A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r A S- 3.4.5. 9 phone: 9-8-444, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: 6 6.. 3... 3. 4. 5. 7 Feynman Lectures on Physics, vol. 3 URL http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html web page (http://www.gakushuin.ac.jp/~8879/mathbook/ A
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r A =.6 +.4 A 5 : 5 = max{ A, } A A A A B A, B A A A %
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 3 web page
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 4 E-mail hara@math.kyushu-u.ac.jp a < b n Z N Q R C x R x A Z Q a < b a = b a b a > b a = b a b a < x < b (a, b a x b [a, b n! = n (n (n n! = unique, uniquely 9 4 5
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 5 4 office hour 5/8 4/ x R x R 4 5 (i (,, (,, (ii A(,,, B(,,, C(,, (i, (ii 3 3 x x A A 3 A V, W, X V, W, X (i V x + x = x, x, x x y x + y x x = x y = y x + y = x + y y 3 + y 3 kx k x kx = kx k (ii W x + x = x, x,
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 6 x x x (i x = x y = y y y 3 x + y = x + y kx x + y k x kx = kx + y 3 k V, W x, x, (iii X x x x + x = x, x, x y x + y x = x y = y x + y = x + y kx k x kx = kx y 3 + y 3 k 4* x ax + bx + c a, b, c X p = ax + bx + c q = dx + ex + f p + q = (a + dx + (b + ex + (c + f. k R p = ax + bx + c kp = (kax + (kbx + (kc. X 5** 4 X x = Y Y x x = Y 4 5 :55 4 4 A4 B5
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 7 5 8 a R a R, C, Z 6 6 a,..., d, x 4 a :=, b :=, c :=, d :=, x := 3 3 x. a, b. b, c 3. a, c, d 3 7* x V a, b, c, p a =, b = x +, c = (x +, p = x x + p. a, b. b, c 3. a, b, c 3 5 3 :3 4 4 A4 B5 (i n = (,, x = (,, n (x x = (x x + (y y (z z = x + y z = x + y z =
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 8 (ii ax + by + cz = d a b c = d a + b c = d a + b c = d d a = d, b = d, c = d d = d d x + ( d y + ( d z = d x + y + z = AB AC (i (i A(,, y =, z = y =, z = x [ (i B( 4,, C(,, AB = ( 5,, AC = (,, x = (,, x x = s AB + t AC, x 5 y = + s + t z s, t A, B, C x = x + sp + tq p, q,,,, 3,, 5, 5, 3, 4 3, 5, 3 3, (ii 3 A, B, C x x A = s AB + t AC (s, t R x A A AB AC x y = + s 3 + t z s, t
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 9 A, B, C x = x + sp + tq p, q 3 3,,, 3, 4,,, 3, 3, (i, (ii x = s, y = t z s, t (ii x + y + z = x = s, y = t, z = s t s, t 3 V 4 W W W W X X 4 p = ax + bx + c, q = dx + ex + f p + q = (a + dx + (b + ex + (c + f X x X kp = (kax + (kbx + (kc x X 8 p + q = (a + dx + (b + ex + (c + f q + p = (d + ax + (e + bx + (f + c a + d = d + a O.K. r = gx + hx + i (p + q + r = {(a + dx + (b + ex + (c + f} + {gx + hx + i} = {(a + d + g}x + {(b + e + h}x + {(c + f + i} p + (q + r = {ax + bx + c} + {(d + gx + (e + hx + (f + i} = {a + (d + g}x + {b + (e + h}x + {c + (f + i} O.K. V p + = + p = p O.K. p = ( ax + ( bx + c p + p = {( a + a}x + {( b + b}x + {( c + c} =
