電気電子数学入門 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/073471 このサンプルページの内容は, 初版 1 刷発行当時のものです.
i 14 (tool) [ ] IT ( ) PC (EXCEL) HP() 1 1 4 15 3 010 9
ii 1... 1 1.1 1 1. 1.3 3 1.4 4... 8.1 8. 10.3 11.4 13 3... 17 3.1 17 3. 18 4... 3 4.1 3 4. 3 4.3 4 4.4 9 5 1... 3 5.1 3 5. 33 5.3 34 6... 41 6.1 41 6. 4 6.3 43 7... 49 7.1 49 7. 50 7.3 50 7.4 5 7.5 53 7.6 53 7.7 55 8... 58 8.1 58 8. 59 8.3 59 8.4 6
iii 9... 66 9.1 66 9. 66 9.3 70 9.4 71 9.5 7 10... 77 10.1 77 10. 78 10.3 3 79 10.4 81 11... 86 11.1 86 11. 88 11.3 89 11.4 90 11.5 91 1... 93 1.1 93 1. 96 1.3 97 1.4 97 1.5 98 1.6 99 13 1... 101 13.1 101 13. 103 13.3 105 14... 109 14.1 109 14. 110 14.3 113 14.4 113 14.5 116 15... 118 15.1 118 15. 10 15.3 11
iv 16... 15 16.1 15 16. 16 16.3 16 16.4 17 16.5 18 16.6 130 16.7 131 17... 135 17.1 135 17. 137 17.3 140 17.4 141 18... 144 18.1 144 18. 147 18.3 148 19 1... 151 19.1 151 19. 15 19.3 K 153 19.4 153 19.5 D 155 19.6 1 156 0... 160 0.1 1 160 0. 16 1... 168 1.1 168 1. 169 1.3 171 1.4 173 1.5 176
v... 180.1 180. 180.3 181.4 18.5 184.6 186 3... 191 3.1 191 3. 193 3.3 193 3.4 194 4... 199 4.1 199 4. 00 4.3 00 4.4 0... 08... 7
3 sin, cos, tan 5.1 x OP O OP 360 5.1 r l θ θ = l/r [rad] ( ) l l = πr (πr)/r = π 180 180 = π [rad] 1 = π/180 [rad] π/180
5. 33 1 [rad] = 180 /π 180/π [rad] 5.1 (1) 150 = 150 π 180 = 5 6 π [rad] () 7 6 π [rad] = 7 6 π 180 π = 10 (0 <θ<π/) θ r 5. x y () ( ) () sin θ = y r, cos θ = x r, tan θ = y x (x 0) 5. sin θ, cosθ, tanθ θ ( )θ 5.3 θ r P (x, y) sin θ, cosθ, tanθ sin θ = y r, cos θ = x r, tan θ = y x (x 0) (5.1) θ tan θ x =0 θ 5.3 5.4
34 5 ( 1) r =1 ( ) 5.4 y sin θ x cos θ θ 0 θ π/ 5.1 θ 5.3 θ 5. 5.1 5. θ 0(0 ) sin θ 0 cos θ 1 tan θ 0 π 6 (30 ) 1 3 π 4 (45 ) 1 3 1 π 3 (60 ) 1 3 1 1 3 π (90 ) 1 0 θ 1 3 4 sin θ + + cos θ + + tan θ + + π/ π/ + π/ π/ θ = π/ (5.1) θ sin θ, cosθ, tan θ tan θ = sin θ cos θ (5.) sin θ +cos θ =1 (5.3) (5.3) x + y = r n θ +nπ θ sin(θ +nπ) =sinθ cos(θ +nπ) =cosθ (5.4) tan(θ +nπ) =tanθ
5.3 35 5.5(a) θ θ x sin( θ) = sin θ cos( θ) =cosθ (5.5) tan( θ) = tan θ 5.5 5.5(b) θ θ + π sin(θ + π) = sin θ cos(θ + π) = cos θ (5.6) tan(θ + π) =tanθ 5.5(c) θ θ + π/ ( sin θ + π ) =cosθ ( cos θ + π ) = sin θ (5.7) ( tan θ + π ) = 1 tan θ 5. (5.4) (5.7) (1) sin ( 53 ) ( ) 5 ( π = sin 3 π = sin π π ) ( = sin π ) ( π ) =sin 3 3 3 3 =
