() (, 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

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1 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 = y () z = y z y = (3) z = cos( y) z y = cos( y) (4) z = y sin y z y = sin y (5) z = 4y 4y z y = 4y (6) z = y + y z y = y( + y ) z = ( + + y + y ) + + y z = + y (7) z = + y z y = ( + ) ( + + y + y ) + + y [. ] z = (,, ) y ( ) z = 3 y [.3 ] z = 4 y (,, ) y + y (8) ( ) + y + z = 4. [.4 ] z z m U(z) U(z) = mgz z z F (z) ( ) F (z) = (0, 0, mg) [.5 ] M Φ(, y, z) = G M r r = + y + z () (, y, z) G(, y, z) () G(, y, z) (3) ( ) () G(, y, z) = G M r 3 (, y, z) () G(, y, z) = GM r (3) (, y, z) r [.6 ] q (0, 0) ϕ(, y) = kq log + y k r = + y

2 () (, 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 ) 3 () f y = f y = + y ( + y ) [.8 ] z = f(, y) z y = z y () z = y ( y) () z = e sin y (3) z = sin y (4) z = e y (5) z = log + y (6) z = tan y ( ) () z = y (3 y) z y = z y = 6y( y) z yy = ( 6y) () z = e sin y z y = z y = e cos y z yy = e sin y (3) z = 4y sin y z y = z y = cos y 4y sin y z yy = 4 sin y (4) z = y e y z y = z y = (y )e y z yy = e y (5) z = + y ( + y ) z y = z y = y ( + y ) z yy = y ( + y ) (6) z y 3 = ( + y ) z y = z y = y ( + y ) z 3 y yy = ( + y ) [.9 ] f(, y) = + y f + f y ( ) + y [.0 ] f(, y, z) = + y + z f + f y + f z ( ) + y + z.4 [. ] z = f(, y) () z = y () z = e sin y (3) z = log + y ( ) () dz = d ydy y () dz = d + ydy e (sin yd + cos ydy) (3) dz = + y [. ] yd + dy f(, y) = y ( )

3 .5 [.3 ] z z u z v u v y y () z = y + y, = u cos v, y = u sin v () z = ey, = u v, y = uv (3) z = tan y, = u + v, y = uv sin v cos v ( ) () z u = sin v + cos v z v = u(cos3 v sin 3 v) (cos v + sin v) () z u = (uv v )e (u v)uv z v = (u uv)e (u v)uv (3) z u = v(u + v ) u 4 u v + v 4 z v = u(u + v ) u 4 u v + v 4 [.4 ] z z z y y u v u v () z = uv, u = + y, v = 3 + y () z = sin u v, u = y, v = + y (3) z = e sin u+cos v, u = y, v = + y ( ) () z = 6 + 5y z y = 5 + 4y () z = { z y = ( + y ) cos y y sin y z y = z{ cos y sin( + y)} { ( + y )y cos y + 3 sin y } / ( + y ) } /( + y ) (3) z = z{y cos y sin( + y)} [.5 ] z dz t y y dt () z = y, = e t, y = e t () z = e sin y, = t, y = t ( ) () e t + e t () e t (t sin(t ) + cos(t )) [.6 ] dy d () 3 3ay + y 3 = 0 () a + y + by = (3) log( + y ) = tan y ( ) () ay a y + y () a + by (4) e y = y sin + y (3) y (4) e y y cos e y + sin [.7 ] z = f(, y), = e u cos v, y = e u sin v () u =, v = y () y u = y, y v = (3) () () z uu + z vv = ( + y )(z + z yy ) z, z y, z, z y, z y, z yy f(, y) ( ).6 [.8 ] () () 3 3

4 ( ) () e ρ = (cos ϕ, sin ϕ, 0), e ϕ = ( sin ϕ, cos ϕ, 0), e z = (0, 0, ), () e r = (cos ϕ sin θ, sin ϕ sin θ, cos θ), e θ = (cos ϕ cos θ, sin ϕ cos θ, sin θ), e ϕ = ( sin ϕ, cos ϕ, 0) [.9 ] f(, y) ( ) ( ) f + ( ) f = y ( ) f + r r [.0 ] (3 ) () f = sin θ cos ϕ f r () f y (3) f z = sin θ sin ϕ f r ( ) f θ cos θ cos ϕ f + r θ sin ϕ f r sin θ ϕ cos θ sin ϕ f + r θ + cos ϕ f r sin θ ϕ f = cos θ r sin θ f r θ (4) ()-(3) f = f + f y + f z = f r + f r r + f r θ + cot θ f r θ + f r sin θ ϕ ( ) [. ] f = f + f y = f r + r () + y () + y + y ( ) () + y () + y r 3 (3) 0 (3) + y + y f r + f r θ [. ] 3 f = f + f y + f z = f r + f r r + f r θ + cot θ f r θ + f r sin θ ϕ () + y + z () + y + z (3) + y + z + y + z + y + z ( ) () + y + z) ( + y + z) () ( r r 3 (3) r 4.7 [.3 ] A B [A, B] = AB BA [, ] p = i h p y = i h y p z = i h ( h ) z () [, p ] = [y, p y ] = [z, p z ] = i h () [, y] = [y, z] = [z, ] = [p, p y ] = [p y, p z ] = [p z, p ] = 0 4