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r p p 4 5 p = ax + bx + c x = 4a + b + c = 4a + b + c = p Y Y = { ax + bx + c a, b, c 4a + b + c = } Y 8 p + q = (a + dx + (b + ex + (c + f x = 4(a + d + (b + e + c + f = (4a + b + c + (4d + e + f = + = Y Y kp x = k(4a + b + c = Y Y 8 4 Y x = (x (x (ax + b a, b (x (ax + b x = Y (x (ax + b a, b Y = { (x (ax + b a, b R } 4 p = (x (ax + b p = ax + (b ax b q = (x (cx + d = cx + (d cx d p + q = (a + cx + (b a + d cx b d = (x {(a + cx + (b + d} Y Y kp = (kax + {k(b a}x + k( b = (x (kax + kb Y Y 8 (x (ax + b 4 Y p + q ax + bx + c
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 5 3 5 9 3 5/3, 5:4 3 8 a,..., e 4 a :=, b :=, c :=, d :=, e := 3 3 6 6 x e R 3. a, b, c. a, c, d 3. b, c, d 4. a, c, d, e 9* V a,..., e a =, b = x +, c = (x +, d = x V. a, b, c. b, c, d 3. a, c, d 3 * cos x, cos(x, (cos x V V := {α + β cos x + γ cos(x + δ(cos x α, β, γ, δ R} V V a =, b = cos x, c = cos(x, d = (cos x V a, b, c, d. V. d a, b, c 3. V
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 5 7 4: 4 4 A4 B5 6. c, c x = c a + c b c + c + = 4, c + c = 3, c 3c = c =, c = x = a + b. x = c b + c 3c = c + = 4, c + c 3 = 3, 3c + c 3 = 3 c i x b, c 3. x = c a + c 3c + c 4d = c + c 4 = 4, c + c 3 c 4 = 3, c + c 3 + c 4 = c j c = 6, c 3 = 5, c 4 = x x = 6a + ( 5c + ( d = 6a 5c d 7 6. p = αa + βb x x + = α + β(x + x p a, b. p = βb + γc x x + = β(x + + γ(x + = γx + (β + γx + (β + γ γ =, β + γ =, β + γ = p c, b 3. p = αa + βb + γc x x + = α + β(x + + γ(x + = γx + (β + γx + (β + γ + α γ =, β = 3, α = 3 p = 3a 3b + c x a, b, c x = a b, c x a, b, c 7 x b = (x+ b 6 7 3 A4
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 3 5 5/9 3 9 6/3 6/ 6/ 4 ( x x R 3 W W R 3 ( W x + x = ( W x + x = 3 (3 W x ( 3 = (4 W x ( x x ( x x ( x x ( x x (5 W x = x = ( x x * V V f, g (f + g(x = f(x + g(x f + g x f(x + g(x (kf(x = kf(x kf x f(x k V W V (3,(4,(5 ( W f(4 = f V ( W f( = f V (3 W 4 x (4 W f(4 = 4 x (5 W cos x, cos(x, cos(3x, (cos x, (cos 5 4 4: 4 4 A4 B5 8. xa + yb + zc =
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 4 x, y, z x + y =, x + y + z =, x 3y + z = 3 x = y = z = 3 R X, Y, Z ( X Y = xa + yb + zc Z x, y, z x + y = X, x + y + z = Y, x 3y + z = Z x = X Y Z, y = Y Z, z = X + 5Y Z 4 4 R 3 a, b, c a, b, c ( ( R 3 R 3. a, c, d R 3 x = (X + Y Z/, y = Z X, z = (Z Y /. 3. b, c, d R 3 x = (X + Y Z/8, y = (X + 5Y + 3Z/8, z = (X Y + Z/4. 4. xa + yc + zd + ue = x = 6u, y = 5u, z = u u a, c, d, e R 3 R 3 a, c, d, e ( ( 9 ( /3 R 3 3 3 3 4 9 8. α, β, γ = αa + βb + γc α + β(x + + γ(x + x α + β + γ =, β + γ =, γ = α = β = γ = 3 p, q, r px + qx + r = α + β(x + + γ(x + α, β, γ γ = p, β = p + q/, α = q/ + r α, β, γ. = β(x + + γ(x + + δx
