[, + f : f = [, +, f 4 = =. 3 f 5 =,. f 3, f 4, f 5 R, {, }, {, } 3 R.3. I = π, π tn f I R f R f = f { R } =,, +, +.4. f 3, f 4,

Transcription

1 473.. f f f f f f f f 3 R 4 X X R X X 5 Y X Y X 6 7 Y X Y X * : function; :the domin; : the rnge; : the imge. 3 set; 4 : rel numbers; R R R 5 : n element; : subset; X Y Y 6 Y X Y X X X 7 A B A B A B A if nd onl if B A is equivlent to B R, b = { R < < b}, [, b] = { R b},, = { R < },, + = { R > },, + = R,, b] = { R < b}, [, b = { R < b},, ] = { R }, [, + = { R }, [, ] = {}., b < b, b [, b] 8.. f f R R f : R R f f :, f = f f f =, f = 4, f =, fs = s, f =. f 9 f [, + 8 : n intervl; n open intervl; : closed intervl., b ], b[ ±, + ] 9 : nonnegtive rel number; the set of nonnegtive rel numbers [, + the set of positive rel numbers, + the set of nonpositive negtive rel numbers, ],,.

2 [, + f : f = [, +, f 4 = =. 3 f 5 =,. f 3, f 4, f 5 R, {, }, {, } 3 R.3. I = π, π tn f I R f R f = f { R } =,, +, +.4. f 3, f 4, f 5 R sin f 3 = +, =. the union A B = { A B}. n n R n R = R R n = {,..., n,..., n R}. R = {,, R} = {,, } R 3 = {,, z,, z } R R R 3 R n R n 4 R n,..., n f,..., n f n f 5 n. f R n f : R : rtionl number; : n irrtionl number. 3 f 3 R 4 : the number line; : the coordinte plne, the Crtesin plne; : the coordinte spce; : point. 5 R n 3 7 4

3 , R f, = + f R 6 f : R R f : R, f, = + R Π Π 4 Π 4 Π f, =, f, =, f, =.6. f, f, f, =.7. f p, f p, 7 R I f : I R {, f I } R f R..3,.4 f f 5 f : R R R 3 {,, f,, } f f R 3 f : R c {, f, = c} 6 R,, f, 7 f : the grph of function f. f = tn b f = c f 3.4 d f 4.4 e f ,.4 f 5 f c 8 f c z = c f c 7 n f : R n R c {,,..., n, f,..., n,..., n } R n+, {,,..., n f,..., n = c } R n f c.8. R f, = + [, + c, + {, f, = c} c f c 8 : the contour, the level curve, the level set.

4 f..8 b f c f z. R f, = R c {, f, = c} c =..6,.7 f R f R : sclr field = 4 = tn 5 = tn π, π - I R f ,, f, = +, =, : f,, f,, f,, f, 3. f, 4, f3, 6, f4, 8. f, m m, f,, F f, = F + f f f -8-4 f -9 3, 4...

5 473.. I R f I. lim h f + h f h f. f f I f f : I f R f.. f =. f = 3 R f fh f h = 3 + h h f. b 3.4 sin f = + = f R sin f = cos + = f =.. f = 3 sin f fh f = lim = lim h sin h h h h + = f = f f = d d f f f f f f f = f n f n = dn d n f = f * : differentible; : the differentil coefficient; : the derivtive; f : f-prime dsh. : the squeeze theorem. 3 : the second derivtive; 3 : the third derivtive; n : the n-th derivtive.

6 R f :, f, R, b f f + h, b f, b, b = lim, h h f f, b + k f, b, b = lim k k f, b f f, b, b f, b f f f :,, R f 5 f f.. d f = f, f = f. f f, f,, f f, f, f, 4 domin 3 5 : prtill differentible; : the prtil derivtive with respect to f, f,, f, f = f = f, f = f = f, f = f = f, f = f = f f 6.3. f, = f, = 3 + 6, f, = 3 +. f = 6 + 6, f = 6, f = 6, f =..3 f f f f f f 3 f f, 3 3 f = f, 3 f = f, 3 f = f, f 3, 3 f, 3 f, 3 f : the second prtil derivtives. 7 : the commuttivit of prtil differentils.

