M3 x y f(x, y) (= x) (= y) x + y f(x, y) = x + y + *. f(x, y) π y f(x, y) x f(x + x, y) f(x, y) lim x x () f(x,y) x 3 -
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1 M (Lgrnge) x y (Jcobi) (Jcobin) n! n Eulr-Lgrnge
2 M3 x y f(x, y) (= x) (= y) x + y f(x, y) = x + y + *. f(x, y) π y f(x, y) x f(x + x, y) f(x, y) lim x x () f(x,y) x 3 - y x + y x + y + f(x, y) x f(x, y) x x y 4 f(x, y) x f(x, y) y f(x + x, y) f(x, y) = lim x x = lim y f(x, y + y) f(x, y) y x y f(x) = x f (x) = x x + y f(x, y) = x + y x + y = C x y = C () (3) * 3
3 M3 3 f(x,y) x C y 3 4 f(x, C) x y = C f(x, C) f(x, C). f(x, y) x f(x, y) x f x (x, y), x f(x, y) f xx (x, y) = x f(x, y), f xy (x, y) = x y f(x, y).3 y = f(x) x x x + x f(x) y x y = f(x + x) f(x) = f (x) x (4) z = f(x, y) x, y df = f(x + x, y + y) f(x, y) f(x, y + y) f(x, y + y) =
4 M3 4 dz = = { } { } f(x + x, y + y) f(x, y + y) + f(x, y + y) f(x, y) f(x, y + y) x + x f(x, y) y (5) y (5) f(x, y + y) x x f(x, y) x (6) x (6) df = f(x + x, y + y) f(x, y) = f(x, y) x + x f(x, y) y (7) y f(x, y + y) f(x, y) y x x f(x, y + y) x = f(x, y) x + o( y) o( y) y * { } f(x, y + y) f(x, y) x = + o( y) x x x f(x, y) = x + o( y x) x x, y (6) df(x, y) = f(x, y) ɛ x + x f(x, y) ɛ y y f(x, y) f(x, y) x f(x, y) y () x + by + c (b) (x + y) (c) (d) sin(xy) x + xy (e) e (x+y) *
5 M3 5 t x = x(t), y = y(t) f ( x(t), y(t) ) t d dt f( x(t), y(t) ) f(x, y) dx f(x, y) dy = + x dt y dt d dt f`x(t), f`x(t + t), y(t + t) f`x(t), y(t) y(t) = lim t t x(t + t) = x(t) + x (t) t (7)
6 M3 6 f(x, y) (x, y ) *3 ɛ ɛ x [ ɛ, ɛ] x = x + ɛ x x x x f(x) x f(x) > f(x ), f(x) < f(x ), f(x) x f (x ) = f (x ) < f(x) x = x f (x ) > f(x) x = x f (x ) = x = x f(x) f(x) = x 3 f (x) = 3x x = x 3 x =. ɛ x ɛ y (x, y ) (x, y) = (x + ɛ x, y + ɛ y ) ( ) f(x, y) > f(x, y ) f(x, y) < f(x, y ) (8) f(x, y ) f(x, y) (x, y ) f(x, y) (x, y ) f x (x, y ) =, f y (x, y ) =. *3
7 M3 7 x y 3 z = x y ( ) 3 z = x y (, ). f(x + ɛ) = f(x) + df(x) dx ɛ + d f(x)! dx ɛ + d 3 f(x) 3! dx 3 ɛ 3 + f(x + ɛ) = { + ɛ d dx + d ɛ! dx + } d3 ɛ3 3! dx 3 + f(x) e x e x = + x +! x + 3! x3 + f(x + ɛ) = e ɛ d dx f(x) (9) f(x, y) ɛ x ɛ y (9) f(x + ɛ x, y) = e ɛ x x f(x, y) e ɛ y y e ɛ y y f(x + ɛx, y) = e ɛ y y e ɛ x x f(x, y) e ɛ y y f(x + ɛx, y) = f(x + ɛ x, y + ɛ y ) e ɛ y y e ɛ x x = e ɛ x x +ɛ y y f(x + ɛ x, y + ɛ y ) = e ɛ x x +ɛ y y f(x, y)
8 M3 8 { ( ) f(x + ɛ x, y + ɛ y ) = + ɛ x x + ɛ y + ( ) } ɛ x y! x + ɛ y + f(x, y) y ɛ f(x + ɛ x, y + ɛ y ) = f(x, y) + f x (x, y)ɛ x + f y (x, y)ɛ y + { } f xx (x, y)ɛ x + f xy (x, y)ɛ x ɛ y + f yy (x, y)ɛ y ().3 f(x, y) (x, y ) f x = f x (x, y ) f xx = f xx (x, y ) f x = f y = () f(x + ɛ x, y + ɛ y ) = f(x, y ) + { } f xx ɛ x + f xy ɛ x ɛ y + f yy ɛ y ɛ x ɛ y ( ) (x, y ) ( ) (x, y ) x + bxy + cy, b, c g(x, y) x = sy g(x, sy) = (sy) + bsy + cy = y (s + bs + c) () (s + bs + c) D = b c D > 4() D < 4(b) 4 (s + bs + c)
