Documento sobre operaciones con números y variables | Apuntes de Matemáticas



                                      

                             

                  

                  

                  

                           

                      

                

                    



                                                                                                            

                                                                                                             

                                             

                               

        

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

 −−→→

                           

 −−→→

bb 11−−→→



                                                        

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                       

 −−→→

     

ZriyZriy

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bb −−→→

ll  11 −−→→

ll µµ−−→→



−−→→

bb 



22 µµ−−→→

                                              

                                  

                 



                                          

                                          

               

ZriyZriy

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ff −−→→

ll ++ −−→→

bb   11  Zriy Zriy−−→→

ff −−→→

ll  ++

ZriyZri y−−→→

ff −−→→

ZriyZriy

−−→→

bbdd−−→→

ll  11 d.Zriyd.Zriy−−→→

bb −−→→

ZriyZriy

dd..−−→→

bb −−→→

ll  11 Z Z rir iyy−−→→

bb −−→→

                  

                                                  

                                                  

                                                   

                                             

FiapFiap−−→→

bb −−→→

ll  11 −−→→

ll µµ−−→→



−−→→

bb 



               

FiapFiap−−→→

ff −−→→

ll  ++ −−→→

bb  11  Fiap Fiap−−→→

ff −−→→

ll  ++

FiapFiap−−→→

ff −−→→

FiapFiap−−→→

bbdd−−→→

ll  11 d.Fiapd.Fiap−−→→

bb −−→→

FiapFiapdd−−→→

bb −−→→

ll  11 FiapFiap−−→→

bb −−→→



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                             

                       

       −−→→ll 

bb                         

bb  11

−−→→ll                      −−→→bb      

ZriyZriy−−→→bb^ −−→→ll 11

 −−→→ll μμ−−→→bb

−−→→bb

μμ −−→→bb

ZriyZriy−−→→ff

−−→→ll ++ −−→→bb

11 ZriyZriy−−→→ff^ −−→→ll ++

ZriyZriy−−→→ff

bb

ZriyZriy−−→→bb^ dd −−→→ll 11 d.Zriyd.Zriy−−→→bb^ −−→→ll

ZriyZriydd ..−−→→bb^ −−→→ll 11 ZZ ririyy−−→→bb^ −−→→ll

FiapFiap−−→→bb^ −−→→ll^11

−−→→ll μμ−−→→bb

−−→→bb^ 

FiapFiap−−→→ff

−−→→ll ++ −−→→bb

11 FiapFiap−−→→ff^ −−→→ll ++

FiapFiap−−→→ff

bb

FiapFiap−−→→bb dd −−→→ll 11 d.Fiapd.Fiap−−→→bb^ −−→→ll

FiapFiapdd −−→→bb^ −−→→ll 11 FiapFiap−−→→bb^ −−→→ll

                 −−→→ll^1 1 ((ll<< ,,^ ll 22 ,,^ ll 66 )) 

bb 1 1 ((bb<< ,, bb 22 ,, bb 66 ))                                      

−−→→ll ÕÕ −−→→bb 11

kk mm dd

ll<< ll 22 ll 66

bb<< bb 22 bb 66

−−→→ll ÕÕ −−→→bb 11

ll 22 ll 66

bb 22 bb 66

kk −−

ll<< ll 66

bb<< bb 66

mm^ ++

ll<< ll 22

bb<< bb 22

dd

                                               −−→→uu ÕÕ −−→→vv

−−→→ll ÕÕ −−→→bb

11 −−→→ll 

bb

sg`ξsg`ξ

    ^ ˏˏkk 1 1 ((<< ,, :: ,, :: )4)4 ˏˏmm ((:: ,, << ,, :: )4)4 ˏˏdd 1 1 ((:: ,, :: ,, << ))

kk ÕÕ kk 11

::  mm ÕÕ mm 11

::  dd ÕÕ dd 11

kk

ÕÕ

mm 11 dd  mm

ÕÕ

dd 11 kk  dd

ÕÕ

kk 11 mm

mm ÕÕ kk 11 −−dd  dd ÕÕ mm 11 −−kk  kk ÕÕ dd 11 −−mm

−−→→ll ÕÕ −−→→ll 11 →→−−::