7 n f,..., n i i =,..., n f i f = f k, l n. k l l k ].3 I R f I 8 lim f = f 9 f I f I.4..4 {, f = =. lim f = lim + f f = lim f = f = f = { cos, =. { n }, { n } n = nπ, n = n+π lim f n =, lim f n = n n lim f n =,, : continuous; : continuous function. 9 f f.4.5. f lim f f = lim f f = lim f + h f h f + h f f + h f = lim h = lim lim h h h h h h = f =. f I I I f.6.6 C r - I f f I f I C - f I C -f I f I k f I C k -f k f k I f k C k -f C -.6. m,..., m f = m m + m m + + k = k f = C -. 3 f R C -.4 f C - : of clss C ; C r - : of clss C r ; C - : of clss C C-infinit. : polnomil; : constnt function.

8 f,, z - -4,, f, = +, =, - t, ut, u t u = ut, = t e 4t -3 t, ut, u t u = ut, = sint + + b sint -4 f, f + f =, b f + f + f zz = f 3 F t f,, z = F + + z 3 f F -6 f, -7 f + f = + f + f + f + f f f, = log + + +,,, f, = +, =, g, = log cos cos. 7-8 n -9 n m f f, = log +, 3 : the het eqution; : the wve eqution; : hrmonic function; : miniml surfce. 8

9 R R {, R + < r }, {, R < < b, c < < d} r,, b, c, d r >, < b, c < d f R, b 3 3. lim f, = A,,b f, A,, b,, b f, A 4,, b f, A 5 + h, b + k, b h, k, 3. lim f, = lim f + h, b + k.,,b h,k, 3.. α, β, f lim αh, k =, lim h,k, βh, k =, lim h,k, f, = A,,b lim f + αh, k, b + βh, k = A. h,k, {h n }, {k n } 6 lim f + h n, b + k n = A n * : domin; : n open disc; : n open rectngle rectngulr domin. 3,, b f, b 4 5 the limit. 6 : rbitrr; X P : P holds for n rbitrr X. 3. {f + h n, b + k n } A {h n }, {k n } 3.3. R,, f, = +, =, -4, - h n = /n, k n = /n, k n = /n 3 {h n }, {k n }, {k n} lim fh n, k n =, lim fh n, k n = n n 3.,, f, f,,, f, lim f, = lim = lim lim + f, =, lim f, = lim = lim lim f, =. + f, = / +,, lim lim f, =, lim lim f, =. 3 R, f, = / +,, r, θ = r cos θ, = r sin θ,, r = + = f, = fr cos θ, r sin θ = r cos θ sin θcos θ sin θ

10 = r sin θ cos θ = 4 r sin 4θ sin 4θ r sin 4θ 4 r 4 r r 4 r r sin 4θ R f,, b + h, b + k h, k f + h, b + k f, b = Ah + Bk + εh, k h + k lim εh, k = h,k, A, B R f, b lim f, = f, b,,b f f f, f, = f, = ,, f, = +, =,, f,, b f, b A, B A = f, b, B = f, b k = f + h, b f, b h εh, εh, h h = Ah + εh, h h = A + εh, h h εh, h εh, f + h, b f, b A = lim = f, b. h h h = B = f, b 3.8. f, b, b h, k, f, f I I + h I h f + h f = f + θhh < θ < 7 : the men vlue theorem.

11 F = F 3. f h f F h = F h F = F + θhh = F θhh = f + θh, b + kh < θ < f f h θ Gk = f, b + k f, b k Gk = G δkk = f, b + δkk < δ < δ Θh h h Θh εh, k = F h + Gk f, bh f, bk h + k h > h < = f + θh, b + k f, b h h + k + f, b + δk f, b k h + k θh < h, δk < k h/ h + k, k/ h + k h, k, θ 89 h + h + θh < θ < 3. f + h f = hf c + h c sin,, f, = +, =,, f, f 3.. f f, f f, b A = f, b, B = f, b εh, k h, k, h, k εh, k = f + h, b + k f, b f, bh f, bk h + k k F h := f + h, b + k f, b + k f F h F h = f +h, b+k, 8 f θ h, h θ 9 3., + h f := 3.3. R f f, f, f = f, b f, b f, b f + h, b + k f, b + k f + h, b + f, b V = V h, k := hk h, k V = F h F F t := f + t, b + k f + t, b k h F t = f + t, b + k f + t, b 3. V = k F θ h = k f + θ h, b + k f + θ h, b = k F k F F t := f + θ h, b + t