5 (3) [, p y ] = [, p z ] = [y, p z ] = [y, p ] = [z, p ] = [z, p y ] = 0 ( ) [.4 ] L = yp z zp y L y = zp p z L z = p y yp [A+B, C] = [A, C] + [B, C] [AB, C] = A[B, C] + [A, C]B.3 () [L, L y ] [L y, L z ] [L z, L ] () L = (L ) + (L y ) + (L z ) [L, L ] = [L, L y ] = [L, L z ] = 0 ( ) () i hl z i hl i hl y () [.5 ].4 L L y L z 3 ( () L = i h sin ϕ ) cot θ cos ϕ θ ϕ ( () L y = i h cos ϕ ) cot θ sin ϕ θ ϕ (3) L z = i h ϕ (.0 ) ( ) ( [.6 ].5 L + L y + L z = h θ + cot θ θ + ) sin θ ϕ ( ).8 [.7 ] (0, 0) y 3 () f(, y) = + y () f(, y) = (3) f(, y) = cos y + + y (4) f(, y) = sin cos y (5) f(, y) = e sin y (6) f(, y) = log( + y) ( ) () ++y +(+y)+ (+y)+ () (+y)+(+y) (+y) 3 + (3) + (4) 6 (3 + 3y ) + (5) y + y + 6 (3 y y 3 ) + (6) y ( y) + 3 ( y)3 [.8 ] () f(, y) = sin( y) ( π, 0) y 4 () f(, y) = log( + y) (, ) y 3 5

6 ( ) () ( π ) + ( π )y y + 4 ( π )4 6 ( π )3 y + 4 ( π ) y 6 ( π )y3 + 4 y4 + () log + {( ) + (y )} 8 {( ) ( )(y ) + (y ) } + 4 {( )3 3( ) (y ) 3( )(y ) + (y ) 3 } + [.9 ] m M Mm G U(, y, z) = G + y + z R 0 N (0, 0, R 0 ) () U(, y, z) N y z () N N U(0, 0, R 0 ) U(, y, z) h h = z R 0 () U E (h) U E (h) h () + (3) g h m h mgh () U E (h) g G R 0 M ( ) () G Mm R 0 + G Mm R0 (z R 0 ) + () (3) G M R0.9 [.30 ] z = f(, y) () z = y () z = 3 + y y (3) z = + y + y 4 y (4) z = y 3y (5) z = e ( +y ) (6) z = sin sin y ( π π, π y π ) ( ) () () (0, 0) 0 (3) (, 0) -4 (4) (5) (0, 0) (6) (± π, ± π ) (± π, π ) - [.3 ] z = f(, y) ( ) () z = + y ( + y = ) () z = + 4y + 4y ( + y = ) (3) z = + y (y = ) (4) z = y (y = ) ( ( ) () 5, ( ) 5 ( 5,, ) () ( 5 5 (, (3) (, ) (, ) (4) ), ( ), ( ) 5, ( ), ) 5 ) [.3 ] 6

7 ( (, y) + y = y, y > 0 ) ( ) [.33 ] O a 3 ABC AOB= θ BOC= ϕ (0 < θ ϕ) B () 3 3 θ ϕ a () O a A ( ) () a(sin θ + sin ϕ + sin(θ + ϕ)) () C [.34 ].33 a 3 () 3 θ ϕ a () 3 3 ( ) () a (sin θ + sin ϕ sin (θ + ϕ)) () [.35 ] + y = 4 (, y) (, 4) ( ) ( 4 5, 8 ) [.36 ] n /n n n n A θ n θ B () A B l a θ θ l(θ, θ ) () A B L a, n, n A θ θ L(θ, θ ) n a (3) () l(θ, θ ) b L(θ, θ ) sin θ = n sin θ n n B b a ( ) () a(tan θ + tan θ ), ()n a + n a cos θ, (3) cos θ 7