36 5 ( 1) ( ) 10 () cos 3 π =cos (4π 3 ) π =cos ( 3 ) π = cos ( 3 ) π + π (3) tan ( 133 ) π ( π ) = cos 3 ( ) 13 = tan 3 π = 1 ( = tan 4π + π ) = tan π 3 3 = 3 α β α + β α β α β sin(α ± β) =sinαcos β ± cos α sin β [ 1) ] cos(α ± β) =cosα cos β sin α sin β tan(α ± β) = tan α ± tan β 1 tan α tan β [ ] [ ] (5.8) http://www.morikita.co.jp/soft/ 73471/index.html 5.1 sin 75, cos 15, tan( 15 ) sin 75 =sin(45 +30 )=sin45 cos 30 + cos 45 sin 30 = 1 3 + 1 1 3+1 = cos 15 =cos(45 30 )=cos45 cos 30 +sin45 sin 30 = 1 3 + 1 1 3+1 = tan( 15 )=tan(30 45 )= tan 30 tan 45 = (1/ 3) 1 1+tan30 tan 45 1+(1/ 3) 1 = 1 3 = (1 3)( 3 1) 3+1 ( 3+1)( 3 1) = 4+ 3 = + 3 1) +/
5.3 37 α = β sin α =sin(α + α) =sinαcos α +cosαsin α =sinαcos α (5.9) cos α =cos(α + α) =cosαcos α sin α sin α =cos α sin α (5.10) (5.10) (5.3) cos α =1 sin α cos α =(1 sin α) sin α =1 sin α (5.11) (5.10) sin α =1 cos α cos α =cos α (1 cos α)=cos α 1 (5.1) (5.11), (5.1) sin α = 1 cos α, cos α = 1+cosα (5.13) 5.3 cos π 8 = 1+cos(π/4) = 1+(1/ ) = +1 = + 4 5. cos α =3/5 0 <α<π/ sin α, tanα, sinα, cosα sin α +cos α =1sin α =1 cos α =1 9/5 = 16/5 α 1 sin α = 4 5, sin α tan α = cos α = 4 3 sin α =sinαcos α = 4 5 3 5 = 4 5 cos α =cos α sin α = 9 5 16 5 = 7 5
38 5 ( 1) sin α cos β = 1 {sin(α + β)+sin(α β)} cos α sin β = 1 {sin(α + β) sin(α β)} cos α cos β = 1 {cos(α + β)+cos(α β)} (5.14) sin α sin β = 1 {cos(α + β) cos(α β)} 5.3 (5.8) (5.14) 1 ( ) = 1 {sin(α + β)+sin(α β)} = 1 {(sin α cos β +cosαsin β)+(sinαcos β cos α sin β)} =sinα cos β ( ) 1 ( http://www.morikita.co.jp /soft/73471/index.html ) (5.14) α = (A + B)/, β =(A B)/ sin A +sinb =sin A + B sin A sin B =cos A + B cos A +cosb =cos A + B cos A cos B = sin A + B cos A B sin A B cos A B sin A B (5.15)
93 11 y = f(x) x a b x Δx Δx = b a y Δy Δy = f(b) f(a) x a b f(x) Δy f(b) f(a) = Δx b a (1.1) y = f(x) A(a, f(a)) B(b, f(b)) [a, b] 1) y = f(x) 1.1 f(x) =x x 4 x Δx Δx =4 = y = f(x) y Δy Δy = f(4) f() = 4 =1 1) a x b [a, b]
94 1 Δy Δx = 16 =8 [, 4] f(x) =x 8 y = f(x) x a a + h Δy f(a + h) f(a) f(a + h) f(a) = = Δx a + h a h (1.) a h 0 11 lim h 0 f(a + h) f(a) lim h 0 h (1.3) (1.3) f(x) x = a ( ) f Δy (a) = lim h 0 Δx = lim f(a + h) f(a) h 0 h (1.4) f (a) 1.1 1. 1.1 1. 1.1 b = a + h B (a + h, f(a + h)) (1.) A B AB a h 0 B A AB 1. B A A f(x) A f (a) A(a, f(a))
1.1 95 1. f(x) =3x x = a f 3(a + h) 3a 6ah +3h (a) = lim = lim = lim (6a +3h) h 0 h h 0 h h 0 =6a y = f(x) x dy dx = lim f(x + h) f(x) h 0 h (1.5) dy/dx df (x)/dx, f (x),y 1.3 (1) f(x) =x f (x + h) x h (x) = lim = lim h 0 h t 0 h =1 () f(x) =x f (x + h) x xh + h (x) = lim = lim = lim (x + h) h 0 h h 0 h h 0 =x (3) f(x) =x 3 f (x + h) 3 x 3 x 3 +3x h +3xh + h 3 x 3 (x) = lim = lim h 0 h h 0 h = lim h 0 (3x +3xh + h ) =3x 1.3 (1) (3) f(x) =x r f (x) =rx r 1 (1.6) (1.6) r