12 θ, F t = f + θ h, b + t θ, θ V = f + θ h, b + θ k θ, θ,. V = Gk G/hk Gt := f + h, b + t f, b + t V = f + φ h, b + φ k φ, φ, φ, φ f, f, h, k, f, b = f, b C k - R f f f C - f C - f, f f C -f f, f, f, f k f C k -f k f C - k C k R f C C k m C m - C k - 4 C - f = f 3.3. R I = [, b], γ =, : I t γt = t, t R R γ γb γ,. R P, Q P Q γ : [, b] R γt t b. R P =, b ε U ε P := {, R + b < ε } R R P ε P ε- the ε-disc centered t P. R P U ε P ε. F : R R {, R F, > } R.. R 3-3.3, 3.5, C - f, = +, 3-3 I R f f I f I 3-4 I R f f f I f I I, < f < f mpping γ R : n open set; : connected set; : disc disk.

13 cosine, sine, tngent 4. cot := cos, sec :=, csc := sin cos sin. cotngent, secnt, cosecnt 4. + tn = sec d d tn = + tn, d d sec = sec tn, d tn d = log cos, d d cot = + cot, csc = csc cot, d cot d = log sin. 4- sec d = + sin log sin = log + tn tn, 4.4 csc d = cos tn log + cos = log. * rtionl function; rdicl root; the squre root; the cubic root; n-the n-th root the eponentil function; the logrithmic function; the trigonometric functions, the circulr functions. cosec sec = cos cos = cos, π = cos = sin, π π = sin = tn, π < < π = tn cos, sin, tn 4 5 cos = π 6 + 3, sin 4 = 5 π, tn = π 8. rccos, rcsin, rctn 4.3. cos + sin = π sin π cos = cos cos = cos π π π cos π sin = π cos tn + tn 3 = π 4 4 tn 5 tn 39 = π 4. α = tn, β = tn 3 tn α = tn β = 3 tnα + β = tn < β < α = tn < tn = π 4. < α + β < π α + β = tn = π rc cosine; rc sine; : rc tngent; inverse trigonometric functions Mchin

14 d d cos =, d d tn = +. d d sin =, = cos = cos d d = d/d = sin. π sin sin = cos = 4.3 tn cos, sin = ± 4.5 d = tn, d = sin tn =, sin = 4.7 tn = dt + t, sin = dt t α 7 C elementr functions. 8 n α { > } 4.5. cosh = e + e, sinh = e e, tnh = sinh cosh = e e e + e cosh sinh = cosh + = cosh cosh + sinh sinh, sinh + = sinh cosh + cosh sinh, tnh + tnh tnh + = + tnh tnh. 3 d cosh = sinh, d d sinh = cosh, d d d tnh = tnh. 4 cosh d = sinh, sinh d = cosh, tnh d = log cosh. 9 hperbolic cosine; hperbolic sine; hperbolic tngent; hperbolic functions. cosh t ht cos ht cosh cosh t, t = cosh t, sinh t =

15 t, t t + t 4 + N t N = t N+ = + t + t + N+ t N+ + t t = 4.7 tn = R N = R N = dt + t = N t N+ + t dt t N+ + t dt N + N+ + R N R N = t N+ dt = N+3 N + 3 N+ t N+ + t dt t N+ + dt = N+3 N R N 4.8 R N 3 N π = N + 3 R N, N + 3 R 4 N N + 3 R N N 3 N 3! α = /5, β = /39 4. π = 4 M k= 4 k α k+ k + N j= j β j+ + R M,N, j + R M,N 6αM+3 M βN+3 N + 3 R M,N M =, N = tn = N N + N+ + R N N k k+ = + R N, k + k= N+3 N R N N+3 N + 3 lim R N = N 4.9 tn = = 4.9 =, 4. k= k k+ k + π 4 = , cosh, < tnh < cosh sinh, tnh 3 = cosh, = sinh, = tnh t = tnh u cosh u, sinh u t 7 A, B A cos t + B sin t r cost + α, r sint + β 8 = cosh = cosh cosh = log +

