βιβλιο μαθηματικών β κατεύθυνσης

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<p>11.1 1.2 - 1.3 1.4 1.5 8 1 - 91.1 . . , . - .- .- .., u, v ., . 0. 0 = . .- - 1 = - 0 &gt;, - 0 0 =- 0 0 = = , . () () 10 () . 0 = ( ) ( ) ( ) / / ' . . , . , / / () ., || () ., || , , . () 1 - 11 : `) = - = - == == - = : `) = = = , . = , = . ( ), ( ), .- 0 s s ( ) , = - . 12 -( ) , 0 || = ( ) , || = - , ( ),2= - , , 0 s s . , , . 1 - 13 = : = / = ( ) 1 = / = ( ) 2 ( ) 1 ( ) 2 := - = , . ,., .- , , = ) : = ( ) 1,= = . . 1 === , :) = ) = 3 . = . 2 14 : = ( ) 2) ( ) 1 ( ) 2 = = ) ) = ) = ( ) = .// = . = . . , . 4 1 - 151) ,. = = , :) = ) 2) ,,,. = = , . 3) 4) . () ().) || ) || ) = ) = 5) . :) ( ),. ) ( ),. ) ( ),. ) ( ),. 16 1- 171.2 ,. = = . + . + = ) .) .) . + + + + = + + = ) + 1. + = + ( )2.( ) ( ) + + = + + ( )3. 0 + = ( 0 )4.( ) 0 + = ( , ) + 18 - . , . ( ) = + x += x = x x += = - = , = = + = . . . . = + = + = = - : + = - : = - - 1- 19, + . s + s + - + = + || - + = || + + + 20 - ) :x OX = , = , = , = , = , = :x OX O = = + + + + = - - - = + + ) ) :x OX = , = , = , = :x OX O = = + + = - - = + . , = = : ++x=+ x=- + + = x : 1)xx) x x. = = , 2 x 1- 211 0 + + = + + + + = + 0 0 0+ = + + = = 0 0 = 1- + = - = 2 ( ) : = ( ) , , , , , . :++ ++ = 3 22 - 2 . : + + = + + + + = + + 0 0 = 1 0 0 0+ + + + + = + + + + + = + + + = + + 0 0 = = 2 0+ + + + + = + + + + 0 0 0 = = = 0 = , , , , , , = +++++ 0 41- 23 // = = . . . : + = + = = 22 3 3== = = == *+e) 0 3 ++= + = + = + = = + (1)( ) 1 +2 3 + = + = + = + || ) 0 ++= += + = + + = = (2)( ) 2 2-3 = = = || ,,, . +=+ , , . 5 ,, + + = 0 = = 2 3 ) || ) || 6 24 - 1) x :2) . :) .......... + = ) .......... = ) .......... + = ) .......... + = ) .......... + = ) .......... = 3) , , , , . : + = 4) ,,,, + = + . .5) + = . , .6) . : + = xxx 1- 257) . = + + . .8) . ( ) f 2 5 3 = + , . 26 - 1- 271.3 = 0 . : ,&gt; 0 - , &lt; 0 . = = 0 = 0 = 0 1-( )+ = + - ( ) + = + ,e- ( )( )= :- = = = 0 0 0-( ) ( ) ( ) = = -( ) = ,e- ( ) = - = = 0 = - = = 0 = 22 28 - , ,, u = + u e , . . = + u 5 3 -, = + v 2 3 , , ., 0 = / / = , e= = 0 / / . / / = 0 . =, = - || , = (:= )- || , = (:= )- = 0 , = 0 (:= 0 = : 2+ = = = = + 2 + = 2 1- 29 ) 2 3 2 5 3 = 2 3 2 5 3 0 + = ( ) ( )2 3 5 0 + = 2 3 5 0 + = 5 5 0 = 0 0 = ) 3 2 3 2 + + = + + ( ) ( )3 2 3 2 + + = + + 3 2 2 3 2 2 + + = + + 3 3 2 3 3 2 + = + 0 0 = 111 5 6 6 5 5 6 = + + = + 6 6 5 5 = ( ) ( )6 5 = 6 5 6 5 6 5 = = = ) 2 3 2 5 3 = ) 3 2 3 2 + + = + + 15 6 = 11 5 6 = + 2 -,- . 