$( # A1= m(I) < h Lp(I)#A 2 #4#% $ $

    *$( hn 0

    1 x I hn(x) h(x) T ( B ; x 6$ n U hn $ *$( T( #+#,U B $ '( *$( 1 An $# $ )( ; #+#. limn hn hp = 0 1= g B 1 ghn ghp g hn hp 0# A1= T B $

    T (h) =

    g(x)h(x) dx .

    A K > 0 IK = {x : g(x) K} h = gq1IK # A1= p q $ T( #-#,U ( T#+U 1

    p(q 1) = p/(p 1) = 1(p 1)/p =

    1

    1/q= q .

    A; 1

    IK

    g(x)q = T (h) T(

    IK

    g(q1)p dx) 1

    p

    = T(

    IK

    gq dx

    )1 1q

    .

    IK

    g(x)q (IK

    g(x)q) 1

    q

    T .

  • ,%

    A1= g B K * IK = {x : g(x) K} I 6$

    (IK

    g(x)q) 1

    q gq T# P$ g Lq(I)# P$ (#

    .1 $ ) 1 X 6 X 7 '$ X X )

    4" * ,

    g X

    f X g, f Tg(f))4" A ( X = Lp# X '$# * f X g X (

    f g, f f $ *$( $ $ X $$ f $ #+#-#

    A8 $ B $ (

    ! R%, )# , ( .S#

    .1 7 I 5

    )

    4" $ Lp #+#- $ 1 1 p < Lp B I# L B I! $ $ B L1 B 8 $ '$ B 8 L

    # ( #%#

    , 0 2

    .1 - ' m1 X m2 Y ) $ m1m2 X Y 5 E1 E2 6 Ei 5 + mi i = 1, 27

    m1 m2(E1 E2) = m1(E1)m2(E2).

  • ,

    4" 1 E X Y x X y Y Ex = {y : (x, y) E} Ey = {x : (x, y) E})D Ex Ey ( E)

    f 5 XY

    fx(y) = f(x, y) f y(x) = f(x, y) fx : Y C f y : X C

    )

    $ B $ 1 T R%% )# 3 . -SU#

    / ' f 5 X Y m1 m2 fx f y m1 m2

    ) E X Y

    (x) = m2(Ex) (y) = m1(Ey) m1

    m2

    X dm1 =

    Y dm2) g 5 R

    (x, y) g(x y) 5 R2)

    '

    . '' 6$

    $ 5$ (# ' ( #+#3 1 > 0 5 E :={(x, y) : g(x, y) > } B $'# A B := {t R : g(t) > }#P$ B $' 1= g B $ *$( $'T $ ( #+#3U# E E = {(x, y) : x+y B}# 1 6$ B $' $ * 1B ( T$ *$( $U $ $' TBU# ! 6$ * B $ 1 $(

    $ 1 (#

    :( 2 #%# 1 $!

    + 2

    (i) ' m1 X m2 Y f : X Y R+

    m1m2)

    (x) =

    Yfx dm2 (y) =

    Xfy dm1 6

    ))? !

    (7)

    ' f 0 5 m1 5 m2

    *$ $( 6

    7.

  • ,,

    X

    dm1 =

    XY

    fd(m1 m2) =Y

    dm2 . T#.%U

    (ii) 2 ' f L1(m1 m2) fx L1(m2) + x X 6 m17 f y L1(m1) 6m27+ y Y L1(m1) L1(m2) );)

    1#

    (i) ) *$( 1 ST S ' $ m1 T 6$ m2# 6$ $ B $ 2#%## A '( $ $ *$( m1 m2$'#A ! f 0 B $ *$( m1 m2 $'# A ) #4#. $ $

    sn *$ ( $ X Y m1 m2$' 1 f$$ ;! 0 s1(x, y) s2(x, y) . . . limn sn(x, y) = f(x, y) x y# n n ( 5$

    sn# 7 1Xn(x)dm1(x) =

    XY sn(x, y)d(m1 m2) =

    Yn(y)dm2(y)# A

    x *$( n n T )( ; #+#.U 6$ T#.%U $ $ ( )( ;#

    (ii) A ' *$( 1 6$ '( $ *$( T U# 5

    *$( f B 9( $ *$( f+ f T ( 2 #+#.U f+ f |f |# + (

    f+ f# E (i) $ 1 + L1(m1)#E 1 fx = (f+)x (f)x (i) *$( (f+)x (f)x 1 x 1 + TU# A1= + L1(m1) 6$

    m16$

    $6$# A 6$ x 1 (x) = +(x) (x) 6$ L1(m1)# A; *$ $( T#.%U

  • ,.