16 sinh, tnh sinh = log + +, tnh = log M =, N = 4-5 log = log log cos, sin, tn 4-6 n I n = π/ cos n d n I n = n I n n m m 3 π/ I n = cos n m m... π n = m, d = m m m + m... n = m / + = tn θ 3 = sinh u 4-3 4, 3, m sin n = sin θ u = f = + /f 4-9, b / + b b =, / + b > 3 b < / + u t = tn t 3 = cos cos cos sin u = sin = cosh u 4- / + d = log + + = sinh. +

17 R f,, b f, b 5. lim εh, k =, h,k, εh, k := f + h, b + k f, b f, bh f, bk h + k f, P =, b f f df P =, b,, b df P f P, f,, f, 5. df = f f, f 5.. φ, =, ψ, = dφ =,, dψ =, d =,, d =, df = f f d + d * row vector; column vector; totl differentil; differentil f P =, b 5.4 f + h, b + k f, b = df P h + εh h h h =, h = h + k k lim h εh = df P h 3 = t, h h, k,, 5.5 f f f, b +, b, f = f +, b + f, b I t, t t, t I R γ : I t γt = t, t R. mtri; sclr. 3,, b h, k t h, k = t, f, f 5.4 f + h f = dfh + εh h,

18 t, t 5 γ γt = t, t γt = dγ dt t = ẋt, ẏt = d d t, dt dt t t, t 6 t γt t γt γt t 5.5. v = t v, v P =, b γt = + tv, b + tv t = P v P v 5. s σs = cos s, sin s π < s < π, 7 sin s, cos s 5. 3, t σt := + t, t + t < t <. t = tn s 5. f, γt = t, t 5.6 F t = f t, t b+v b+v b O v P v v +v + O s coss,sins t O s t +t, t +t 5.6. f, γt = t, t 5.6 df f d f d t = t, t t + t, t dt dt dt t t δ ε ε t + δ t t + δ t ε δ := ẋt, ε δ := ẏt δ δ t, t δ ε j δ j =, hδ := δ ẋt + ε δ, kδ := δ ẏt + ε δ δ h, k f F t + δ F t = f t + δ, t + δ f t, t = f t + hδ, t + kδ f t, t = f f t, t hδ + t, t kδ + ε hδ, kδ hδ + kδ 4 curve; prmetric representtion of the curve. 5 γ γt = t sin t, cos t the ccloid C - t = nπ nπ, n =, ±, ±,... 6 the velocit vector; the speed. 7 line; circle.

19 εh, k h, k, F t + δ F t δ = f ẋt t, t + ε δ + f ẏt t, t + ε δ + ε hδ, kδ δ ẋt + ε δ + ẏt + ε δ. δ δ ε j δ, j =, hδ, kδ, εhδ, kδ δ /δ = F F t + δ F t t = lim = f ẋt f ẏt t, t + t, t δ δ 5.6 df dt = f d dt + f d = df γ dt df γ v = t v, v 5.5 P =, b γt = + v t, b + v t 5.7. R f P =, b v = t v, v F t := f + v t, b + v t t = F f P v 8 f P v v f, b = df P e, 5.6 f, b = df P e e = t,, e = t, 5.8. R f P =, b f P v = t v, v 5.7 df P v = f, bv + f, bv R f S P =, b S P S := {,, f,, } R 3, P :=, b, f, b. t = P, γt := t, t =, = b S 5.8 ˆγt = t, t, f t, t t = P 5.6 ˆγ t = 5.9 dˆγ dt = ẋ, ẏ, f, bẋ + f, bẏ = ẋ,, f, b + ẏ,, f, b. P S f S P ˆγ 8 the directionl derivtive. 9 the tngent plne.