30 - 2 . :( ) ( )5 6 5 6 = = 5 5 6 6 = 5 6 6 5 + = + 11 5 6 = + 1 B = ( )B = B + = B 0 = 2B B = = B B = + = , , . u 2A+3B 5 = . 4B = . 3 - :) B 0 = ) = 1- 311 u 2A+3B 5 = = ( ) ( )2A+3 B 5 = = 2A+3B 3 5 5 = = 3B 5 = u2u 2A+3B 5 = = 2A+3B 3 2 = = ( ) ( )2 A +3 B = = 2A 3B = + u , , , , A+3 2 3 = + + ) , , ) , , . 5u u - - . :) -----) --- . - . - - -. 3B 5 2 3 = + 32 - ) / / :A 3 2 3 + = + + ( ) ( ) 3 2 3 = + + 3 2 2 3 3 = + + 3 = 3 3 + + 0 3 = + 3 = / / ) 3 = , 2 0 2 0 + + = + + = 2 2 0 + + = 2 2 = + 1 2 = + , , . : 2 0 + + = 6 ,,-- B, , . - - B/ / .B = - B/ / .B = - / / . = - . 12 --- .1- 33 = = = / / 1 6= 1 6= (1) ( ) ( )1 1 1 7 7 7= = + = + 1 1 7 7= + (2)( )( ) 121 1 1 1 1 7 7 6 42 7= = + = + ( ) 1=1 1 6 6= = = 1 1 42 642 742 6 716 6 6= + = + = = = + = + / / ( || ) , 1 6= 1 7= , , 7 -B = = - - ,. --- . 34 - () += += + + + = + + + + +`= +) (1) = (2) = (3) = (4) (1) (2), (3), (4) : + + = + + + + + + + = + + 0 + + = , , 2 2 = = 2 = + = + 2 = + 2+ = -,., . 8 . = = . . 9 - . 1- 35 : = + + (1) : = + + (2) (1) (2) :2 = + + + + + 2 = = = + + + + + 2 = + + + + 2 = + + + + 2 = + 2+ = x, y - , .( )() x 6 3y x 3y 6 2x 6y 12 2 26y 7x 11 66 7x 6y 66 11 7x 6y 66 11+ += + = = + = + = + + = + 9x 9 54 x 6 = = (2) ( ) ( ) 231 6 6 3y 2 12 3y 6 y2+ = + = = (3) (2) (3) 3x y2= x y || . , - ( )1 2= + 10 , x, , y x 6 3y + = + 6y 7x 11 66 + = x, y 11 (1) 36 - ) . = , = , r = || = = = = = ( ) ( ) = = r r = ( ) r r r r + + = + + = + =+ ) . = , = , r = || = = = BM= = ( ) ( ) = = r r = ( ) r r r r = = = , r, MA MB = , ) r +=+ ) r = 12 rr1- 37 0 = 0 + = = / /= 0 = 0 0 0 0=+ = == 0 = = ( ) 1/ / 2 2 2 2 2 = = + = + (1)( ) / / + + = = (2) (1) (2) : 1 2 2 2 2 = + = + 2 2 0 = ( ) ( ) 2 1 2 0 + = , : 2 2 0 211 2 0 2 1 2= = = = = = 2 = (1)( )1 2 2 2 2 2 / /22= + = + = , , ( )/ / 2 ( ) / / + ( ) / / 2 14 , - 0 + = , e. 0 = = 13 38 - , , / / = (1) : 4 1 23 3| | | |= = + + = ||\ . \ . 4 1 23 3| |+ = |\ . 4 3 3 3| | | |= + ||\ . \ . 2 2 23 3| | | |= = + + = ||\ . \ . 2 2 23 3| |+ = |\ . 2 4 3 3| | | |= + ||\ . \ . (1) 4 3 2 4 3 3 3 3 | | | | | | | | + = + |||| \ . \ . \ . \ . 4 3 2 4 3 3 3 3| | | | | | | | + = + ||||\ . \ . \ . \ . 4 3 2 4 03 3 3 3| | | | | | | | + = ||||\ . \ . \ . \ . 4 2 3 4 03 3 3 3| | | | + + + = ||\ . \ . () 4 2 012 6 03 312 6 9 12 0 9 12 03 4 03 3+ + = + = + = + = + =13 6 0 6 3 2+ = = = (1) 9 6 0 2 18 12 0 3 30 102 + = + = = = , 0 = . , , 23= + , 4 1 3| |= + |\ . , 2 2 3| |= + |\ . - , , -. 15(1) 13 37.1- 39 1) ) 2 3 2 3 + = + ) 3 5 8 16 3 5 8 + + = + + + ) 7 2 5 2 7 + = +) 2 3 5 3 2 + = 2) = + = + = 3) 3 2 = 2 = 4) 3 2 = 5 3 2 = + 5) = 3 = 2 3 = + 6) ( ) f K 2AB 3K = + () f 2AB 3 = + = ( ) f 2AB 3 = + = . 