Ðåñéå÷üìåíá ÅéóáãùãÞ: Áðü ôïí ÈáëÞ ôïí Ìçëßóéï óôïí Êùíóôáíôßíï ÊáñáèåïäùñÞ .................19 ÌÅÑÏÓ ÐÑÙÔÏ • ÂÁÓÉÊÅÓ ÅÍÍÏÉÅÓ ÌÁÈÇÌÁÔÉÊÙÍ ÊåöÜëáéï 1: Óõìâïëéóìïß áðü ôç ìáèçìáôéêÞ ëïãéêÞ êáé ôá óýíïëá......27 1.1 Óõìâïëéóìïß áðü ôç ìáèçìáôéêÞ ëïãéêÞ ..........................................................27 1.2 Óõìâïëéóìïß áðü ôç èåùñßá ôùí óõíüëùí ......................................................28 1.3 Óõìâïëéóìïß áðü ôçí Üëãåâñá .........................................................................29
ÊåöÜëáéï 2: Óýíïëá áñéèìþí......................................................................31 2.1 Ôï óýíïëï ôùí ðñáãìáôéêþí áñéèìþí (R) .......................................................31 2.1.1 Õðïóýíïëá ôïõ óõíüëïõ ôùí ðñáãìáôéêþí áñéèìþí ...........................31 2.1.2 Ç óõìðáãÞò åõèåßá ôùí ðñáãìáôéêþí áñéèìþí....................................32 2.1.3 ÄéÜóôçìá ðñáãìáôéêþí áñéèìþí............................................................33 2.1.4 Ðåñéï÷Þ ðñáãìáôéêïý áñéèìïý ...............................................................34 2.2Ôï óýíïëï ôùí ìéãáäéêþí áñéèìþí C ..................................................................35 2.2.1 Ïñéóìüò êáé Üëãåâñá ôùí ìéãáäéêþí áñéèìþí .......................................35 2.2.2 Ç öáíôáóôéêÞ ìïíÜäá .............................................................................37 2.2.3 Óõæõãåßò ìéãáäéêïß áñéèìïß.......................................................................39 2.2.4 ÃåùìåôñéêÞ ðáñÜóôáóç ôùí ìéãáäéêþí áñéèìþí...................................41 2.2.5 ÃåùìåôñéêÞ ðáñÜóôáóç ôïõ áèñïßóìáôïò êáé ôçò äéáöïñÜò ìéãáäéêþí áñéèìþí...................................................................................42 2.2.6 ÌÝôñï ìéãáäéêïý áñéèìïý........................................................................43 2.2.7 ÔñéãùíïìåôñéêÞ ìïñöÞ ìéãáäéêïý áñéèìïý... ..........................................47 2.2.8 Èåþñçìá: Éóüôçôá Ìéãáäéêþí................................................................51 2.2.9 Èåþñçìá: Ãéíüìåíï Ìéãáäéêþí ...............................................................51 2.2.10 Èåþñçìá: Ðçëßêï Ìéãáäéêþí..................................................................52 2.2.11 Èåþñçìá: Áíôßóôñïöïò Ìéãáäéêïý ........................................................52 2.2.12 Èåþñçìá De Moivre ................................................................................53 2.2.13 Ñßæåò ìéãáäéêþí áñéèìþí .........................................................................54 2.2.14 ÃåùìåôñéêÞ óçìáóßá ôùí ñéæþí ìéãáäéêïý áñéèìïý ...............................54 2.2.15 Í-ïóôÝò ñßæåò ôçò ìïíÜäáò .....................................................................55 2.2.16 ÅêèåôéêÞ ìïñöÞ ìéãáäéêïý áñéèìïý.........................................................55 2.2.17 ËïãÜñéèìïò (íåðÝñéïò) ìéãáäéêïý áñéèìïý..............................................56 2.3 ÁóêÞóåéò.............................................................................................................57
ÊåöÜëáéï 3: Ïñßæïõóåò êáé ãñáììéêÜ óõóôÞìáôá ................................... 61 3.1 Ïñßæïõóåò ..........................................................................................................61 3.1.1 Ïñéóìüò: Ïñßæïõóá äåõôÝñáò ôÜîåùò....................................................61
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ÌÁÈÇÌÁÔÉÊÇ ÁÍÁËÕÓÇ
3.1.2 Ãåíßêåõóç: Ïñßæïõóá í-ïóôÞò ôÜîåùò ....................................................61 3.1.3 Êáíüíáò ôïõ Sarrus ..................................................................................63 3.1.4 Éäéüôçôåò ôùí ïñéæïõóþí.........................................................................63 3.2 ÃñáììéêÜ ÓõóôÞìáôá ........................................................................................66 3.2.1 Åðßëõóç ãåíéêþí ãñáììéêþí óõóôçìÜôùí ............................................. 66 3.2.2 ÏìïãåíÞ ãñáììéêÜ óõóôÞìáôá................................................................69 3.3 ÁóêÞóåéò.............................................................................................................70