96 1 f(x) =x 4 x + x f(x) =(x + 1)(x 3 ) f(x) =(x 1)/(x +1) ( ) (1.6) x n y = kf(x) y = kf (x) (1.7) y = f(x) ± g(x) y = f (x) ± g (x) [ ] (1.8) y = f(x)g(x) y = f (x)g(x)+f(x)g (x) (1.9) y = f(x) g(x) y = f (x)g(x) f(x)g (x) {g(x)} (1.10) f(x) =1 y = 1 g(x) y = g (x) {g(x)} (1.11) y = {f(x)} n y = n{f(x)} n 1 f (x) (n ) (1.1) (1.7) {kf(x)} kf(x + h) kf(x) f(x + h) f(x) = lim = k lim = kf (x) h 0 h h 0 h (1.8) {f(x)+g(x)} {f(x + h)+g(x + h)} {f(x)+g(x)} = lim h 0 h f(x + h) f(x) g(x + h) g(x) = lim + lim h 0 h h 0 h = f (x)+g (x) {f(x) g(x)} = f (x) g (x) (1.9) (1.1) http://www.morikita.co.jp/soft /73471/index.html 1.4 f(x) =x 3 +5x x +5 f(x) =(x 3 +5x x +5) =(x 3 ) +(5x ) (x) +(5)
1.4 97 = 3x +5 x 1=6x +10x 1 1.1 (1) y =(x + 1)(x 1) () y = x +1 x + (1) y =(x +1) (x 1) + (x +1) (x 1) =(x 1) + (x +1) =x 1+x +=4x +1 () y = (x +) (x +1) x = x x +4 (x +)(x 1) = (x +) (x +) (x +) y = f(u) u u = g(x) y = f(g(x)) y = f(g(x)) dy dx = dy du du dx dy dx = f (u) g (x) (1.13) f(g(x)) u = g(x) (1.13) f (u) g (x) 1.5 y =(ax + b) u = ax + b y = u dy/du =u, du/dx = a dy dx = dy du =u a =a(ax + b) du dx 1. (K) =0 (K ) (sin x) =cosx
98 1 (cos x) = sin x (sin ax) = a cos ax (cos ax) = a sin ax (e x ) = e x (e ±ax ) = ±ae ±ax (a ) [ ] (log x ) = 1 x (log f(x) ) = f (x) f(x) (a x ) = a x log a (a >0, a 1) 1. (cos x) = sin x (5.15) (cos x) = lim h 0 cos(x + h) cos x h (cos x) = lim h 0 sin(x + h/) sin h/ h (11.10) (cos x) sin(h/) = lim sin(x + h/) = 1 sin x = sin x h 0 h/ sin x (5.15) y = f(x) y = f (x) x f(x) f (x), y, d y dx, d dx f(x) 3 1.6 y = x 3 f (x) =3x, f (x) =6x
180 3 (scalar) (vector ) () O P.1 O P OP A A OP OP A A 1 u ( u =1) A A = Au = A u (.1). A = B
.3 181.1. PQ RS A A A () 0 0 OO AA 0 0( ).1.3 A = B, C = D.3.4(a) x y A x () i y j A A = ia x + ja y =(A x,a y ) (.) A x,a y x y A (A x,a y ) A A x i + A y j ia x + ja y.4(b) x y z 3 A z k A A = ia x + ja y + ka z =(A x,a y,a z ) (.3)
18.4 A ( ) A = A = A x + A y + A z (.4) A = ia x + ja y + ka z B = ib x + jb y + kb z A = B A x = B x, A y = B y, A z = B z. i = i1+j0+k0 i =(1, 0, 0) j =(0, 1, 0), k =(0, 0, 1).1 (1) A = i+j3 () B = i+j4+k4 (1) A = A = A x + A y = +3 = 13 () B = B = B x + B y + B z = +4 +4 =6 A B C D C = A + B = i(a x + B x )+j(a y + B y )+k(a z + B z ) =(A x + B x,a y + B y,a z + B z )=(C x,c y,c z ) (.5) D = A B = i(a x B x )+j(a y B y )+k(a z B z )
.4 183 =(A x B x,a y B y,a z B z )=(D x,d y,d z ) (.6) C D.5 (.5 ).5.3 A = i3 j k4 B = i4+j3 k C = A + B D = A B C = A + B = i{3+( 4)} + j{( ) + 3} + k{( 4) + ( )} = i + j k6 D = A B = i{3 ( 4)} + j{( ) 3} + k{( 4) ( )} = i7 j5 k (.5) A + B = B + A (A + B)+C = A +(B + C) A +( A) =0, A + 0 = A (.6) A B = A +( B) A A = 0 A (m ) A