3 0 + = .7) ) = ) 2 = ) 3 + = ) + = 40 - 8) ,,, + = + . . 9) , , , , + + = + + . 10) 2 3 2 3 + + = + + . 11) . u 4 2 3 = + . 12) ,,, , , . ( ) ( ) u = + . 13) 2 3 2 + + = + + || 14) , , 2 3 2 3 = 15) , , 3 7 5 2 + = 16) , , , 3 4 0 + =) , , ) , , 1- 4117) , , , :3OA 4OB 7O + = ) ) , , ) 18) , , . 2A 0 + + = 19) . 2 3 2 0 + + = 20) . :)( )1 22= ) 4 = + + + 21) ,,. :22) ' '. ' ' 2 ' ' ' + = . 42 - 23) ,,- = = = . : 5 + + + = 24) - . :) 2 + + + = ) 2 + + + = 25) . 2 = . :) 2 = + ) = 26) ,,,, = , = , 2 = + 2 = . , , ,, - .27) 4v 2 3= + 3 3u 2 4= + . 28) u, v, w , , , u v 2w = + , B u 2v w = + u 5v 2w = , , .1- 4329) , , = , = , 3 2 = , -.) , .) , , .30) -2 5= . = = ( )2 7= 5 27+= .31) = , = 2 = .- , , , , , .32) = , = 2 = . , , :) , , , ) 2 = 33) ,1 4= = . = , = :) , ) 44 - 34) , , 3 = , 2 = 5 3 = , , . 35) - 66-. 2 = . - :) , .) .36) = = . ,, ,, ) , ) , . 37) . 3 = , 3 = , = = ) - , , ) , , -.38) . = + = + ,- / / . 1- 4539) , , , :( ) 4 2 4 3 + = , , - .40) : = + 1 + = , , . 41) ,, ( ) ( ) ( )2 2 2 2 2 0 + + = . 42) x , , ( ) ( ) ( )2 2 2x 3x 2 x 5x 3 x 13x 3 + + + + = + 43) , , 3 5 = + , = 4 = + , -3 = , , . 44) : = = , . + = , , . 46 - 45) 0 = . -:( )2 5+10 3 = + 46) , , :( ) / /2 ( ) 6 / /2 ( ) 6 = 47) , , :( ) 12 / /3 ( )6 / /4 + ( ) / / 2 48) , , :( ) / / 2 + ( )/ / 2 + 4 2 = 49) , , :( ) / / + ( )/ / + 0 + += 1- 4750) 3u 4v 4u 3v + = = u,v , .51) u 2v u v =+ = u,v , . 48 - 1- 491.4 xx -, i = 1(-). .xx. M/ / i xe M xi = xe xx. x () , xx yy i, j. xy x xix xixxyyij50 - xy.M1xx 2 yy.:- x 1, xx.- y 2, yy -.,(x,y) :xx yy 12-.1xx 2 yy . . x y (x,y)xy -i, j. , x, ye : x i y j = + OA = 1,2xx yy. yxijM(x,y)M1M21- 51 1 2OA OA OA = + (1) x, y , 1OA x i = 2OA y j = (1) OA x i y j x i y j = + = + (2) x, y . x, y x i y j ' ' = + (3) x x ' = (2) (3) ( ) ( )x i y j x i y j x x i y y j ' ' ' ' + = + = (4)y yi jx x' = ' i / / j x x ' = (4) x x ' = y y ' =1. x y x y 2. x i y j i j.3. x - y, ( ) ,y x = jiA1A2A52 - 4. ( ) y , x = .: 0yx 00 == = 5. ( )1 1xy , =( )2 2x y , = - .: 1 21 2 y yxx== = ( )1 1x ,y =( )2 2x y , = :1. ( )1 1 2 2y y x , x + + + = 2. ( )1 1x y , =3. ( )1 1 2 2 x x y y , + = + + ( )1 1 1 1 x ,y x i y j = = + ( )2 2 2 2 x ,y x i y j = = + 1.( ) ( )( ) ( )1 1 2 2 1 2 1 2 x i y j x i y j x x i y y j + = + + + = + + + = ( )1 2 1 2x x ,y y = + +2.