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 5 β = γ, δ = γ γ 3. 3 3 3 x x. V x = α + β cos x + γ cos(x + δ (cos x, y = α + β cos x + γ cos(x + δ (cos x x + y = α + β cos x + γ cos(x + δ (cos x + α + β cos x + γ cos(x + δ (cos x = (α + α + (β + β cos x + (γ + γ cos(x + (δ + δ (cos x V k kx = k { α + β cos x + γ cos(x + δ (cos x } = (kα + (kβ cos x + (kγ cos(x + (kδ (cos x V V V 8 x = x x x V x x 8 V 8. α + β cos x + γ cos(x = (cos x x α, β, γ x =, π/, π cos(x = (cos x d = (cos x = cos(x + = a + c 3. V α + β cos x + γ cos(x + δ(cos x α + β cos x + γ cos(x + δ(cos x = (α + δ a + βb + (γ + δ c V a, b, c a, b, c a, b, c α + β cos x + γ cos(x x α, β, γ x =, π/, π α + β + γ =, α γ =, α β + γ = α = β = γ = ( ( V a, b, c V
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 6 5 7 5 9 9 6/ 38 (5/7 5 3 x x x 4 R 4 W W R 4 ( ( (a W x + x + x 4 = (b W x + x + x 4 = x x x 4 = (b 4 V V, V W V ( W f(x = f( x f V ( W (i x 4 (ii (i(ii (3 W (i x 4 (ii (iii f( = (i(ii(iii 5 x V 4 f = 3 + x + x ( V, x, x f ( V x, x, f (3 V, (x +, (x + f f 5 3 4: 4 4 A4 B5
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 7 5/7 t (x, y, z ( x = ( x x y = ( y y y 3 W x + x =, y + y y 3 = ( x + y x + y = x + y + y 3 W W (x + y + (x + y ( + y 3 (x + y + (x + y ( + y 3 = x + x + y + y y 3 = + = x + y W ( kx k kx = kx k W kx + kx k kx + kx + k = k(x + x = W W x + x = = x + x W ( x x = ( x x x + x = x ( + x ( (, ( ( ( ( ( (,,,,,... 3 W W W
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 8 ( ( ( W W x + x = 3, y + y y 3 = 3 (x + y + (x + y ( + y 3 = x + x + y + y y 3 = 3 + 3 = 6, kx + kx + k = k(x + x = 3k t (,, W k = (3 x = ( 3 3 W t (,, t (,, W (4 W W (5 W t (,, + t (,, W. ( f, g W f(4 = g(4 = h = f +g h(4 = f(4+g(4 = + = p = kf p(4 = kf(4 = k = W f(x = (x 4 4 g(x = (x 4 3 W ( h( = ( +( = p( = k ( = k ( f(x = g(x h(x (3 4 4 W 4 OK., x, x,, x 4 5 4 5 4 V (4 ( (3 W ( (3 W OK. (3, x W f(4 = f f(4 = f (x 4 W (x 4 f(x = (x 4g(x f 4 g 3 f W g 3 W f 4 f(4 = O.K. f W 3 g, x, x, (x 4 (x 4, (x 4x, (x 4x, (x 4 4 (x 4, (x 4, (x 4 3, (x 4 4 x 4 = y y (5 W 3 (cos x n cos(x = (cos x, cos(3x = 4(cos 3 cos x
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 9 cos(x, cos(3x, cos x, (cos x, (cos 4 c + c cos x + c (cos x + c 3 (cos x =, π/4, π/, π c j W, cos x, (cos x, (cos W = (cos x cos(x W, cos x, cos(x, cos(3x (cos x, cos x, cos(x, cos(3x n R n A, B A a A B A B A B a A a A a A a A W W W W W cos x V, x, x x V V ( ( (,, ( ( (, x, x (,, (, x, cos x
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 6 3 6/ 38 3 3 (a x + x + x 4 = x 4 = x + x + x, x, W x x x + x + = x + x +, x, x, 3 W 3 3,, (b x + x + x 4 = x x x 4 = x + = x x 4 = x j W x x x x = x + x,, x, x (a W W a (b W w b W b W a why? W a W b W b W a W a 3 W b (b t (,,,,, W a