8 . [. ] y y = () y = ae a () y(0) = 0 ( ) () () y = e [. ] y 3y + y = 0 () y = ae + be a b () y(0) = 0, y (0) = ( ) () () y = e + e [.3 ] f(, t) = v f(, t) v t () f(, t) f(, t) = g(x) g(x) X k ωt X k ω v = ω/k f(, t) = k d g(x) dx () () t f(, t) = ω d g(x) dx (3) g(k ωt) () () g(k ωt) (4) h(k + ωt) (5) f(, t) = g(k ωt) t t = 0 f(, 0) = g(k) φ() φ() = g(k) t = t f(, t) f(, t) = φ( ω t) k (6) ω/k > 0 φ( ω t) φ() k (7) g(k) = φ() g(k ω t) = φ( ω t) t = 0 t = t k g(k) t g(k ω t) (8) f(, t) = h(k + ωt) (5) (7) ( ) () (4) (5) f(, t) = g(k ω t) = g(k( ω k t)) = φ( ω k t) (6) φ( ω k t) φ() ω t (7) k ω/k = v (8) ( ) v g(k ωt) h(k + ωt) v = ω/k g(k ωt) + h(k + ωt) 8

9 ( [.4 ] + y + z v ) f(, y, z, t) = v f(, y, z, t) t () 4 f(, y, z, t) f(, y, z, t) = g(x) r = (, y, z) k = (k, k y, k z ) g(x) X k r ωt X k k y k z ω ω > 0 v = ω/ k + ky + kz f(, y, z, t) = d g(x) k dx y z () () t f(, y, z, t) = ω d g(x) dx (3) g(k r ωt) () () g(k r ωt) (4) h(k r + ωt) ( ) ( ) v 3 f(, y, z, t) e k r ωt ( ) [.5 ].4 + y + z f(, y, z, t) = v f(, y, z, t) t () r = + y + z sin(kr ωt) r [.0](4) k ω v = ω/k () t = 0 () r (3) r () ( ) () ( ). [.6 ] y = + () () y(0) = 0, y (0) = 0 ( ) () y = tan log( + ) + C + C () y = tan log( + ) [.7 ] m v(t) f(t) () 9

10 () t t v v () t t mv mv = t t f(t)dt (3) f(t) f(t) = f t = t t mv mv = f t ( ) a(t) = (a (t), a y (t), a z (t)) a(t)dt ( a (t)dt, a y (t)dt, a z (t)dt) ( ) [.8 ] t v A (t) m A t v B (t) m B ( ) () m A m B f m A m B () m A t t v A (= v A (t )) v A (= v A (t )) m B t t v B (= v B (t )) v B (= v B (t )) () t t t m A v A + m B v B = m A v A + m B v B ( ) [.9 ] U((t), y(t), z(t)) m v(t) = (v (t), v y (t), v z (t)) () ( ) d () dt mv = mv v v v v v v = v v = v + vy + vz (3) du dt = v U (4) () v mv + U(, y, z) = C C ( ) () m v = U, () (4).3 [.0 ] y = () y = y () ( + )y = (3) y = y y (4) (y )y = y (5) y cos 3 + sin cos y = 0 (6) y + = + y ( ) () y = Ce / () y = log(+ )+C (3) y = C+ (4) y = Ce y (5) tan y+sec = C (6) + y = + + C 0

11 [. ] N(t) α () N(t) () () (3) t = 0 N 0 t (4) T (3) N 0 T t ( ) () dn dt = αn () N = ( Ce αt (3) N = N 0 e αt (4) N = N ) t/t 0.4 [. ] () y + y = log () y y = (3) y y = 3 (4) y 3y = e (5) y + y = sin (6) y sin y cos = tan ( ) () y = log 4 + C () y = + Ce / (3) y = C (4) y = e /4 + Ce 3 (5) y = ( cos + sin + C)/ (6) y = sin (tan + C).5 ( ) [.3 ] t = 0 V 0 sin ωt t I(t) () t = 0 I(0) () I(t) (3) () () I(t) (4) (3) ( ) () I(0) = 0 () V 0 sin ωt RI(t) LI(t) = 0 V 0 (3) I(t) = R + ω L (R sin ωt ωl cos ωt) + ωlv 0 R + ω L e R L t (4).6 [.4 ] a a 0 () y + 3y 4y = 0 () y y + y = 0 (3) y 0y + 5y = 0 (4) y + a y = 0 (5) y a y = 0 (6) y ay = 0 ( ) () y = C e 4 + C e () y = e (C cos + C sin ) (3) y = (C + C )e 5 (4) y = C cos a + C sin a (5) y = C e a + C e a (6) y = C + C e a