ÊåöÜëáéï 4: ÐñáãìáôéêÝò óõíáñôÞóåéò......................................................75 4.1 4.2 4.3 4.4 4.5 4.6
Ïñéóìüò óõíÜñôçóçò ........................................................................................75 ÃñáöéêÞ ðáñÜóôáóç óõíÜñôçóçò ....................................................................77 Áñôéåò êáé ðåñéôôÝò óõíáñôÞóåéò......................................................................80 ÐåñéïäéêÝò óõíáñôÞóåéò ....................................................................................80 Ìïíïôïíßá óõíÜñôçóçò ....................................................................................81 Áëãåâñá ôùí óõíáñôÞóåùí ..............................................................................81 4.6.1 Éóüôçôá óõíáñôÞóåùí .............................................................................81 4.6.2 Óýíèåóç óõíáñôÞóåùí ........................................................................... 82 4.6.3 ¢èñïéóìá óõíáñôÞóåùí ..........................................................................82 4.6.4 ÄéáöïñÜ óõíáñôÞóåùí............................................................................82 4.6.5 Ãéíüìåíï óõíáñôÞóåùí ............................................................................82 4.6.6 Ðçëßêï óõíáñôÞóåùí...............................................................................82 4.6.7 Áíôßóôñïöç óõíÜñôçóç ...........................................................................82 4.7 Áêñüôáôá óõíÜñôçóçò......................................................................................85 4.8 Êáôçãïñßåò óõíáñôÞóåùí .................................................................................87 4.8.1 ÁëãåâñéêÝò óõíáñôÞóåéò ..........................................................................87 4.8.2 ÕðåñâáôéêÝò óõíáñôÞóåéò........................................................................88 4.9 ÁóêÞóåéò.............................................................................................................97
ÌÅÑÏÓ ÄÅÕÔÅÑÏ ÄÉÁÍÕÓÌÁÔÉÊÏÓ ËÏÃÉÓÌÏÓ ÊÁÉ ÁÍÁËÕÔÉÊÇ ÃÅÙÌÅÔÑÉÁ Á. ÄÉÁÍÕÓÌÁÔÉÊÏÓ ËÏÃÉÓÌÏÓ................................................................103 ÊåöÜëáéï 5: Ïñéóìïß êáé ìåôñéêÝò ó÷Ýóåéò óôá äéáíýóìáôá ..................105 5.1 5.2 5.3 5.4 5.5
ÂáóéêÝò Ýííïéåò óôá äéáíýóìáôá ....................................................................105 ÓõíôåôáãìÝíåò äéáíýóìáôïò ...........................................................................107 ¢ëãåâñá ôùí äéáíõóìÜôùí.............................................................................109 ÓõíôåôáãìÝíåò óçìåßïõ ðïõ ÷ùñßæåé äéÜíõóìá óå ëüãï ë .............................110 Ç ãùíßá óôá äéáíýóìáôá ................................................................................111 5.5.1 Äéáíýóìáôá óôï åðßðåäï (R2) ...............................................................111 5.5.2 Äéáíýóìáôá óôïí ÷þñï (R3) ..................................................................115 5.6 Áðüóôáóç äýï óçìåßùí ..................................................................................117 5.7 Åóùôåñéêü ãéíüìåíï äýï äéáíõóìÜôùí (áñéèìçôéêü ãéíüìåíï) ..................... 118
ÐÅÑÉÅ×ÏÌÅÍÁ 5.8 5.9 5.10 5.11
13
Åîùôåñéêü ãéíüìåíï äõï äéáíõóìÜôùí (äéáíõóìáôéêü ãéíüìåíï) ...................120 Ìéêôü ãéíüìåíï ôñéþí äéáíõóìÜôùí ...............................................................125 ÄéÜöïñåò áíáãêáßåò êáé éêáíÝò óõíèÞêåò ......................................................127 ÁóêÞóåéò...........................................................................................................128
ÊåöÜëáéï 6: ÓõóôÞìáôá óõíôåôáãìÝíùí ..................................................131 6.1 6.2 6.3 6.4
