Download to read offline
![φ−Ω=→Ω= WVHQ;W[WVHQ)WI
/LQHDOL]DFLyQ GH ORV VLVWHPDV SDUD SHTXHxDV RVFLODFLRQHV UHVSHFWR D VX SRVLFLyQ GH
HTXLOLEULR
→
→
→
HWF
FRV
VHQ
θθθ
θ
θθ
(MHPSOR
( )
=−+−+
=+
=∑
θθθθ
θθ
θ
O
PJ,
VHQ
O
PJ,
,0
3DUD SHTXHxDV RVFLODFLRQHV
=+ θθ
O
PJ,](https://image.slidesharecdn.com/apuntespsg-171004055506/85/Apuntes-psg-14-320.jpg)
![IXHU]DV R PRPHQWRV
[EW](https://image.slidesharecdn.com/apuntespsg-171004055506/85/Apuntes-psg-19-320.jpg)
![PRYLPLHQWR GH OD EDVH
5LJLGH]
[
)
[[
)
Q'HIRUPDFLy
)XHU]D
N
∆
=
−
==
O($O)N
OO($)
(
=∆=→
∆=
= εσ
/-*7N
*-7/
==→
=
θ
θ
O(,)N
(,)O
==→
=
δ
δ
NNNHT
+=
NNNHT +=](https://image.slidesharecdn.com/apuntespsg-171004055506/85/Apuntes-psg-20-320.jpg)
!['HWHUPLQDU HO PRYLPLHQWR GHO VLVWHPD LGHDO FXDQGR VH OH Gi XQ GHVSOD]DPLHQWR
YHORFLGDG LQLFLDO [ [](https://image.slidesharecdn.com/apuntespsg-171004055506/85/Apuntes-psg-27-320.jpg)
![; $PSOLWXG GHO GHVSOD]DPLHQWR YDORU SLFR P](https://image.slidesharecdn.com/apuntespsg-171004055506/85/Apuntes-psg-35-320.jpg)
![I 7 )UHFXHQFLD GHO PRYLPLHQWR +]](https://image.slidesharecdn.com/apuntespsg-171004055506/85/Apuntes-psg-37-320.jpg)
![$QiOLVLV GH 6LVWHPDV 'LQiPLFRV
9 $PSOLWXG GH OD YHORFLGDG
$ $PSOLWXG GH OD DFHOHUDFLyQ
/D YHORFLGDG DFHOHUDFLyQ HVWiQ DGHODQWDGRV UHVSHFWR DO GHVSOD]DPLHQWR HQ ƒ
ƒ UHVSHFWLYDPHQWH
ωQ
P
N
6ROR GHSHQGH GH ODV FDUDFWHUtVWLFDV GHO VLVWHPD N P](https://image.slidesharecdn.com/apuntespsg-171004055506/85/Apuntes-psg-52-320.jpg)
![6L GLVPLQXH OD ULJLGH] GHO VLVWHPD GLVPLQXH ωQ
6L DXPHQWD OD PDVD GHO VLVWHPD GLVPLQXH ωQ
9LEUDFLRQHV OLEUHV DPRUWLJXDGDV
5HVROYHU
39, =++ N[[F[P](https://image.slidesharecdn.com/apuntespsg-171004055506/85/Apuntes-psg-53-320.jpg)
![LQWURGXFLHQGR ODV FRQGLFLRQHV LQLFLDOHV
( )[ ] W
Q
Q
H[W[W[ ξω
ω −
++=](https://image.slidesharecdn.com/apuntespsg-171004055506/85/Apuntes-psg-69-320.jpg)
![+=
( ) ( )[ ]](https://image.slidesharecdn.com/apuntespsg-171004055506/85/Apuntes-psg-72-320.jpg)
![6H REVHUYD TXH OD UHVSXHVWD QR HV RVFLODWRULD (O FXHUSR VROR UHWRUQD KDFLD OD
SRVLFLyQ GH HTXLOLEULR
(VWDELOLGDG GH XQ VLVWHPD
/D HVWDELOLGDG GH XQ VLVWHPD VH DQDOL]D SDUD VXV YLEUDFLRQHV OLEUHV
/D VROXFLyQ GH OD HFXDFLyQ =++ N[[F[P
HV VW