184 m>0 ma A m m =1 A m<0 ma A m m = 1 A m =0 ma m(na) =(mn)a (m + n)a = ma + na m(a + B) =ma + mb 0 (m ) 0 m0 = 0. (1) 3A +5B A 4B () 3(A + B) (4A +B) (3) 1 (A +B) 1 (A B) 3 (1) 3A +5B A 4B =(3 )A +(5 4)B = A + B () 3(A + B) (4A +B) =6A +3B 8A 4B = A B (3) 1 (A +B) 1 3 (A B) = 1 A + B 3 A + 1 3 B = 1 6 A + 4 3 B A, B θ (0 θ π) (.6 ) (.7) AB A B C C = AB = A B.6 C = AB = A B = A B cos θ = AB cos θ (.7) (.7)
.5 185 θ 0 A B = AB cos 0 = AB (.8) A = B A A = AA cos 0 = A θ π/ A B = AB cos π = 0 (.9) A B = B A (.10) A (B + C) =A B + A C (.11) (ka) B = k(a B) =A (kb) (.1) i, j, k 1 1 0 i i =1, j j =1, k k = 1 (.13) i j =0, j k =0, k i =0, j i =0, k j =0, i k = 0 (.14) (.7) C = AB = A B = A x B x + A y B y + A z B z (.15) (.7) (.15) θ cos θ = A B AB = C AB = A xb x + A y B y + A z B z AB ( ) θ =Cos 1 Ax B x + A y B y + A z B z AB (.16) (.17)
186.3 C = AB = A B = A x B x +A y B y +A z B z C = AB = A B =(ia x + ja y + ka z ) (ib x + jb y + kb z ) (.11) (.1) A x B x (i i)+a x B y (i j)+a x B z (i k)+a y B x (j i) + A y B y (j j)+a y B z (j k)+a z B x (k i)+a z B y (k j) + A z B z (k k) (.13) (.14) i, j, k A x B x 1+A x B y 0+A x B z 0+A y B x 0+A y B y 1+A y B z 0 + A z B x 0+A z B y 0+A z B z 1 = A x B x + A y B y + A z B z C = AB = A B = A x B x + A y B y + A z B z.4 A = i3 j k4 B = i4+j3 k θ C = A B =3 ( 4) + ( ) 3+( 4) ( ) = 1 6+8= 10 A = 3 +( ) +( 4) = 9, B = ( 4) +3 +( ) = 9 cos θ = C AB = 10 9 ( θ =Cos 1 10 ) 9 110. A, B C = A B
7 1 160 98 10 1 EXCEL 30, 47, 13, 168, 197, 06 y 137 15 3 3, 6 6 3 6, 103 7 60, 116 176 199 199 186 78 44 191 194, 03 55 135 16 36 3 91, 99 137 194 181 71, 78 4 () 4 ( ) 4 181 ( ) 4 66 66 153 3 (conjucate complex number) 10, 58 66 (determinant) 7 86 58 103 103 103 59 1, 18 9 (imaginary part) 9, 58 137 193 8 68, 183 15 137 183, 185 98 45 97 14 171 157 3 9 11 6, 105 6, 105 4 171 73 33 34 34 191 191 1 50, 49 5 5, 113 6, 103 148 8 1, 17 (real part) 9, 58 1, 17 4, 147 169
8 4 8 73 77 4, 34 135 5 5 44 153 169 3 50 44 173 168 173 18 180 184 7 0 () 33 43 1 8 () 33 71 3 66, 181 66 19 135 15, 15 94, 99, 101 60 70 181 171 104 199 70 173 71 50 5, 53 55 8 1 118 70 180, 181 1 103 103 3 140 18 171 3 4 58 59 50 3, 105 1 156 1 161 135 11 53 70 95 3 157 1 80 153 193 199 199 3 60, 116 184 16 148 75 169 37, 13, 148 89 187 193 1 199 156 95 155, 164 94 151 03 196, 01 36 (complex number) 9 58 90 15 3 141 130 17 131
9 44 173 196, 00 13, 68, 185, 187 10 147, 194 109 93 9 9 14 180 186 58 153, 160 119 119 119 101 6 157 113 157, 164 113 8 144 8 8 66 ( ) 33 3 168 55 70 181 66 66 18, 77 110 19
010 Printed in Japan ISBN978-4-67-73471-5