ÃåíéêÜ ...............................................................................................................131 ÁëëáãÞ óõóôçìÜôùí .......................................................................................134 Ìåôáó÷çìáôéóìüò óõóôÞìáôïò óõíôåôáãìÝíùí... ........................................136 ÁóêÞóåéò...........................................................................................................139
ÊåöÜëáéï 7: ÅéóáãùãÞ óôïõò äéáíõóìáôéêïýò ÷þñïõò ..........................141 7.1 ÃåíéêÜ ...............................................................................................................141 7.2 Ïñéóìüò äéáíõóìáôéêïý ÷þñïõ ......................................................................142 7.3 ÃñáììéêÞ áíåîáñôçóßá äéáíõóìÜôùí..............................................................144 7.3.1 Éäéüôçôåò.................................................................................................147 7.4 ÄéÜóôáóç êáé âÜóç äéáíõóìáôéêïý ÷þñïõ ....................................................148 7.5 ÁóêÞóåéò...........................................................................................................150 7.6 ÅöáñìïãÝò ôïõ Äéáíõóìáôéêïý Ëïãéóìïý óôéò ÈåôéêÝò ÅðéóôÞìåò...............152 7.7 ÁóêÞóåéò...........................................................................................................158
Â. ÁÍÁËÕÔÉÊÇ ÃÅÙÌÅÔÑÉÁ.........................................................................161 ÊåöÜëáéï 8: ÃñáììÝò óôï åðßðåäï (R2).................................................... 165 8.1 Ç åõèåßá ...........................................................................................................166 8.1.1 Ç åîßóùóç ôçò åõèåßáò ..........................................................................166 8.1.2 ÓõíôåëåóôÞò äéåýèõíóçò åõèåßáò ..........................................................171 8.1.3 ÓõíåõèåéáêÜ óçìåßá ...............................................................................173 8.1.4 Ãùíßá äýï åõèåéþí .................................................................................174 8.1.5 Ó÷åôéêÝò èÝóåéò äýï åõèåéþí..................................................................176 8.1.6 Ðñïóáíáôïëéóìüò åðéðÝäïõ ùò ðñïò åõèåßá Þ ãñáììéêÞ áíéóüôçôá ùò ðñïò ÷ êáé ø ...............................................178 8.1.7 ÅöáñìïãÞ ôçò ãñáììéêÞò áíéóüôçôáò -Ãñáììéêüò ðñïãñáììáôéóìüò .179 8.1.9 Áðüóôáóç óçìåßïõ áðü åõèåßá.............................................................184 8.1.10 Åîßóùóç åõèåßáò óå ðïëéêÝò óõíôåôáãìÝíåò.........................................185 8.2 Ï êýêëïò ..........................................................................................................187 8.2.1 Ç åîßóùóç ôïõ êýêëïõ.......................................................................... 187 8.2.2 Åîßóùóç êýêëïõ ðïõ ðåñíÜ áðü ôñßá óçìåßá......................................190 8.2.3 Ó÷åôéêÞ èÝóç ôïõ êýêëïõ ìå Üëëç ãñáììÞ............................................192 8.2.4 Åîßóùóç åõèåßáò åöáðôüìåíçò óå óçìåßï êýêëïõ................................193 8.3 Ç Ýëëåéøç .........................................................................................................195 8.3.1 Óôïé÷åßá ôçò Ýëëåéøçò ............................................................................195 8.3.2 Ç åîßóùóç ôçò Ýëëåéøçò ........................................................................197 8.3.3 ÐáñáìåôñéêÝò åîéóþóåéò ôçò Ýëëåéøçò................................................. 200
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ÌÁÈÇÌÁÔÉÊÇ ÁÍÁËÕÓÇ