$HW[ =](https://image.slidesharecdn.com/apuntespsg-171004055506/85/Apuntes-psg-74-320.jpg)
![$QiOLVLV GH 6LVWHPDV 'LQiPLFRV
ULWHULRV GH HVWDELOLGDG
8QD FRQGLFLyQ QHFHVDULD VXILFLHQWH SDUD TXH XQ VLVWHPD VHD HVWDEOH HV TXH
WRGDV ODV UDtFHV GH OD HFXDFLyQ FDUDFWHUtVWLFD WHQJDQ SDUWHV UHDOHV QHJDWLYDV
$QDOL]DQGR
P
N
P
F
P
F
U −
±−=
(VWR VXFHGHUi FXDQGR N R F VHDQ QHJDWLYRV HV GHFLU QR VH RSRQHQ DO
PRYLPLHQWR VLQR TXH OR DXGDQ
(MHPSOR $QDOLFH OD HVWDELOLGDG GHO SpQGXOR LQYHUWLGR GH OD ILJXUD
, ( )
PO
PJNO
P
N
U
WPJNOWPO
PJO
NO
PO
POPJO
OO
N
,0
N
P](https://image.slidesharecdn.com/apuntespsg-171004055506/85/Apuntes-psg-76-320.jpg)
![6L OD ULJLGH] HIHFWLYD N
HV QHJDWLYD HV GHFLU VL NO PJ HO PRYLPLHQWR GHO
SpQGXOR VHUi LQHVWDEOH SRU GLYHUJHQFLD](https://image.slidesharecdn.com/apuntespsg-171004055506/85/Apuntes-psg-79-320.jpg)
![$QiOLVLV GH 6LVWHPDV 'LQiPLFRV
5HODFLRQHV HQWUH IXHU]DV H[FLWDFLRQHV
9LEUDFLRQHV IRU]DGDV FRQ H[FLWDFLyQ DUPyQLFD
WVHQ)N[[F[P Ω=++
( )
( )
( )
( )](https://image.slidesharecdn.com/apuntespsg-171004055506/85/Apuntes-psg-80-320.jpg)
![VH REWLHQH
ξξ −
= N
)
PD[; SDUD
ξω −=Ω Q
3DUD ξ SHTXHxRV 4;HVWN
)
;
Pi[
==
ξ
SDUD Qω≈Ω
N
)
;HVW = HV OD UHVSXHVWD HVWiWLFD HV GHFLU OD UHVSXHVWD GHO VLVWHPD D XQD IXHU]D
FRQVWDQWH )
ξ
=4 HV HO IDFWRU GH DPSOLILFDFLyQ](https://image.slidesharecdn.com/apuntespsg-171004055506/85/Apuntes-psg-95-320.jpg)
![5HHPSOD]DQGR HQ ODV HFXDFLRQHV GHO PRYLPLHQWR VH REWLHQH
( ) WMWM
)H;HNFMP ΩΩ
=+Ω+Ω−
NFMP
H)
H;[
)M
[M
+Ω+Ω−
==
φ
φ
SRU OR WDQWR HO PyGXOR LJXDO D HFXDFLyQ](https://image.slidesharecdn.com/apuntespsg-171004055506/85/Apuntes-psg-105-320.jpg)
![[HRD] MATH_NOTES.pdf](https://cdn.slidesharecdn.com/ss_thumbnails/hrdmathnotes-220908025845-75eba44f-thumbnail.jpg?width=640&height=640&fit=bounds)
Apuntes psg
- 1. 'HSDUWDPHQWR GH ,QJHQLHUtD 0HFiQLFD $3817(6 $1È/,6,6'( 6,67(0$6 ',1È0,&26 352)(625 3('52 6$$9('5$ * (',,Ï1 5,67Ï%$/ 6+((/ / ,/8675$,21(6 -8$1 3$55$ 2 5,67Ï%$/ 6+((/ /
- 2. $QiOLVLV GH 6LVWHPDV'LQiPLFRV DStWXOR 6LVWHPDV GH XQ JUDGR GH OLEHUWDG 6LVWHPD OLQHDO (FXDFLyQ GLIHUHQFLDO TXH ULJH VX PRYLPLHQWR HV OLQHDO (MHPSOR
- 3. WIN[[F[P =++ 3URSLHGDGHV 3ULQFLSLRGH VXSHUSRVLFLyQ 6L HW
- 4.
- 5.
- 6.
- 7.
- 8.
- 9.
- 10.