9 M3 9 D s + bs + c s D > g(x, y) D < > g(x, y) < g(x, y) () H = f xy f xx f yy. H < f(x, y) (x, y ) f xx < f(x, y) (x, y ) b f xx > f(x, y) (x, y ). H > (x, y ) H = 3 H H = f xx f yx f xy f yy () f(x, y) = x y f x = x =, f y = y = (, ) H = 4 > (, ) f(x, y) = x 3 + y 3 3xy f x = 3x 3y =, f y = 3y 3x = (, ) (, ) H = 9 36xy f xx = 6x (, ) H = 7 <, f xx = 6 > (, ) H = 9 > (x, y ). f(x, y) = x y + x + x. f(x, y) = x xy
10 M3 3 f(x, y) g(x, y) = c f(x, y) 6 8 z = f(x, y) y = h(x) f(x, y) z = f(x, y) 6 y = h(x) 8 3. y = /x z = x + y y = /x 5? z = x + x z = x = x = ± y = /x x3 y = ± (, ) (, ) x y x + y = S = xy x + y = xy y = x ds dx S = x( x) = x = x = / x + y = y / d S = < x = y dx g(x, y) = c y( x) x 3 + x 4 y 3 + 4y =, sin(x) cos(y) = /
11 M3 y x y x 3. (Lgrnge) y = h(x) f(x, y) : g(x, y) = c y = h(x) y h(x) = y h(x) = g(x, y) c = g(x, y) = c (x, y ) dy dx = g x(x, y ) g y (x, y ) : g(x, y) = c (x, y ) x = x *4 n t t = ( dy dx ) = g x(x, y ) g y (x, y ) = g y (x, y ) ( ) gy (x t =, y ) g x (x, y ) n t n = n = ( gx (x, y ) g y (x, y ) ) g(x, y) = c f(x, y) (x, y ) (x, y ) g(x, y) = c 6 g(x, y) = c f(x, y) = h h (x, y ) n f *4 6
12 M3 g(x, y) = c n g λ n f = λn g (3) ( ) ( fx (x n f =, y ) gx (x, n f y (x, y ) f =, y ) g y (x, y ) (3) f x (x, y ) λg x (x, y ) = (4) f y (x, y ) λg y (x, y ) = (5) λ g(x, y) = x + y = f(x, y) = xy x y λ f(x, y) λg(x, y) (4) (5) y λ =, x λ = x + y = x = y = / y = x + /x l = x + y y x /x = ( ) x + y λ y x x ) x y x λ ( ) x =, y λ = y x /x = x = ±, y = ± (4) (5) g(x, y, z) = f(x, y, z) f x (x, y, z) λg x (x, y, z) = f y (x, y, z) λg y (x, y, z) = f z (x, y, z) λg z (x, y, z) =
13 M3 3 3 h(x, y, z) = f x (x, y, z) λ g x (x, y, z) λ h x (x, y, z) = f y (x, y, z) λ g y (x, y, z) λ h x (x, y, z) = f z (x, y, z) λ g z (x, y, z) λ h x (x, y, z) =. x + y = x + y. A x B y f(x, y) = x + y A B l : l S = p l(l x)(l y)(l z) S
14 M h(t) k m (t) = kh(t)/m *5 t t = t v(t) dv(t) dt = (t) t v(t + t) = v(t) + (t) t (6) t = t v(t ) (6) t = t v(t + t) = v(t ) + (t ) t t = t + t t t + t v(t + t) t v(t + t) + (t + t) t = v(t ) + (t ) t + (t + t) t { } = v(t ) + t (t ) + (t + t) { } v(t + t) = v(t ) + t (t ) + (t + t) (7) t T (7) t t + t { } v(t + 3 t) = v(t ) + t (t ) + (t + t) + (t + t) *5 ipod
15 M3 5 (t) t (p -) p t (p+) t t t +T (p -) t (p+) t 7 n { } v(t + n t) = v(t ) + t (t ) + (t + t) + (t + t) + + (t + (n ) t) n = v(t ) + t (t + p t) (8) p= n T = n t t n n n p= p= T/ t v(t + T ) = v(t ) + lim t (t + p t) t v(t + T ) t (t + p t) p t 7 t (t) t (t) [t, t + T ] (t) lim (9) T/ t lim t p= p= t (t + p t) = v(t + T ) v(t ) = t +T t t +T t (t)dt (9) (t)dt *6 t + T t t b f( )d Z *6 S
16 M3 6 b f(t)dt = b f(z)dz = b f(q)dq t t = t + T (t) dv(t) dt = (t) () v(t) v(t ) = t t (t)dt () f(x) g(x) df(x) = g(x) () dx () 4. f(x) df (x) dx = f(x) F (x) f(x) F (x) f(x) df (x) dx = f(x) F (x) = f(x)dx + C log x /x cos x sin x f(x) F (x) C F (x) + C