8.3.4 ÐïëéêÞ åîßóùóç ôçò Ýëëåéøçò................................................................201 8.3.5 Åîßóùóç åöáðôüìåíçò ó' Ýíá óçìåßï ôçò Ýëëåéøçò..............................202 8.4 Ç õðåñâïëÞ .....................................................................................................203 8.4.1 Óôïé÷åßá ôçò õðåñâïëÞò ........................................................................204 8.4.2 Ç åîßóùóç ôçò õðåñâïëÞò.....................................................................206 8.4.3 ÐáñáìåôñéêÝò åîéóþóåéò ôçò õðåñâïëÞò ..............................................210 8.4.4 ÐïëéêÞ åîßóùóç ôçò õðåñâïëÞò ............................................................211 8.4.5 Åîßóùóç åõèåßáò åöáðôüìåíçò óå óçìåßï õðåñâïëÞò .........................212 8.5 ÐáñáâïëÞ ........................................................................................................213 8.5.1 Óôïé÷åßá ôçò ðáñáâïëÞò .......................................................................214 8.5.2 Ç åîßóùóç ôçò ðáñáâïëÞò....................................................................215 8.5.3 Óõããåíåßò ðáñáâïëÝò.............................................................................216 8.5.4 ÐáñáìåôñéêÝò åîéóþóåéò ôçò ðáñáâïëÞò .............................................217 8.5.5 ÐïëéêÞ åîßóùóç ôçò ðáñáâïëÞò ...........................................................217 8.5.6 Åîßóùóç åõèåßáò åöáðôüìåíçò óå óçìåßï ðáñáâïëÞò ........................218 8.6 ÁóêÞóåéò...........................................................................................................219
ÊåöÜëáéï 9: ÃåíéêÜ ðåñß êáìðõëþí 2ïõ âáèìïý ....................................229 9.1 9.2 9.3 9.4 9.5 9.6
ÃåíéêÞ åîßóùóç êáìðýëçò 2ïõ âáèìïý...........................................................229 Ó÷åôéêÞ èÝóç äýï êáìðõëþí...........................................................................230 Äéåñåýíçóç ôçò ãåíéêÞò åîßóùóçò êáìðýëçò 2ïõ âáèìïý... .........................231 ÊÝíôñï êáìðýëçò ............................................................................................233 ¢îïíåò êáìðýëçò.............................................................................................233 ÁóêÞóåéò...........................................................................................................234
ÊåöÜëáéï 10: Óôïé÷åßá áðü ôç ìåëÝôç ôïõ R3 ...........................................237
10.1 ÅðéöÜíåéåò êáé êáìðýëåò ôïõ R3 ....................................................................237 10.2 Ç åõèåßá óôïí R3 .............................................................................................238 10.2.1 Åîéóþóåéò ôçò åõèåßáò............................................................................238 10.2.2 Áðüóôáóç óçìåßïõ áðü åõèåßá.............................................................242 10.3 Ôï åðßðåäï.......................................................................................................242 10.3.2 Óõíåðßðåäá óçìåßá ................................................................................247 10.3.3 Ó÷åôéêÞ èÝóç äéáíýóìáôïò êáé åðéðÝäïõ..............................................248 10.3.4 Ó÷åôéêÞ èÝóç äýï åðéðÝäùí ..................................................................249 10.3.5 Ðñïóáíáôïëéóìüò ôïõ R3 ùò ðñïò åðßðåäï.........................................251 10.3.6 Áðüóôáóç óçìåßïõ áðü åðßðåäï ..........................................................252 10.3.7 Áðüóôáóç åðéðÝäïõ áðü åðßðåäï .......................................................253 10.4 Ç óöáßñá. ........................................................................................................253 10.4.1 Åîéóþóåéò ôçò óöáßñáò ..........................................................................254 10.4.2 Åîßóùóç óöáßñáò ðïõ ðåñíÜ áðü 4 óçìåßá ........................................255 10.4.3 Ó÷åôéêÞ èÝóç åðéðÝäïõ ùò ðñïò óöáßñá .............................................256 10.4.4 Åðßðåäï åöáðôüìåíï óöáßñáò .............................................................258 10.5 Ï êýëéíäñïò .....................................................................................................259 10.5.1 ÊõëéíäñéêÞ åðéöÜíåéá.............................................................................259