- 11. /D SUHVHQFLD GHXQD H[FLWDFLyQ QR DIHFWD OD UHVSXHVWD GHO VLVWHPD D RWUDV H[FLWDFLRQHV (Q HO HVWDGR SHUPDQHQWH /D UHVSXHVWD GH XQ VLVWHPD OLQHDO D XQD H[FLWDFLyQ DUPyQLFD Ω HV D OD PLVPD IUHFXHQFLD Ω 6L
- 14. φ−Ω=→Ω= WVHQ;W[WVHQ)WI /LQHDOL]DFLyQ GHORV VLVWHPDV SDUD SHTXHxDV RVFLODFLRQHV UHVSHFWR D VX SRVLFLyQ GH HTXLOLEULR → → → HWF FRV VHQ θθθ θ θθ (MHPSOR ( ) =−+−+ =+ =∑ θθθθ θθ θ O PJ, VHQ O PJ, ,0 3DUD SHTXHxDV RVFLODFLRQHV =+ θθ O PJ,
- 15. $QiOLVLV GH 6LVWHPDV'LQiPLFRV RPSRQHQWHV GHO PRGHOR GLQiPLFR HOHPHQWDO 0DVD P FRQFHQWUDGD HQ XQ EORTXH UtJLGR
- 16. 3URSLHGDGHV HOiVWLFDV NUHVRUWH VLQ PDVD
- 17. 'LVLSDFLyQ GH HQHUJtDR DPRUWLJXDPLHQWR F VXSXHVWR YLVFRVR
- 18.
- 19.
- 20. PRYLPLHQWR GH ODEDVH 5LJLGH] [ ) [[ ) Q'HIRUPDFLy )XHU]D N ∆ = − == O($O)N OO($) ( =∆=→ ∆= = εσ /-*7N *-7/ ==→ = θ θ O(,)N (,)O ==→ = δ δ NNNHT += NNNHT +=
- 21. $QiOLVLV GH 6LVWHPDV'LQiPLFRV $PRUWLJXDPLHQWR YLVFRVR
- 22. YYF) −= F RHILFLHQWH GH DPRUWLJXDPLHQWR YLVFRVR (FXDFLRQHV GHO PRYLPLHQWR
- 23.
- 24. $QiOLVLV GH 6LVWHPDV'LQiPLFRV 9LEUDFLRQHV OLEUHV IW
- 25. 9LEUDFLRQHV OLEUHV QR DPRUWLJXDGDV IW
- 26.
- 27. 'HWHUPLQDU HO PRYLPLHQWRGHO VLVWHPD LGHDO FXDQGR VH OH Gi XQ GHVSOD]DPLHQWR YHORFLGDG LQLFLDO [ [
- 28.
- 29.
- 30. += (FXDFLyQ FDUDFWHUtVWLFD P N MUNPU ±=→=+ W P N VHQ%W P N FRV$W[ +=
- 31. PRYLPLHQWR DUPyQLFR VLPSOH $ % VH REWLHQHQ GH ODV FRQGLFLRQHV LQLFLDOHV PNWVHQ [ WFRV[W[ QQ Q Q =+= ωω ω ω
- 32.
- 34.
- 35. ; $PSOLWXG GHOGHVSOD]DPLHQWR YDORU SLFR P
- 36. 7 3HUtRGR GHOPRYLPLHQWR V
- 37. I 7 )UHFXHQFLDGHO PRYLPLHQWR +]
- 38. ωQ π IQ)UHFXHQFLD FLUFXODU
- 39.
- 40. ϕ ÈQJXOR GHIDVH UDG
- 41.
- 42.
- 43.
- 44.
- 45.
- 46.
- 47.
- 48.
- 49.
- 50.
- 52. $QiOLVLV GH 6LVWHPDV'LQiPLFRV 9 $PSOLWXG GH OD YHORFLGDG $ $PSOLWXG GH OD DFHOHUDFLyQ /D YHORFLGDG DFHOHUDFLyQ HVWiQ DGHODQWDGRV UHVSHFWR DO GHVSOD]DPLHQWR HQ ƒ ƒ UHVSHFWLYDPHQWH ωQ P N 6ROR GHSHQGH GH ODV FDUDFWHUtVWLFDV GHO VLVWHPD N P
- 53. 6L GLVPLQXH ODULJLGH] GHO VLVWHPD GLVPLQXH ωQ 6L DXPHQWD OD PDVD GHO VLVWHPD GLVPLQXH ωQ 9LEUDFLRQHV OLEUHV DPRUWLJXDGDV 5HVROYHU 39, =++ N[[F[P
- 54.