17 M3 7. x [,] b e x [,] c sin x [, π] d cos x [, π] y (i) y (ii) x x /4 /4 3/4 /4 /4 3/4 8. y = x x [, ] (i) 4 S 4 (ii) 4 S 4 S 4 S 4 b [, ] n S n,s n c n S n,s n S n = S n (b) NX k= k = N(N + )(N + ) 6 ( S = lim n n + n + + n ) n + n () S S = lim n n k= + k n n (b) S = log k n = x, n = dx
18 M3 8 5 f(x) F (x) f(x) Z b f(x)dx = F () F (b) f(x) 5. I = x n e x dx (n =,, 3, ) n α x n e αx dx = I α = x n e αx dx x n e αx = ( ) n dn dα n (e αx ) ( ) n dn dα n e αx dx = ( ) n dn dα n e αx dx /α ( ) I α = ( ) n dn dα n = n!α n α I = lim α I α = n! 5. f(x) x = g(t) x t g(t) f(x) = x + 3 f(x)dx x + 3 = t x = g(t) = t 3
19 M3 9 f(x) f(g(t)) f(t) t log t x t x dx x = g(t) x t dx dt = dx dx = dt dx dx dt t 3 5 x + 3 dx = 5 3 t dt b f(x)dx = β α f(g(t))g (t)dt () f(x) F (x) F (x) = f(x) f(x) b f(x)dx = [F (x)] b = F (b) F () F (x) x = g(t) t d dt F (g(t)) = F (g(t))g (t) = f(g(t))g (t) (3) g(α) =, g(β) = b α β (3) β α b () d F (g(t))dt = F (g(β)) F (g(α)) = F (b) F () dt f(x)dx (3) β α f(g(t))g (t)dt x I = + x dx + x = t x = t x > x = t dx = dt t ( ) t I = t t dt = t dt = log
20 M3 I = + x dx + tn x = cos x x = tn θ x = θ = x = θ = π/4 dx = (tn θ) dθ = dθ/ cos θ 5.3 I = π/4 + tn θ dθ π/4 cos θ = cos θ dθ cos θ = π 4 f(x)g(x) x [, b] I = π π d { } f(x)g(x) = f (x)g(x) + f(x)g (x) (4) dx [ ] b b f(x)g(x) = f (x)g(x)dx + b I = f (x)g(x)dx = [ ] b f(x)g(x) b b f(x)g (x)dx f(x)g (x)dx (5) x sin xdx ( cos x) = sin x π x sin xdx = = π + (x) = log x dx = [ ] π x(cos x) dx = x cos x [ sin x ] π = π π + cos xdx log x dx log x (log x) (x) log x dx = x log x = x log x x x(log x) dx
21 M3. I = e π/ x log xdx ( cos x + π ) dx xe x dx, x rctn xdx ( ), I n = π x n sin xdx I n = π n n(n )I n I =, I = π I = π x 3 sin xdx I 4 = π x 4 sin xdx π x sin xdx I 3 =, x + y = y y = ± x f(x) = x I = S = π x dx
22 M x m t x(t) v(t) t = x, v dx(t) dt m dv(t) dt = v (6) = F (7) v x m = F (7) m = F m F (6) (7) t = t x(t) x }{{} m{v(t) v } = }{{} = t t v(t )dt }{{} F (t )dt }{{} (8) (9) x() = x, v() = v t = t v(t)dt dt t = t F dt dt t = t F (t) = (9) v(t) = v (8) x(t) x = v t (8) x(t) x = F = mg (9) v(t) = v gt t (v gt)dt = [ v t ] t gt = v t gt
23 M3 3 m F = sin t x() = v() = (9) mv(t) = t sin tdt = cos t + v(t) = ( cos t)/m (8) x(t) = (t sin t) m x 6. (7) v(t) mv(t) dv(t) dt = F v(t) t = t m t v(t ) dv(t ) dt dt = t F v(t )dt d { dt v(t ) } t m d { dt v(t ) } dt = [ m v(t ) ] t = mv(t) mv v(t ) = dx(t ) dt t F dx(t ) dt dt = x(t) x F dx mv(t) x(t) mv = F dx (3) x mv(t) F dx (3) F = mg (3) F = mg v = mv(t) = mg{x(t) x }
24 M3 4 x(t) x h mv(t) = mgh mgh mgh (3) k m F = kx mv(t) + kx(t) = mv + kx (3). m F = cos t (8) (9) x(t) bf = sin t. F = k x k 3 x 3 m M m F = G Mm x G x G = 6.7 [m 3 /s kg] M = 6. 4 [kg] R = [m]. t = v = x = R (3). lim x(t) = t v(t) = lim t
25 M x y x y f(x) I = f(x)dx (3) f(x) = 3x =, = I = x f(x) x f(x, y) 7. f(x, y) x y *7 [, ] f(x, y) I(y) = f(x, y)dx (33) y I(y) [b.b ] y I I = b b I(y)dy (33) x y I = b { b } f(x, y)dx dy (34) I = b b dy f(x, y)dx (35) *8 *7 *8 I = Z b Z b f(x, y)dxdy x y [, ]