12 [.5 ] m k A t = 0 0 () (t) () t = 0 (3) () ( ) () mẍ(t) = k(t) () (0) = A, ẋ(0) = 0 (3) (t) = k A cos m t 0 A m [.6 ].5 aẋ(t) () (t) () t = 0 (3) () a mk (4) (3) a = mk τ (5) (3) ( ) () mẍ(t) = aẋ(t) k(t) () (0) = A, ẋ(0) = 0 (3) (a) a > ( mk (t) = A a a 4mk (b) a = mk a ( k m = m (t) = A + (c) 0 < a < mk (t) = Ae a m t {cos ) e a+ a 4mk m t +A ) k m t e k m t ( + a a 4mk } 4mk a a 4mk a t + m sin t 4mk a m ) e a (4) (a), (b), (c) (b) τ τ = m/k (5) a 4mk m t.7 [.7 ] () y + y y = 0 cos () y 6y + 5y = (3) y + y + y = e (4) y + y = sin (5) y + y = e (cos sin ) ( ) () y = C e + C e 3 cos + sin () y = e ( 3/ C cos + C ) sin + 5 (5 4) (3) y = (C + C )e + e /9 (4) y = C cos + C sin cos (5) y = C + C e + 0 e (3 cos sin ) [.8 ] m q k t = 0 E 0 sin ωt 0

13 () (t) t = 0 () ω k/m () (3) ω = k/m () E0sin t m, q 0 (4) (3) () ( ) () mẍ(t) = k(t) + qe 0 sin ωt (0) = 0, ẋ(0) = 0 ( () (t) = qe ) 0 m k k mω ω k sin t + sin ωt (3) (t) = qe 0 m ω m sin ωt qe 0 t cos ωt ωm (4) [.9 ].8 aẋ(t) () (t) () () 0 < a < mk (3) () (t) = A sin ωt + B cos ωt A B (4) ω (> 0) (3) ω ω > 0 ( ) () mẍ(t) = aẋ(t) k(t) + qe 0 sin ωt () qe 0 { (3) (t) = (k mω (k mω ) + a ω ) sin ωt aω cos ωt } qe 0 = (k mω ) + a ω sin(ωt ϕ) (ϕ = aω tan k mω ) mk a (4) ω = m, mk > a 3

14 3 3. [ 3. ] () yddy : 0, 0 y () sin( + y)ddy : 0 π/, 0 y π/ (3) ye ( +y ) ddy : 0, 0 y (4) ddy : 0, 0 y + + y + y ( ) () /4 () (3) /4 (4) (log ) [ 3. ] f(, y)ddy y y () : 0, 0 y, + y () :, 0 y, y (3) : 0, 0 y, y (4) : 0, 0 y, ( ) + (y ) (5) : y, y ( ) () () (4) (5) 0 / f(, y)dyd = y f(, y)ddy f(, y)dyd = f(, y)ddy (3) f(, y)dyd = 0 y 0 0 y y f(, y)dyd = f(, y)ddy 0 0 y f(, y)dyd = 0 0 y f(, y)ddy 0 y f(, y)ddy [ 3.3 ] () yddy : 0, y () ddy : 0 π, 0 y sin (3) ddy : 0, 0 y, + y (4) ddy : 0 y + y (5) + y ddy : 0 y ( ) () /3 () π (3) /3 (4) 4( )/3 (5) log + π/ 4

15 3. [ 3.4 ] () + y ddy : a + y b, a > 0, b > 0 () log( + + y )ddy : + y (3) (4) (5) ddydz : a + y b + z, a > 0, b > 0, c > 0 c X = a, Y = y b, Z = z c (X, Y, Z) zddydz : + y + z, z 0 y + + y + z ddydz : + y + z, 0, 0 y, 0 z ( ) () π log b 4πabc a () π( log ) (3) 3 (4) π/4 (5) 9 (3π/4 ) [ 3.5 ] : + y R R σ z 0 σ () 4πϵ 0 ( + y + z0 σ y () 4πϵ 0 ( + y + z0 σ z 0 (3) 4πϵ 0 ( + y + z0 ( ) () 0 () 0 (3) σz0 )3/ ddy )3/ ddy )3/ ddy ( ϵ 0 z 0 R +z0 ) σ ϵ 0 [ 3.6 ] + y + z R z 0 z 0 > R ρ () ddydz 4πϵ 0 ( + y + (z z 0 ) ) 3/ ρ y () ddydz 4πϵ 0 ( + y + (z z 0 ) ) 3/ ρ z 0 z (3) ddydz 4πϵ 0 ( + y + (z z 0 ) ) 3/ (3) (i) cos θ = t θ (ii) z 0 z ( + y + (z z 0 ) ) 3/ = u ( + y + (z u) ) / u=z0 ( ) () 0 () 0 (3) ρ 4πR 3 4πϵ 0 3 z0 5

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,

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.