- 55. [[ = (FXDFLyQ FDUDFWHUtVWLFD =++ NFUPU P N P F P F UU − ±−= 3DUD TXH HO VLVWHPD YLEUH UL GHEHQ VHU LPDJLQDULRV P F P N 'HILQLFLyQ (O Pi[LPR YDORU GH F SDUD TXH HO VLVWHPD YLEUH VH OODPD DPRUWLJXDPLHQWR FUtWLFR FF YDOH = P F P N QF PNPF ω ==
- 56. (V PiV FyPRGRH[SUHVDU HQ IXQFLyQ GH SDUiPHWURV IiFLOHV GH PHGLU Q ωξ ξ )DFWRU GH DPRUWLJXDPLHQWR FF F
- 57. −±−= ξωξω QQUU DVR, $PRUWLJXDPLHQWR VXEFUtWLFR ξ
- 58. ⇒ 5DtFHV FRPSOHMDVFRQMXJDGDV ⇒ ([LVWH YLEUDFLyQ ( )WFRV%WVHQ$HW[ GG WQ ωωξω += −
- 59. $ %GHWHUPLQDGRV GH ODV FRQGLFLRQHV LQLFLDOHV [ [ /XHJR + + = − WFRV[WVHQ [[ HW[ GG G QWQ ωω ω ξωξω
- 61. $QiOLVLV GH 6LVWHPDV'LQiPLFRV ξωω −= QG IUHFXHQFLD QDWXUDO GH YLEUDU DPRUWLJXDGD ( )GG W WVHQH;W[ Q ϕωξω += −
- 62. ( ) G Q [[ [; ω ξω+ += WJ [[ [ Q G G ξω ω ϕ + = (IHFWRGHO DPRUWLJXDPLHQWR HQ ODV YLEUDFLRQHV OLEUHV 'LVPLQXLU VHFXHQFLDOPHQWH OD DPSOLWXG GH ODV YLEUDFLRQHV OLEUHV 'LVPLQXLU OD IUHFXHQFLD QDWXUDO GH Qω D Gω (Q OD SUiFWLFD JHQHUDOPHQWH ξ Qω ≈ Gω 'HFUHFLPLHQWR ORJDUtWPLFR δ
- 63. 8QD IRUPD SUiFWLFDGH GHWHUPLQDU HO DPRUWLJXDPLHQWR HV D SDUWLU GH XQ UHJLVWUR GH YLEUDFLRQHV FRPR HO LQGLFDGR HQ OD ILJXUD PLGLHQGR HO FXRFLHQWH HQWUH GRV DPSOLWXGHV ;Q ;Q 'HILQLFLyQ = +Q Q ; ; OQδ
- 64. /RV YDORUHV GH;Q ;Q RFXUUHQ HQ XQ WLHPSR W W 7G UHVSHFWLYDPHQWH 3DUD DPERV FDVRV VHQ ωGW ϕG
- 65. ≈ (QWRQFHV ( )GQ 7 7W W 7HOQ H H OQ GQ GQ Q ξωδ ξω ξω ξω == = +− − ξ πξ δ − =
- 66. 3DUD SHTXHxRV YDORUHVGH ξ δ ≈ πξ
- 67. $QiOLVLV GH 6LVWHPDV'LQiPLFRV DVR ,, $PRUWLJXDPLHQWR FUtWLFR ξ
- 68. ⇒ UDtFHV UHDOHVH LJXDOHV ⇒ QR H[LVWH YLEUDFLyQ QUU ξω−== ( ) WQ HW$$W[ ξω− +=
- 69. LQWURGXFLHQGR ODV FRQGLFLRQHVLQLFLDOHV ( )[ ] W Q Q H[W[W[ ξω ω − ++=
- 70.
- 71. ⇒ UDtFHV UHDOHV GHVLJXDOHV ⇒ QR H[LVWH YLEUDFLyQ −±−= ξωξω QQUU WUWU HHW[
- 72. += ( ) ()[ ]
- 73.
- 74. 6H REVHUYD TXHOD UHVSXHVWD QR HV RVFLODWRULD (O FXHUSR VROR UHWRUQD KDFLD OD SRVLFLyQ GH HTXLOLEULR (VWDELOLGDG GH XQ VLVWHPD /D HVWDELOLGDG GH XQ VLVWHPD VH DQDOL]D SDUD VXV YLEUDFLRQHV OLEUHV /D VROXFLyQ GH OD HFXDFLyQ =++ N[[F[P HV VW $HW[ =
- 75. GRQGH ωσ MV+= 3ODQRV V
- 76. $QiOLVLV GH 6LVWHPDV'LQiPLFRV ULWHULRV GH HVWDELOLGDG 8QD FRQGLFLyQ QHFHVDULD VXILFLHQWH SDUD TXH XQ VLVWHPD VHD HVWDEOH HV TXH WRGDV ODV UDtFHV GH OD HFXDFLyQ FDUDFWHUtVWLFD WHQJDQ SDUWHV UHDOHV QHJDWLYDV $QDOL]DQGR P N P F P F U − ±−= (VWR VXFHGHUi FXDQGR N R F VHDQ QHJDWLYRV HV GHFLU QR VH RSRQHQ DO PRYLPLHQWR VLQR TXH OR DXGDQ (MHPSOR $QDOLFH OD HVWDELOLGDG GHO SpQGXOR LQYHUWLGR GH OD ILJXUD , ( ) PO PJNO P N U WPJNOWPO PJO NO PO POPJO OO N ,0 N P
- 77.