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 4 ( V, x, x 4,..., x n x n, x,..., x n n V x, x 4, x 6,... a + bx ( (i, x, x,, x 4 f(x = a + a x + a x + a 3 + a 4 x 4 a j (ii f(x f( x a = a 3 = a, a, a 4, x, x 4 3, x, x 4 3 (3 f(x = a + a x + a 4 x 4 (iii a = x, x 4 3 x, x 4 5 [ 3 ( f = 3 + x + x ( f = x + x + 3 [ 3 (3, (x +, (x + f = b + b (x + + b (x + b, b, b f = (x + + = + (x + + (x + [ ( x, x, R 4 4 x, x, x, x, x x, x, ( [, [, [ [ [ [, x, x 4, x, x 4, x 6,... 3 [ x 4 3 x 4, x, x 4, x 6 x sin x
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 6 7 7 5/9 6 6 p.5 f : R R f(3 = f(5 f f(x = x 7 p.5 ([ 3 ([ 3 g : R R g g g(x = = [ 3 g = [ [ 3 x R 5 ([ ([ g g 8 p.5 4 4 ( [ [ ( [ [ ( [ [ ( [ h : R 3 R h = h = h = h ( [ h h h(x = [ 3 x R 3 5 6 4: 4 4 A4 B5
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 3 6 4 7 5/9 7 9 f f (a f f( =, f(3 = 4 ( [ ( [ (b f R R f = f = ( [ ( [ (c f R 3 R f = f = f ( [ ( [ (d f R 3 R f = f = f A R 3 E, F A = [, E = [, [, [, F = [ [ [,, ( [ ( [ = 3 = 4 x R 3 Ax R 3 f : R 3 R 3 Ax A x ( E, E f ( F, F f F (3 F, E f (((3 F, E f F E f : X Y X F Y E V x f p(x p (x p (x p(x V, x, x f 7 5 4: 4 4 A4 B5 6 f k, x R f(kx = kf(x k k x x ( 5 f(5 = f 3 3 = 5 3 f(3 = 5 3. f(x = x = k 3 k x k k = x = 3 = f(x = f(k 3 = kf(3 = k
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 4 7 6 x = αa + βb f(x = αf(a + βf(b x, y [ [ = + [, ( [ f = ( [ f + ( [ f = [ [ [ 3 x = α + β f(x = 5 [ ( [ 3 = f(x = αf + βf 5 [ 3, f 4 [ = ( [ = α [ [ ( [ = ( [ f ( [ f = [ 3 + β 4 [ [ = 3α + β 4α + β 3α + β = 6, 4α + β = α =, β = 7 x = 8 6,7 [ = [ + [ [, [ [ [ [ = + 3 ( [ ( [ ( [ [ ( [ f = f f =, f = f [ 3 f(x = x 5 [ [ [ [ ( [ 3 x = α + β + γ = = f(x = αf + βf 5 ( [ f ( [ + γf [ [ 7 ( [ [ = ( [ = α [ + β [ + γ α + γ = 3, β + γ = 5 β = α + 4, γ = α 3 α x α R x = α [ + (α + 4 [ + ( α 3 [ = α [ α t (,, f 7,8 R n R m f m n A f f(x = A f x A f 7 A f(x = Ax A A [ = A [ 3, A [ = [ = [ 3 [ + [ 4 A [ [ [ ( A = = [ = 3 3 [ [ f(x = [ [ 3 Ax = = x = A 3 5 5 [ [ = 3 [ 3 4 [ 3 x 5 [
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 5 7 8 7/5 7/6 8 [ A A = x R 3 Ax R 3 f : R 3 R 3 Ax A x A f ( f ( f 3 X = R 3 Y = R 3 f f = f = ( [ [ ( [ [ f ( [ [ = 3 3 ( f ( f (3 f 7 4: 4 4 A4 B5 9 (a f(3 = 3 f( = 6 (b t (, t (, (c (b 3 ( ( (d ( + = f (a (d (b (c (b(c
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 6 E, E E = v, v,..., v n [ [f(v E, [f(v E,..., [f(v n E ( ( E v = ( ( ( f(v =, f(v =, f(v 3 = ( ( v =, v =, v = (, v 3 = ( f(v j E (. ( ( ( 3, v 3 = f(v = 6 ( v, f(v 3 = = v 3 F ( (3 F v = = 3v, f(v = ( ( 3 (, v =, v 3 = ( = ( ( ( 3 ( f(v =, f(v =, f(v 3 = 6 ( E E ( 3 3 6 [ v =, v = x, v 3 = x f(v =, f(v = = v, f(v 3 = x = v ( [f(v E X Y f X E = e, e,..., e n Y F = f, f,..., f m E, F f x X E [x E x = x e + x e + + x n e n x, x,..., x n y Y F [y F y = y f + y f + + y mf m y, y,..., y m y a a a n y a = a a n y m a m a m a mn x x x n
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 7 7 6 7/ 7/9 7/9 8/5 5 A4 A4 A4 9 4 X = R 3 Y = R 3 f f = f = ( [ [ ( [ [ 4 ( [ f = [ ( f ( f (3 f ( [ (4 f