−−→→ll ÕÕ −−→→bb

11 −−→→ll ^22

bb

→→−−ll μμ −−→→bb

−−→→ll ÕÕ −−→→bb

11 −−→→ll ^22

bb

Vg`Vg`^22 ξξ    

−−→→ll ÕÕ −−→→bb

11 −−→→ll 

bb

sg`ξsg`ξ

                 ^ LL^11

−−→→ll ÕÕ −−→→bb

LL 11 << 22

−−→→ll ÕÕ −−→→bb

ll 1 1 ((

          ^ −−→→bb 1 1 (6(6,, −−<<,, :):)

         ^ −−6;6; <:<: ,, <6<6 <:<: ,, :):)

    −−→→                    ^ −−→→ll^11 kk^ ++ 2^2 mm^ ++ 6^6 dd 

bb 1 1 8 8 kk ++ 7 7 mm ++ ??dd                        

          ^ −−→→ll^ ÕÕ −−→→bb 

                         −−→→ll 1 1 (<(<,, −− 66 ,, 8)8) 

bb 1 1 (6(6,, −− 88 ,, 2)2)  −−→→ff 1 1 ((−−<<,, <<,, 88      

       ZriyZriy−−→→bb ++−−→→ff^ −−→→ll

<7<

6:6: 55

        −−→→LL 1 1 (<(<,, <<,, 2)2) −−→→BB 1 1 ((−−<<,, 22 ,, 6)6)

LL ÕÕ

BB

ÕÕ

LL ++

BB

                           ^ −−→→ll 1 1 2 2 kk ++ 6 6 mm −− 55 dd

bb 1 1 7 7 kk −− 22 mm

2222 uu^22

−−→→ll 1 1 6 6 kk−−mm  −−→→bb 11 mm +2+2dd −−→→ff 11 kk+7+7 mm+8+8dd

         2:2:uu^66

                         −−→→ll^ 

bb                

               −−→→ll  1 1 6 6 

bb

1 1 8^8

−−→→ll ++ −−→→bb

ÕÕ

−−→→ll −− −−→→bb

                       ^ −−→→ll^1 1 (<(<,,^22 ,,^ 6)6)^ 

bb 11

(<(<,, −−<<,, 8)8) ..  −−→→ff 1 1 6 6 mm −− 22 dd               

     ^ −−→→l ,l ,

bb ∎∎ RR          −−→→ll  11 ??

bb

1 1 7 7                   −−→→ll 

bb

     ϊϊ??              −−→→ll ÕÕ −−→→bb 

                           ^ −−→→rr 1 1 ((aa + <+ <,, `` + <+ <,, aa −− qq ))

^ −−→→ss 1 1 (:(:,, `,`, 8)8) ..  −−→→tt 1 1 ;;kk −− <2<2mm + 6+ 6dd

−−→→tt^11 −−→→rr^ ÕÕ −−→→ss ,,

^ ^ ^ ^ ^ ^ 

−−→→ll^11

(2(2a,`,a,`, −− 66 qq )) 

bb 1 1 ((a,a, −− 66 `` ++ q,q, −−qq )) ..

                             ^ −−→→rr ,, −−→→ss ,, −−→→l ,l ,

bb 

                   ^ −−→→ll 

bb

        ^ −−→→rr 1 (1 ( 6464 2424 <)4<)4 −−→→ss 1 (:4<48)1 (:4<48)

−−→→ll 1 (84<41 (84<4 −−;);)  −−→→bb 1 (24:41 (24:4 −−6)6)

LrglLrgl^ ¸¸ 11 ∝∝;<6;<

     −−→→ll 1 1 6 6 kk −− mm −− 22 dd  bb 1 1 2 2 kk ++ 66 ++ dd            −−→→

ll ++ −−→→bb (^)  −−→→

ll −−

−−→→bb           22

∝∝ <>7<>

                        ^ −−→→ll 1 1 (2(2,, −−<<,, 6)6)  −−→→ bb 1 1 (2(2,, −−<<,, 8)8)  −−→→ff 1 1 ((−− 22 ,, <<,, 66              ZriyZriy−−→→bb −−−−→→ff^ −−→→ll