( )( )1 1 1 1 1 1 x i y j x i y j x , y = + = + = 3. ( ) ( )1 1 2 2 x ,y x ,y + = + = ( )1 2 1 2x x ,y y = + + 1- 53 (x,y) (x,y) :(x,y) ABMBMAy yyx22xx+ ==+ ( )A AOA x ,y =, ( )B BOB x ,y =( ) OM x,y = ( ) ( ) ( )M M A A B BOA OB 1OM x ,y x ,y x ,y2 2+= = + ( ) ( )M M A B A B1x ,y x x ,y y2 = + + ( )A B A BM Mx x y yx ,y ,2 2+ +| |= |\ . A B A BM Mx x y yx y2 2+ += = (x,y) (x,y) ( )B B A Ay y , x x = ( ) x,y =( )A AOA x ,y =, ( )B BOB x ,y = ( ) ( ) ( )B B A A OB OA x,y x ,y x ,y = = ( ) ( )B A B Ax,y x x ,y y = B A B Ax x x y y y = = ( )B A B A x x ,y y = yx A(xA,yA)B(xB,yB)A(xA,yA)yx (x,y)B(xB,yB)54 - 1. ( ) x,y =xxy 0 =2. ( ) x,y =yyx 0 =( ) ,y x = 2 2x +y = OA = (x,y)Av 1, 2 xx yy :( )1OA x = ( )2OA y =1:( ) ( ) ( ) ( ) ( )22 2 2 2 21 1 1 2OA OA A A OA OA = + = + 2 22 22 2 2 2 x y x y x y = + = + = + (x,y) (x,y) ( ) ( )2 2B A B A x x y y = + ( )B A B A x x ,y y = ( ) ( )2 2B A B A x x y y = + ==-- yx A(x,y) A1A2 1- 551. xx . ( )Ad ,x x y ' =2. yy . ( )Ad ,y y x ' =( )1 1 x ,y =( )2 2 x ,y = :( ) / / det , 0 = ( )det , :( ) ( )1211 2 12 2det , dxx yyye yx, x t = = - / /( ) det , 0 = xx( ) x,y = OA = xx xx () OA. 0 2 s &lt; ( ) x,y =x 0 = yx= , x 0 = Ayx+56 - , , : / / = ( )1 1 x ,y =( )2 2 x ,y = 11yx= 22yx= ( )1 12 2x y / / det , 0 0x y = = 2 11 2 2 1 2 1y yx y x y x x = = = 1. -( ) x,y =x 0 = yy ( xx)2. -( ) x,y = 0 - xx. / /x x 0 ' =3. ( ) x,y = - xx. = xA=xByA=yB 1- 57 ) xx ( )( )2 6 0 3 2 0 = + = 3 2 = = ) yy ( )( )2 5 6 0 3 2 0 + = = 3 2 = =) ( )( )( )( )22 5 6 0 3 2 0 6 0 3 2 0 + = = = + = 3 2 3 2= = = = ( )2 2 5 6, 6 + e ) xx) To A yy) To A ) 1 1 (,)- xx 0 =- yy 0 =- xx 0 0 = =- yy 0 0 = =- 1 0 0 &gt; &gt;- 2 0 0 &lt; &gt;- 3 0 0 &lt; -2 3 2 6 + - + ( ) ( ) , 2 3, e +2 5 6 0 + &gt; 2 3 2 6 + - +( ) ( ) ,2 3, e +-2 2 3 2-31- 59 (x,y) 3 -3 1, 2 .) x 1 1 x 1 s s s(x,y)1 x 1 s s 1,2( ) .) y 2 2 y 2 &lt; &lt; ) x 1 = y 1 s ) y 4 = x 1 &gt;4) ) 7i 6j = ) 5j 4i = ) 3i = ) 4j = )( ) ( )u 5 i 4j 9 3i 2j = 5) , e ( ) 1,2 + ,( ) 2, + ) ) xx) 1 3 6) 2 ,3 1| |= |\ .,22 , 4 13| |= + |\ .e { }1,1 , .7) ( ) 1, 2 = ( ) ,2 1 = . , :) ) , ) , 8) ( )u 2 3, 1 = + , e , e u .74 - 9) ( )2 2 4 2, 5 1 = + + ( )2 2 1,3 2 2 = + e , 10) ( )2 4 1,2 3 = + ( )2 2 4, = + +, , e. , , = .11) ( )2 2 5 6, 9 = + e. e ) 0 = ) / /x x ' 0 = ) / /y y ' 0 = 12) (2,3) :) AB (3,2)) ( )A 2, 4 = ) 2A 3 = (5,4)13) (x,4), B(5,y) (2,3) x, ye. N - x, y . 14) (-2,-1) , (4,0) (6,1)) 1- 75) 15) (-2,3)) , - (0,1)) , - (-1,0).16) (-3,-4), (2,3), (4,5), (-1,-2)) ) 17) (-3,5),(1,7)(1,1) .- . 18) ( ) 2,3 =,( ) 5,2 = ( ) 3, 4 = ) u 3 = + ) v 2 3 = 19) ( ) 2, 4 = ( ) 3,2 = . ( )u x,y = 76 - ) u = + ) u + = )u = , e ) u = + , , e20) ( ) 2,3 =,( ) 1,7 = ( )u 10, 13 = .- u , .21) (...</p>