ÐÅÑÉÅ×ÏÌÅÍÁ
15
10.5.2 Åîßóùóç êõëéíäñéêÞò åðéöÜíåéáò...........................................................259 10.5.3 Åßäç êõëéíäñéêþí åðéöáíåéþí ................................................................261 10.6 ÁóêÞóåéò...........................................................................................................262 ÌÅÑÏÓ ÔÑÉÔÏ • ÄÉÁÖÏÑÉÊÏÓ ËÏÃÉÓÌÏÓ
ÊåöÜëáéï 11: ¼ñéï óõíÜñôçóçò ...............................................................273 11.1 Ïñéóìüò ïñßïõ óõíÜñôçóçò ............................................................................273 11.2 ÐëåõñéêÜ üñéá..................................................................................................278 11.3 Ãåíßêåõóç ôçò óýãêëéóçò .................................................................................281 11.3.1 Óýãêëéóç (üñéï) ãéá ÷ + , f/(á, ) ...................................................282 11.3.2 Óýãêëéóç (üñéï) ãéá ÷ - , f/(-, â) ....................................................282 11.3.3 Óýãêëéóç (üñéï) ãéá î, f/(á, â) êáé î(á, â) .......................................282 11.4 ÃåùìåôñéêÞ åñìçíåßá ôçò óýãêëéóçò ...............................................................285 11.5 ÈåùñÞìáôá - Éäéüôçôåò ôùí ïñßùí.................................................................288 11.5.2 Èåþñçìá: ¼ñéï óå óôáèåñÜ: ÷lim ÷ = á .............................................289 á 11.5.3 Ðüñéóìá: ¼ñéï óôáèåñÜò: ÷lim c = c...................................................289 á 11.5.4 Èåþñçìá: ¼ñéï óôáèåñÜò åðß óõíÜñôçóç: lim (ê · f(÷)) = ê · ÷lim f(÷), ê 0.........................................................289 ÷á á 11.5.5 Èåþñçìá: ¼ñéï ñßæáò: ÷lim á
÷= á
á 0, ÷ 0 ..............................290
11.5.6 Èåþñçìá: Áëãåâñá ïñßùí óõíáñôÞóåùí .............................................291 11.6 Ï áñéèìüò e ùò üñéï óõíÜñôçóçò ..................................................................296 11.7 ÁóêÞóåéò...........................................................................................................298
ÊåöÜëáéï 12: ÓõíÝ÷åéá óõíÜñôçóçò .........................................................301 12.1 Ïñéóìüò óõíÝ÷åéáò óõíÜñôçóçò .....................................................................301 12.2 Åßäç áóõíÝ÷åéáò ...............................................................................................306 12.3 ÂáóéêÝò éäéüôçôåò óõíå÷þí óõíáñôÞóåùí......................................................311 12.4 ÓõíÝ÷åéá óõíÜñôçóçò óå äéÜóôçìá ................................................................312 12.4.1 ÐëåõñéêÞ óõíÝ÷åéá óõíÜñôçóçò óå óçìåßï ...........................................312 12.4.2 ÓõíÜñôçóç óõíå÷Þò óå äéÜóôçìá..........................................................313 12.5 ÁóêÞóåéò...........................................................................................................315
ÊåöÜëáéï 13: ÐáñÜãùãïò óõíÜñôçóçò....................................................319 13.1 Ïñéóìüò ðáñáãþãïõ.......................................................................................319 13.2 ÐëåõñéêÝò ðáñÜãùãïé .....................................................................................322 13.3 ÃåùìåôñéêÞ åñìçíåßá ôçò ðáñáãþãïõ............................................................323 13.4 Ó÷Ýóç ìåôáîý óõíÝ÷åéáò êáé ðáñáãþãïõ óõíÜñôçóçò .................................326 13.4.1 Èåþñçìá óõíÝ÷åéáò...............................................................................327 13.5 Êáíüíåò ðáñáãþãéóçò ....................................................................................329 13.5.1 ÐáñÜãùãïò óôáèåñÜò ...........................................................................329
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ÌÁÈÇÌÁÔÉÊÇ ÁÍÁËÕÓÇ