26 M3 6 I = dy xy dx x [ ] I = dy x y = 3 y dy y I = / y x 7.. f(x)dx x = x = x f(x) I = b b dy f(x, y)dx 9 f(x, y) dxdy xy f(x, y)dxdy 9() x I(y) = f(x, y)dx (b) I(y)dy y (34) I (c) () f(x, y) (b) f(x, y) y y y y x x x (c) f(x, y) b b y x 9
27 M3 7 ds = dxdy f(x, y)ds S S f(x, y) xy x y b b dx f(x, y)dy = dy f(x, y)dx (36) b b f(x, y) = c z = c z xy c c f(x, y)dxdy = c [ cx ] c dy = c 3 c 7. x y x y y x (35) r r x f(x, y) y x r x r x f(x, y)dy AB x r r r r r x dx f(x, y)dy r x (37) D f(x, y)dxdy, D = {(x, y) r < x < r, r x < y < } r x D (38)
28 M3 8 () y (b) y y = x y = x /4 x y = x x f(x, y) = xy () y = φ (x) = y = φ (x) = x D = {(x, y) < x <, < y < x} D xydxdy = dx x xydy = [ ] y=x xy dx = y= x3 dx /8 f(x, y) = xy y = φ (x) = x y = φ (x) = x /4 (b) D = {(x, y) < x <, x < y < x /4 } D f(x, y)dxdy = x /4 dx xydy (39) x 7/3 () e (x+y) dxdy, D = {(x, y) < x <, < y < } () D D (x y)dxdy, D = {(x, y) < x <, x < y < x }
29 M3 9 xyz (L,, ), (, L, ), (,, h) z = h h (x + y) L x,y,z (L,, ), (, L, ) = z (,, h) x (L,, ) (, L, ) y
30 M3 3 8 x y u v [() ] x = g(t) g(b) g() f(x)dx = b f ( g(t) ) g (t)dt (4) x t x t g (t)dt D f(x, y)dxdy (x, y) (u, v) x = x(u, v), y = y(u, v) f(x, y)dxdy = f ( x(u, v), y(u, v) ) Xdudv D D X (4) g (t)dt X? 8. (Jcobi) (Jcobin) D f(x, y)dxdy x = x(u, v), y = y(u, v) (4) xy uv ds D f ( x(u, v), y(u, v) ) ds xy uv uv ds dudv
31 M3 3 ds (4) *9 dx = x u du + x v dv, dy = y u du + y v dv ( ) dx = dy ( xu y u ) du + (4) = ( xu y u ) du, b = ( xv ( xv y v ) dv (4) y v ) dv u du (x, y) b v dv (x, y) ds b b ( ds = det xu y u x v y v ) dudv = x u y v x v y u dudv (8.) ( ) det xu x v (43) y u y v (Jcobi) J, (x, y) (u, v) f(x, y)dxdy = f ( x(u, v), y(u, v) ) J dudv (44) D D dxdy = J dudv (x + y )dxdy D y *9 (7) x
32 M D x + y = u x y = v D = {(u, v) < u <, < v < } x = u + v, y = u v / / / / = (x + y )dxdy = D = 4 = 8 3 D { (u ) ( ) } + v u v + dudv (45) du (u + v )dv (46) (47) 8. x = r cos θ, y = r sin θ (48) x + y J = r dxdy = rdrdθ rdrdθ 4 4 dxdy drdθ
33 M3 33 I = D xy x + y dxdy D = {(x, y) x + y <, x >, y > } r sin θr cos θ I = D r rdrdθ = sin θ cos θrdrdθ D D D = {(r, θ) r <, < θ < π } D sin θ cos θrdrdθ = π/ sin θ cos θ = sin θ sin θ cos θdθ rdr I = 4. x = r cos θ, y = r sin θ b ( x y ) ( cos θ sin θ = sin θ cos θ ) ( u v ). dx (x + y + ) dy = π 4 xyz x + y + z = R. R x y dxdy D = {(x, y) x + y < R } D. V = 4 3 πr3
34 M Z e x dx = π/ 9. n! e x x n dx = n! (49) Section.5 Γ(n + ) = Γ(n + ) = ( e x ) x n dx [ = e x x n] + n = nγ(n) e x x n dx (5) e x x n dx Γ(n + ) = nγ(n) (5) Γ(n + ) = nγ(n) = n(n )Γ(n ) = n(n )(n )Γ(n ) = = n!γ() Γ() = Γ(n + ) = n! (5) n! = 3 n n Γ(n) n > ( )!, ( 5 )!,
35 M3 35 ( )! (5) ( ) ( ) ( ) 3! = Γ + = Γ = ( ) Γ (5) Γ(/) ( ) Γ = e x x / dx x = y ( ) Γ = ( ) π! = =.886 Γ( 5 ) (5) Γ ( ) ( 5 5 = e y dy = π ( ) (53) = 3 ) { Γ ( 5 Γ )} ( ) = 3 4 π 9. B(m, n) = x m ( x) n dx (n >, m > ) (54) B(m, n) = B(n, m) (55) x = cos θ x = B(m, n) = y + y y = π/ (cos θ) m (sin θ) n dθ (56) x x, dx = ( + y) dy