    f+ + f ! ( f # 2( *$ B #

    ' '

    f 5 (x) =Y|f |xdm2) ' 5 5

    Xdm1 < f L1((m1

    m2))

    1#

  • ,-

    6 * T 5 $ (0, 1) (1, 0)# E 1 T

    $ x = 0 y = 0 x+ y = 1# P$ *$ $(

    T

    f(x, y) dx dy =

    10

    ( 1x0

    f(x, y) dy

    )dx =

    10

    ( 1y0

    f(x, y) dx

    )dy .

    6 V 5 $ (0, 0, 1)(0, 1, 0) (1, 0, 0)# E 1 *

    V

    $ x = 0y = 0 z = 0 x+y+z = 1# & z = z0 * 0 1 (

    z = z0 B z = z0 0 y 1 z0 0 x 1 y z0#

    P$ *$ $( V

    f(x, y, z) dx dy dz =

    10

    ( 1z0

    ( 1yz0

    f(x, y, z) dx

    )dy

    )dz; .

    *$ $( 5 ( T z y xU# ) 5 $ V *$( 1/(x2+y2+z2) R# A 6$ 5 B X ) ' 5 B 1/(x+ y + z)X

  • " ,3

    6 5( ((

    V C ' z = 0 5 1 $ (0, 0, 1)# 1 *$ $(

    C

    f(x, y, z) dx dy dz =

    10

    ( 1z0

    ( 1y2z2

    1y2z2f(x, y, z) dx

    )dy

    )dz; .

    6 5( (( * S 5 1# 1 *$ $(

    S

    f(x, y, z) dx dy dz =

    10

    ( 1z21z2

    ( 1y2z2

    1y2z2f(x, y, z) dx

    )dy

    )dz; .

    + ) 3

    '' J ' 1 1

    $

    6$ *( $ ( $ * $ $ $

    $ 6$#

    Rn B $ ( ( B ( #% 8 # 2 (

    $ B #

    .1 ' x0 A Rn 6

    /

    A7)' f : A C x x0 V C U Rn x0

    f(U A \ {x0} V ) ,-

    limxx0 f(x) = > 0 > 0 x A 0 < x x0 < |f(x) | < )' x0 A 6 + f 5 x0 f 5 x0 limxx0 f(x) = f(x0))

  • ,4

    A 5( n = 2# $ $ B

    !

    ' f A R2(x0, y0) A) ' f (x, y) (x0, y0)

    g(t) = f(x(t), y(t)) f + (x(t), y(t)) 6 0 t 17

    (x0, y0) 6

    x(t) x0 y(t) y0 t 17 limt1 g(t) = )

    Rn)

    1# 2 ( $ * 1

    1 (x(t), y(t)) (x0, y0) t 1 '' 1 $6$ $ U (x0, y0) t $J $(x(t), y(t))

    U # 2 $ #

    6 ) (

    5( 8 $ $ T$ $ 1 $ (x0, y0)# 2

    1 6$ 5( ' $ f $ *$( $ A R2 (x0, y0) A#

    (i)lim

    (x,y)(x0,y0)f(x) = lim

    xx0( limyy0

    f(x, y)) = ,

    $ ' 5 $ #

    P$ $ ' ) 1= $ $ $ 6$ B 6$ (x, y) (x, y0) (x, y0) (x0, y0)#

    (ii)lim

    (x,y)(x0,y0)f(x) = lim

    t0f(x0 + ta, y0 + tb)) =