        

 7272 2<2< 44 −−

2?2? 2<2< 44

<6< 2<2<



                                   kk++mm  mm++dd dd ++ kk

        22 uu^22

                         ^ ll                            ^ −−→→vv 1 1 2^2 kk−−^ mm ++dd  −−→→uu 11 kk +2+2^ mm −−^66 dd ^ −−→→ww^1 1 6^6 kk^ ++^ lmlm^ + 7+ 7dd                                ^ ll^11 −− 88

  <8<8                                            <?<?                                            ?:?:                                            <6<6 

∝∝ 7575

Documento sobre operaciones con números y variables, Apuntes de Matemáticas

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       −−→→ll 

bb                         

bb  11

−−→→ll                      −−→→bb      

ZriyZriy−−→→bb^ −−→→ll 11

 −−→→ll μμ−−→→bb

−−→→bb

μμ −−→→bb

ZriyZriy−−→→ff

−−→→ll ++ −−→→bb

11 ZriyZriy−−→→ff^ −−→→ll ++

ZriyZriy−−→→ff

bb

ZriyZriy−−→→bb^ dd −−→→ll 11 d.Zriyd.Zriy−−→→bb^ −−→→ll

ZriyZriydd ..−−→→bb^ −−→→ll 11 ZZ ririyy−−→→bb^ −−→→ll

FiapFiap−−→→bb^ −−→→ll^11

−−→→ll μμ−−→→bb

−−→→bb^ 

FiapFiap−−→→ff

−−→→ll ++ −−→→bb

11 FiapFiap−−→→ff^ −−→→ll ++

FiapFiap−−→→ff

bb

FiapFiap−−→→bb dd −−→→ll 11 d.Fiapd.Fiap−−→→bb^ −−→→ll

FiapFiapdd −−→→bb^ −−→→ll 11 FiapFiap−−→→bb^ −−→→ll

                 −−→→ll^1 1 ((ll<< ,,^ ll 22 ,,^ ll 66 )) 

bb 1 1 ((bb<< ,, bb 22 ,, bb 66 ))                                      

−−→→ll ÕÕ −−→→bb 11

kk mm dd

ll<< ll 22 ll 66

bb<< bb 22 bb 66

−−→→ll ÕÕ −−→→bb 11

ll 22 ll 66

bb 22 bb 66

kk −−

ll<< ll 66

bb<< bb 66

mm^ ++

ll<< ll 22

bb<< bb 22

dd

                                               −−→→uu ÕÕ −−→→vv

−−→→ll ÕÕ −−→→bb

11 −−→→ll 

bb

sgξsgξ

    ^ ˏˏkk 1 1 ((<< ,, :: ,, :: )4)4 ˏˏmm ((:: ,, << ,, :: )4)4 ˏˏdd 1 1 ((:: ,, :: ,, << ))

kk ÕÕ kk 11

::  mm ÕÕ mm 11

::  dd ÕÕ dd 11

kk

ÕÕ

mm 11 dd  mm

ÕÕ

dd 11 kk  dd

ÕÕ

kk 11 mm

mm ÕÕ kk 11 −−dd  dd ÕÕ mm 11 −−kk  kk ÕÕ dd 11 −−mm

−−→→ll ÕÕ −−→→ll 11 →→−−::

−−→→ll ÕÕ −−→→bb

11 −−→→ll ^22

bb

→→−−ll μμ −−→→bb

−−→→ll ÕÕ −−→→bb

11 −−→→ll ^22

bb

VgVg^22 ξξ    

−−→→ll ÕÕ −−→→bb

11 −−→→ll 

bb

sgξsgξ

                 ^ LL^11

−−→→ll ÕÕ −−→→bb

LL 11 << 22

−−→→ll ÕÕ −−→→bb

ll 1 1 ((

sg`ξsg`ξ

Vg`Vg`^22 ξξ    

sg`ξsg`ξ

^ −−→→ss 1 1 (:(:,, `,`, 8)8) ..  −−→→tt 1 1 ;;kk −− <2<2mm + 6+ 6dd

(2(2a,`,a,`, −− 66 qq )) 