36 M3 36 B(m, n) = y m dy (57) ( + y) n+m B(m, n) = Γ(m)Γ(n) Γ(m + n) m n (58) (58) 9.3 I = x x dx ( I = B 3, ) = Γ(3)Γ ( ) Γ ( ) 7 =! π 5 3 π = 6 5 I = π/ cos 3 θ sin θdθ (56) ( I = B, 3 ) = Γ()Γ ( ) 3 Γ ( ) 7 = π 5 3 π = I 3 = x ( + x) 4 dx (57) I 3 = B(3, ) = Γ(3)Γ () Γ (4) =! 3! = 3 4 I 4 = + x dx x = tn θ π/ x = y I 4 = dy + y y
37 M3 37 y / + y dy B(, ) I 4 = ( B, ) = Γ ( ( ) Γ ) Γ () = π 9.4 (58) x = u Γ(n) = e u u n du (59) (58) Γ(n)Γ(m) (59) ( Γ(n)Γ(m) = 4 = ) ( e u u n du e (u +v ) u n v m dudv u = r cos θ, v = r sin θ, dudv = rdrdθ ) e v v m dv 4 e r r (m+n) dr π/ (59) (56) Γ(m + n)b(n, m) (cos θ) n (sin θ) m dθ (6) Γ(n)Γ(m) = Γ(m + n)b(n, m)
38 M3 38. I = e x dx I = I xy r e (x +y ) = e r dr ( ) () (πrdr) (e r ) < r < e x dx e y dy = dy e (x +y ) dx (6) I = π e r rdr (6) I = π/ I > I = I = sin(x) sin x dx x x x = e x dx = π e xy dy ( ) I = sin(x) e xy dy dx I = = ( + y dy ) e xy sin xdx dy
39 M3 39 y = tn θ sin(x) x dx = π. n 4 r n r V n V = πr, V 3 = 4 3 πr3 n V n = C n r n C n n n x, x, x 3,, x n (x, x, x 3,, x n ) n = r = x + x + + x n (63) I = ( I n = e x dx ( = = ) n ) ( e x dx = π n/ e x dx = π ) ( ) e x dx e x n dxn e (x +x + +x n ) dx dx dx n (64) 3 I n n (6) I I = π e r rdr πr I n I n = e r S n dr
40 M3 4 S n n S n = dv n dr I n = nc n e r r n dr = nc nr n I n = π n/ C n C n = n π n/ e r r n dr r = x e x x n/ dx Γ( n ) V n = n = 4 Γ() = V 4 = π r4 (65) πn/ n ( n )r n (65) Γ (hint: + x = e x π dx = x (hint: x = t...) dx ( + x) x = π I F = e (+x)y dy ) sin(x )dx ( p π/ ( ) )
41 M y y = y(x) y = x = T y(x) T y(x) (functionl) T [y(x)] T [y(x)] (x, y) v (g ) v = gy v = gy dl dl = dx + dy dl dt dt = dl v = dx + dy gy dy dx y dt + y dt = gy dx v x y 5
42 M3 4 T [y(x)] = + y dx (66) gy (66) y(x) J[y(x)] = b F (y, y, x)dx (67) y(x) J[y(x)] y(x) x f(x) f(x) x y(x) J[y(x)] y(x). Eulr-Lgrnge (67) y(x) 6 y(x) y(x) + δy(x) y δj y(x) δy(x) y δy δj δj = b F (y + δy, y + δy, x) F (y, y, x)dx (68) b x δy = d(δy) dx 6 y(x) y(x) + δy(x) δy() = δy(b) = (69)
43 M3 43 δj = y(x) [section (7) ] F (y + δy, y + δy, x) F (y, y, x) = F F δy + y y δy (68) δj = b F F δy + y y δy dx [ ] b F b y δy F + y δy d ( ) F dx y δydx (69) δj = b { F y d ( )} F dx y δydx = (7) δy x 6 [, b] (7) δy F y d ( ) F dx y = (7) Eulr-Lgrnge F F (y, y ) x Eulr-Lgrnge (c ) F y F y = c (7) y = y(x) l l = b + y dx (7) ( ) d dx + y xy ( + y ) / = c y(x) =
44 M3 44 (7) F (y, y) = F y = y gy + y + y gy = c gy + y c = A y( + y ) = A g (73) x = (θ sin θ) (74) y = ( cos θ) (75) (θ sin θ) = ( cos θ) =.57 θ < θ <.77π 7. (74)(75) (73) = A/4g dy dx = dy dθ dx dθ
45 M x l b (66) l = 4km x y 8 J[y(x)] = b F (y, y)dx Eulr-Lgrnge (7) F x df dx = F dy y dx + F dy y dx = F y y + F y y Eulr-Lgrnge F y
46 M3 46. y = y(x) 9 x y x x x S = π x S y = y(x) x (7) F = y + y x y + y dx (76) F y F y = y + y yy + y = y = + y y y = dx dy = (y ) y -x x x 9