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 8 5 3 x V V F p(x V p (x V p p(x x (, x, x, F ( F (3 F 7 9 5: 4 4 A4 B5 { Ax x R 3 }, { x R 3 Ax = } (( x = [ x y z Ax = [ [ [ x x + z y = y z x + y + z x x, y, z x + z =, y = x + y + z = x x + z =, y = y = z = x x { [ } Ker f = x x [ x + z y x + y + z = x [ + y [ + z [ = (x + z [ [ + y [ { Im f = s [ + t [ s, t } [, [ (
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 9 F F = f, f, f 3 ( f(f = 3f, f(f = f, f(f 3 = = f 3 x R 3 x = x f + x f + f 3 ( f(x = f(x f + x f + f 3 = x f(f + x f(f + f(f 3 = 3x f + x f (* x, x Im f = { sf + tf s, t } f, f (* x = x = x = f 3 Ker f = { sf 3 s } f 3 3 a = [, a = [, a 3 = [, b = [, b = [, b 3 = 3 [ 3 ( f f(a = b, f(a = b, f(a 3 = b 3 ( a, a, a 3 R 3 R 3 x R 3 f(x x x = c a + c a + c 3a 3 (3 f(x = f(c a + c a + c 3 a 3 = c f(a + c f(a + c 3 f(a 3 = c b + c b + c 3 b 3 (4 b, b, b 3 c b + c b + c 3b 3 = c = 6c 3, c = 9 c 3 c 3 (5 b = 9b + b 3 (6 x f(x b, b, b 3 (6 b b 3 b, b 3 b, b b, b 3 Im f = {sb + tb 3 s t } (7 b = f(a b = f(a a, a Y Y a, a b, b f(x = x (3 (4 c b + c b + c 3 b 3 = (8 c, c, c 3 c a + c a + c 3a 3 Ker f (8 (?? (5 c 3/ c Ker f = { c( a + 9a + a 3 c } (9
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 3 d = a + 9a + a 3 d [ d = a + 9a + a 3 = 3, 9, t (, 9, a j R 3 a, a, a 3 a + 9a + a 3 t (, 9, t (,, a j a j a j e j [ A = f(e f(e f(e 3 ( e j a, a, a 3 (3 (4 e xa + ya + za 3 = e [ x + z x + y y + z = [ x = z = /, y = /, e = (a a + a 3 ( ( e = (a + a a 3, e 3 = ( a + a + a 3 (3 (4 f(e = (b b + 3b 3 = [ [ [ 3 3, f(e =, f(e 3 = 7 7 5 5 (4 [ 3 3 7 7 5 5 Ax = x Ker f (5
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 3 7 7/9 7/9 8/5 5 E [x E A4 A4 A4 4 3 a = [, a = [, a 3 = [, b = [, b = [ 4, b 3 = [ (6 f f(a = b, f(a = b, f(a 3 = b 3 (7 a, a, a 3 R 3 R 3 a, a, a 3 x R 3 f(x x x = c a + c a + c 3 a 3 (8
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 3 f(x = f(c a + c a + c 3 a 3 = c f(a + c f(a + c 3 f(a 3 = c b + c b + c 3 b 3 (9 f 3 b, b, b 3 c b + c b + c 3 b 3 = ( b = b b 3 b b ( x f(x b, b, b 3 b = b b b 3 b, b b, b 3 f(x = x (8 (9 c b + c b + c 3b 3 = ( c, c, c 3 c a + c a + c 3a 3 Ker f ( ( c = c c 3 = Ker f = { } c( a + a c ( d = a + a d [ d = a + a = ( f (3 e j [ A = f(e f(e f(e 3 (3 e j a, a, a 3 (8 (9 a a = e e = a 3 + e = a 3 + a a e 3 = a e = a a a 3 (9 f(e = b 3 + b b = b 3 + b = [, f(e = b b = b = [ [, f(e 3 = b b b 3 = b 3 = [ Ax = x Ker f b j = Aa j j =,, 3 3 3 [b b b 3 = A [a a a 3 [a a a 3 [ [ [b b b 3 [a a a 3 = A A = 4 = [ (4 (5
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 33 (4 [ 5 ( p V p = p +p x+p x +p 3 p p = p +p x+3p 3 x E =, x, x, p p [p p p E = 3p [p E = 3 p, A = 3 p 3 ( ( p, x, x 3 p p(x = (3 3 3. 5, x, x, x, x,, R 4 ( 4 a + a a, a a + a a + a