13.5.2 ÐáñÜãùãïò ôçò áíåîÜñôçôçò ìåôáâëçôÞò...........................................329 13.5.3 ÐáñÜãùãïò óýíèåôçò óõíÜñôçóçò.......................................................329 13.5.4 ÐáñÜãùãïò áëãåâñéêïý áèñïßóìáôïò óõíáñôÞóåùí...........................330 13.5.5 ÐáñÜãùãïò ãéíïìÝíïõ óõíáñôÞóåùí ...................................................330 13.5.6 ÐáñÜãùãïò ôïõ áíôéóôñüöïõ ìéáò óõíÜñôçóçò .................................332 13.5.7 ÐáñÜãùãïò ðçëßêïõ äýï óõíáñôÞóåùí ...............................................333 13.5.8 ÐáñÜãùãïò ëïãáñéèìéêÞò óõíÜñôçóçò.................................................335 13.5.9 ÐáñÜãùãïò åêèåôéêÞò óõíÜñôçóçò ......................................................335 13.5.10 ÐáñÜãùãïò óýíèåôçò åêèåôéêÞò óõíÜñôçóçò ......................................337 13.5.11 ÐáñÜãùãïò áíôßóôñïöçò óõíÜñôçóçò.................................................338 13.5.12 ÐáñÜãùãïé ôñéãùíïìåôñéêþí óõíáñôÞóåùí ........................................339 13.5.13 ÐáñÜãùãïé áíôßóôñïöùí ôñéãùíïìåôñéêþí óõíáñôÞóåùí.................342 13.5.14 ÐáñÜãùãïé õðåñâïëéêþí óõíáñôÞóåùí...............................................345 13.5.15 ÐáñÜãùãïò óõíÜñôçóçò ðáñáìåôñéêÞò ìïñöÞò. ................................347 13.5.16 ÐáñÜãùãïò óõíÜñôçóçò ðëåãìÝíçò ìïñöÞò.. .....................................349 13.5.17 ÐáñÜãùãïé áíþôåñçò ôÜîçò ................................................................ 350 13.5.18 Ðßíáêáò ðáñáãþãùí âáóéêþí óõíáñôÞóåùí ......................................352 13.6 ÁóêÞóåéò...........................................................................................................357
ÊåöÜëáéï 14: Ðåñß äéáöïñéêþí.................................................................363 14.1 14.2 14.3 14.4 14.5
Ïñéóìüò............................................................................................................363 ÃåùìåôñéêÞ åñìçíåßá ôïõ äéáöïñéêïý ............................................................364 Êáíüíåò äéáöüñéóçò ........................................................................................368 ÄéáöïñéêÜ áíþôåñçò ôÜîçò ............................................................................368 ÁóêÞóåéò...........................................................................................................371
ÊåöÜëáéï 15: ÅöáñìïãÝò ôïõ äéáöïñéêïý ëïãéóìïý...............................373 15.1 Õðïëïãéóìüò áðñïóäéüñéóôùí ìïñöþí (Êáíüíáò L' Hopital*)......................373 15.2 ÌåëÝôç óõíáñôÞóåùí ìå ÷ñÞóç ðáñáãþãùí................................................381 15.2.1 Èåþñçìá ôïõ Fermat* .......................................................................... 381 15.2.2 Èåþñçìá ôïõ Rolle* ..............................................................................382 15.2.3 Èåþñçìá ôïõ Lagrange* (Þ ðåðåñáóìÝíùí áõîÞóåùí).......................383 15.2.4 Èåþñçìá (Ó÷Ýóç ìïíïôïíßáò êáé ðáñáãþãïõ óõíÜñôçóçò) ...............385 15.2.5 Èåþñçìá (Ó÷Ýóç áêñüôáôùí ìå ôçí ðñþôç êáé ôç äåýôåñç ðáñÜãùãï)...............................................................................388 15.2.6 Óçìåßá êáìðÞò ...................................................................................... 389 15.2.7 ÄéÜôáîç êïßëùí êáé ôñßôç ðáñÜãùãïò..................................................390 15.2.8 Áóýìðôùôåò åõèåßåò ôçò êáìðýëçò ìéáò óõíÜñôçóçò ....................... 393 15.2.9 ÂÞìáôá ìåëÝôçò óõíÜñôçóçò ø =f(÷)...................................................397 15.3 ×ñÞóç ðáñÜãùãùí óå äéÜöïñåò åðéóôÞìåò...................................................407 15.3.1 Óôçí ÉáôñéêÞ - ÐáñÜäåéãìá....................................................................407 15.3.2 Óôéò Åðé÷åéñÞóåéò- ÐáñÜäåéãìá .............................................................407 15.3.3 Óôç ÓôáôéóôéêÞ - ÐáñÜäåéãìá.. .............................................................408 15.3.4 Óôç ÖõóéêÞ êáé óôéò Üëëåò èåôéêÝò åðéóôÞìåò......................................409