47 M3 47 y * ( x = cosh y ) + b ( ) x b y(x) = cosh (77), b y( x ) = y(x ) = y(x) b = x = / (77) ( ) = cosh =.35,.8483 = q q = dg dt t q q q, q, t L(q, q, t) * Z dx x = cosh (x) + x = cosh(u) = (e u + e u )/
48 M3 48 t = t t = t S = t t L(q, q, t)dt (78) L(q, q, t) Lgrngin S Lgrngin (78) L = ( ) ( ) S Eulr-Lgrnge (7) F L, x t, y q, y q L q d ( ) L dt q = (79) m q q mq mgq Lgrgin Eulr-Lgrnge L = mq mgq L q d dt ( L q ) = m q = mg θ θ = dθ l, dt m m(lθ ) mgl( cos θ) Lgrngin L = m(lθ ) mgl( cos θ) θ Eulr-Lgrnge q θ L θ d ( ) L dt θ = (8) θ = g sin θ (8) l
49 M3 49 Eulr-Lgrnge (8) (8) A k k m Lgrngin B m θ l Lgrngin θ, θ, l, l 4 Eulr-Lgrnge L θ d dt L l d dt ( ) L θ = (8) ( ) L l = (83)
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A. A. A-- [ ] f(x) x = f 00 (x) f 0 () =0 f 00 () > 0= f(x) x = f 00 () < 0= f(x) x = A--2 [ ] f(x) D f 00 (x) > 0= y = f(x) f 00 (x) < 0= y = f(x) P (, f()) f 00 () =0 A--3 [ ] y = f(x) [, b] x = f (y)
2011de.dvi
211 ( 4 2 1. 3 1.1............................... 3 1.2 1- -......................... 13 1.3 2-1 -................... 19 1.4 3- -......................... 29 2. 37 2.1................................ 37
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[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
.1 z = e x +xy y z y 1 1 x 0 1 z x y α β γ z = αx + βy + γ (.1) ax + by + cz = d (.1') a, b, c, d x-y-z (a, b, c). x-y-z 3 (0,
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0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9
1-1. 1, 2, 3, 4, 5, 6, 7,, 100,, 1000, n, m m m n n 0 n, m m n 1-2. 0 m n m n 0 2 = 1.41421356 π = 3.141516 1-3. 1 0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c),
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h 0 g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)
5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h 0 f(a + h, b) f(a, b) h............................................................... ( ) f(x, y) (a, b) x A (a, b) x
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. () x 3 + dx () (x )(x ) dx + sin x sin x( + cos x) dx () x 3 3 x + + 3 x + 3 x x + x 3 + dx 3 x + dx 6 x x x + dx + 3 log x + 6 log x x + + 3 rctn ( ) dx x + 3 4 ( x 3 ) + C x () t x t tn x dx x. t x
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微分積分 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. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 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. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
( 12 ( ( ( ( Levi-Civita grad div rot ( ( = 4 : 6 3 1 1.1 f(x n f (n (x, d n f(x (1.1 dxn f (2 (x f (x 1.1 f(x = e x f (n (x = e x d dx (fg = f g + fg (1.2 d dx d 2 dx (fg = f g + 2f g + fg 2... d n n
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google