- 79. 6L OD ULJLGH]HIHFWLYD N HV QHJDWLYD HV GHFLU VL NO PJ HO PRYLPLHQWR GHO SpQGXOR VHUi LQHVWDEOH SRU GLYHUJHQFLD
- 80. $QiOLVLV GH 6LVWHPDV'LQiPLFRV 5HODFLRQHV HQWUH IXHU]DV H[FLWDFLRQHV 9LEUDFLRQHV IRU]DGDV FRQ H[FLWDFLyQ DUPyQLFD WVHQ)N[[F[P Ω=++ ( ) ( ) ( ) ( )
- 90.
- 91. KRP $ ϕGVH GHWHUPLQDQ GH ODV FRQGLFLRQHV LQLFLDOHV Ω+ Ω− = QQ N ) ; ω ξ ω
- 92.
- 93.
- 94. $QiOLVLV GH 6LVWHPDV'LQiPLFRV 5HVSXHVWD HVWDFLRQDULD HV DUPyQLFD D OD PLVPD IUHFXHQFLD Ω GH OD H[FLWDFLyQ /D UHVSXHVWD WUDQVLHQWH GHVDSDUHFH UiSLGDPHQWH FRQ HO WLHPSR $ PDRU DPRUWLJXDPLHQWR PiV UiSLGR GHVDSDUHFH 'HULYDQGR OD HFXDFLyQ
- 95. VH REWLHQH ξξ− = N ) PD[; SDUD ξω −=Ω Q 3DUD ξ SHTXHxRV 4;HVWN ) ; Pi[ == ξ SDUD Qω≈Ω N ) ;HVW = HV OD UHVSXHVWD HVWiWLFD HV GHFLU OD UHVSXHVWD GHO VLVWHPD D XQD IXHU]D FRQVWDQWH ) ξ =4 HV HO IDFWRU GH DPSOLILFDFLyQ
- 96. $QiOLVLV GH 6LVWHPDV'LQiPLFRV 0pWRGR GHO iOJHEUD FRPSOHMD (O XVR GHO iOJHEUD FRPSOHMD JHQHUDOPHQWH VLPSOLILFD HO SURFHVR GH VROXFLyQ GH OD HFXDFLyQ GLIHUHQFLDO /D IXQFLyQ ( )φω += WVHQ;W[
- 97. SXHGH VHU UHSUHVHQWDGDFRPR OD SURHFFLyQ GH XQ YHFWRU ; TXH JLUD FRQ YHORFLGDG DQJXODU Ω (O YHFWRU URWDWRULR ; H[SUHVDGR HQ IXQFLyQ GH ORV HMHV 5 , HV ( ) ( ) ( ) WMMWM HH;H;WVHQM;WFRV;; ωφφω φωφω ==+++= + WM ;H; ω = φM H;; = DPSOLWXG FRPSOHMD 3RU OR WDQWR ( )φω += WVHQ;W[
- 98. SXHGH VHU H[SUHVDGRFRPR ( )WM ;H,PW[ ω =
- 99.
- 100. (Q OD SUiFWLFDJHQHUDOPHQWH VH REYLD HVFULELU ,P
- 101. $SOLFDFLyQ HQ GHWHUPLQDUOD UHVSXHVWD HVWDFLRQDULD VL WM )HWI Ω =
- 102. 3RU VHU VLVWHPDOLQHDO OD UHVSXHVWD HV D OD PLVPD IUHFXHQFLD TXH OD H[FLWDFLyQ WM ;HW[ Ω =
- 103.
- 104.
- 105. 5HHPSOD]DQGR HQ ODVHFXDFLRQHV GHO PRYLPLHQWR VH REWLHQH ( ) WMWM )H;HNFMP ΩΩ =+Ω+Ω− NFMP H) H;[ )M [M +Ω+Ω− == φ φ SRU OR WDQWR HO PyGXOR LJXDO D HFXDFLyQ
- 107. ( ) Ω+ Ω− = Ω+Ω− = QQ N ) FPN ) ; ω ξ ω
- 108. $QiOLVLV GH 6LVWHPDV'LQiPLFRV ODV IDVHV LJXDO D HFXDFLyQ