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II 1 3 2 5 3 7 4 8 5 11 6 13 7 16 8 18 2 1 1. x 2 + xy x y (1 lim (x,y (1,1 x 1 x 3 + y 3 (2 lim (x,y (, x 2 + y 2 x 2 (3 lim (x,y (, x 2 + y 2 xy (4 lim (x,y (, x 2 + y 2 x y (5 lim (x,y (, x + y x 3y
= M + M + M + M M + =.,. f = < ρ, > ρ ρ. ρ f. = ρ = = ± = log 4 = = = ± f = k k ρ. k
7 b f n f} d = b f n f d,. 5,. [ ] ɛ >, n ɛ + + n < ɛ. m. n m log + < n m. n lim sin kπ sin kπ } k π sin = n n n. k= 4 f, y = r + s, y = rs f rs = f + r + sf y + rsf yy + f y. f = f =, f = sin. 5 f f =.
II (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x (
II (1 4 ) 1. p.13 1 (x, y) (a, b) ε(x, y; a, b) f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a x a A = f x (a, b) y x 3 3y 3 (x, y) (, ) f (x, y) = x + y (x, y) = (, )
x i [, b], (i 0, 1, 2,, n),, [, b], [, b] [x 0, x 1 ] [x 1, x 2 ] [x n 1, x n ] ( 2 ). x 0 x 1 x 2 x 3 x n 1 x n b 2: [, b].,, (1) x 0, x 1, x 2,, x n
![x i [, b], (i 0, 1, 2,, n),, [, b], [, b] [x 0, x 1 ] [x 1, x 2 ] [x n 1, x n ] ( 2 ). x 0 x 1 x 2 x 3 x n 1 x n b 2: [, b].,, (1) x 0, x 1, x 2,, x n](/thumbs/92/108343288.jpg "x i [, b], (i 0, 1, 2,, n),, [, b], [, b] [x 0, x 1 ] [x 1, x 2 ] [x n 1, x n ] ( 2 ). x 0 x 1 x 2 x 3 x n 1 x n b 2: [, b].,, (1) x 0, x 1, x 2,, x n")
1, R f : R R,.,, b R < b, f(x) [, b] f(x)dx,, [, b] f(x) x ( ) ( 1 ). y y f(x) f(x)dx b x 1: f(x)dx, [, b] f(x) x ( ).,,,,,., f(x)dx,,,, f(x)dx. 1.1 Riemnn,, [, b] f(x) x., x 0 < x 1 < x 2 < < x n 1
y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =
[ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v =
1/1 lim f(x, y) (x,y) (a,b) ( ) ( ) lim limf(x, y) lim lim f(x, y) x a y b y b x a ( ) ( ) xy x lim lim lim lim x y x y x + y y x x + y x x lim x x 1
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(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y
[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
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II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
- II
- II- - -.................................................................................................... 3.3.............................................. 4 6...........................................
f(x) = x (1) f (1) (2) f (2) f(x) x = a y y = f(x) f (a) y = f(x) A(a, f(a)) f(a + h) f(x) = A f(a) A x (3, 3) O a a + h x 1 f(x) x = a
3 3.1 3.1.1 A f(a + h) f(a) f(x) lim f(x) x = a h 0 h f(x) x = a f 0 (a) f 0 (a) = lim h!0 f(a + h) f(a) h = lim x!a f(x) f(a) x a a + h = x h = x a h 0 x a 3.1 f(x) = x x = 3 f 0 (3) f (3) = lim h 0 (
d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r
![d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r](/thumbs/85/92098187.jpg "d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r")
2.4 ( ) U(r) ( ) ( ) U F(r) = x, U y, U = U(r) (2.4.1) z 2 1 K = mv 2 /2 dk = d ( ) 1 2 mv2 = mv dv = v (ma) (2.4.2) ( ) U(r(t)) r(t) r(t) + dr(t) du du = U(r(t) + dr(t)) U(r(t)) = U x = U(r(t)) dr(t)
() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.
![() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.](/thumbs/89/98384840.jpg "() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.")
() 6 f(x) [, b] 6. Riemnn [, b] f(x) S f(x) [, b] (Riemnn) = x 0 < x < x < < x n = b. I = [, b] = {x,, x n } mx(x i x i ) =. i [x i, x i ] ξ i n (f) = f(ξ i )(x i x i ) i=. (ξ i ) (f) 0( ), ξ i, S, ε >
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mugensho.dvi
1 1 f (t) lim t a f (t) = 0 f (t) t a 1.1 (1) lim(t 1) 2 = 0 t 1 (t 1) 2 t 1 (2) lim(t 1) 3 = 0 t 1 (t 1) 3 t 1 2 f (t), g(t) t a lim t a f (t) g(t) g(t) f (t) = o(g(t)) (t a) = 0 f (t) (t 1) 3 1.2 lim
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(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {
![(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {](/thumbs/94/118399854.jpg "(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {")
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1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
n=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x
n= n 2 = π2 6 3 2 28 + 4 + 9 + = π2 6 2 f(z) f(z) 2 f(z) = u(z) + iv(z) * f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x f x = i f y * u, v 3 3. 3 f(t) = u(t) + v(t) [, b] f(t)dt = u(t)dt
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5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h f(a + h, b) f(a, b) h........................................................... ( ) f(x, y) (a, b) x A (a, b) x (a, b)
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ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d
A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9
, x R, f (x),, df dx : R R,, f : R R, f(x) ( ).,, f (a) d f dx (a), f (a) d3 f dx 3 (a),, f (n) (a) dn f dx n (a), f d f dx, f d3 f dx 3,, f (n) dn f
,,,,.,,,. R f : R R R a R, f(a + ) f(a) lim 0 (), df dx (a) f (a), f(x) x a, f (a), f(x) x a ( ). y f(a + ) y f(x) f(a+) f(a) f(a + ) f(a) f(a) x a 0 a a + x 0 a a + x y y f(x) 0 : 0, f(a+) f(a)., f(x)
Gmech08.dvi
145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
1 nakayama/print/ Def (Definition ) Thm (Theorem ) Prop (Proposition ) Lem (Lemma ) Cor (Corollary ) 1. (1) A, B (2) ABC
1 http://www.gem.aoyama.ac.jp/ nakayama/print/ Def (Definition ) Thm (Theorem ) Prop (Proposition ) Lem (Lemma ) Cor (Corollary ) 1. (1) A, B (2) ABC r 1 A B B C C A (1),(2),, (8) A, B, C A,B,C 2 1 ABC
() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y
![() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y](/thumbs/85/92979039.jpg "() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y")
5. [. ] z = f(, y) () z = 3 4 y + y + 3y () z = y (3) z = sin( y) (4) z = cos y (5) z = 4y (6) z = tan y (7) z = log( + y ) (8) z = tan y + + y ( ) () z = 3 8y + y z y = 4 + + 6y () z = y z y = (3) z =
1 2 1 No p. 111 p , 4, 2, f (x, y) = x2 y x 4 + y. 2 (1) y = mx (x, y) (0, 0) f (x, y). m. (2) y = ax 2 (x, y) (0, 0) f (x,
No... p. p. 3, 4,, 5.... f (, y) y 4 + y. () y m (, y) (, ) f (, y). m. () y a (, y) (, ) f (, y). a. (3) lim f (, y). (,y) (,)... (, y) (, ). () f (, y) a + by, a, b. + y () f (, y) 4 + y + y 3 + y..3.
Gmech08.dvi
51 5 5.1 5.1.1 P r P z θ P P P z e r e, z ) r, θ, ) 5.1 z r e θ,, z r, θ, = r sin θ cos = r sin θ sin 5.1) e θ e z = r cos θ r, θ, 5.1: 0 r
1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1
sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω ω α 3 3 2 2V 3 33+.6T m T 5 34m Hz. 34 3.4m 2 36km 5Hz. 36km m 34 m 5 34 + m 5 33 5 =.66m 34m 34 x =.66 55Hz, 35 5 =.7 485.7Hz 2 V 5Hz.5V.5V V
,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.
9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,