20 f, b, f, b, 5.. f S P =, b, f, b P 5. f, b + f, b b z f, b = P =, b f grd f P := f, b f, b f P 5.7 df P v = grd f P v grd f P P f f, = e cos + sin f., f.6 γt t , f, = P =, b f P df P, f P v df P v v grd f P v 5-6 P =, b f P df P, P f γt = t, t γ = P t = γ γ grd f P 5-7 f { = f, =. v = t v, v, f v f 5. the grdient vector df P

21 R f, γt = t, t F t = f t, t df f d f d t = t, t t + t, t dt dt dt t f, = ξ, η, = ξ, η fξ, η = f ξ, η, ξ, η f ξ f η f f ξ, η = ξ, η, ξ, η ξ, η + ξ, η, ξ, η ξ, η ξ ξ f f ξ, η = ξ, η, ξ, η ξ, η + ξ, η, ξ, η ξ, η. η η fξ, η f, f fξ, η 6. f ξ = f ξ + f ξ, f η = f η + f η z = f, = f ξ, η, ξ, η = fξ, η * the chin rule. ξ: i; η: et. ξ, η, ζ zet,, z z ξ = z ξ + z ξ, z η = z z z z z 6. = ξ ξ η ξ R m R n η + z η η η R m F : R n n,..., m R n F,..., m F,..., m R n n,..., n j,..., m j,..., m F : R m R n R m n 6. F : R m,..., m F,..., m,..., F n,..., m R n. F j : R j =,..., n m F 3 = F =,..., m, =,..., n F F j j =,..., n F = F,..., F n F = F,..., F n : R m R n C k - 4 j j =,..., n F j : R C k - 3 ; 3 mp: components. 4 C k -

22 R m C - F = F,..., F n : R n F F... m df =..... F n F n... m n m F 5,..., m R m R m U R n F : R n G: U R k =,..., m F U, G F : R m G F R k G F : R m R k F G F, G C - dg F = dg df, dg F = dg F df R m id : id = 7 R m U R n F : U G: U G F = id, F G = id U 5 the differentilthe Jcobin mtri Jcobi, Crl Gustv Jcob 84 85,. 6 the composition. 7 the identit mp id id G F 8 G = F 6.6. { = r, θ R r >, π < θ < π }, U = {, R > } F : r, θ F r, θ = r cos θ, r sin θ U, G: U, +, tn G = F, F = G, r, θ π < θ < π tn tn θ = θ 4. r > G F r, θ = Gr cos θ, r sin θ = r cos θ + r sin r sin θ θ, tn r cos θ = r, tn tn θ = r, θ = id r, θ. θ = tn / π < θ < π cos θ > > cos tn = cos θ = + tn θ = + tn tn = + = +, sin tn = sin θ = cos θ tn θ = + = +. = + F G, = F +, tn = + cos tn, + sin tn =, = id U,. 6.7., 6.6 r, θ = G,, r, θ, 8 the inverse mp; F : the inverse of F /F -inverse; R m, R n m = n m = n 6.8

23 , { } = r, θ R r >, π < θ < π, Ũ = {, R > } h: Ũ R tn h, := tn + π tn π >, >, < F : r, θ F r, θ = r cos θ, r sin θ Ũ, G: Ũ, +, h, F = G, G = F, r, θ = G,, 6.8. F : R m U R n G = F F F C - m = n df = df df F = df m E F F = id 6.5 df df = E, F F = id U 6.5 df df = E df df m = n 9 the polr coordinte sstem; the orthognonl cooridnte sstem; the Crtesin coordinte sstem; : escrtes, René Rentus Crtesius; h,,, C Fortrn tn, = r, θ = r cos θ, = r, θ = r sin θ. F : r, θ, 6.4 r θ cos θ r sin θ 6.4 df = = sin θ r cos θ r G = F :, r, θ r r 6.5 dg = = + + θ θ θ C - f, z = f = f + f + f, 6.3 r, θ f f r, θ f = r f r + θ f θ = f = = = + f r + f r + f θ + f θ f r f r f r f θ + + f rr + + f θr + + f rθ f θ + f θθ f θ + 6.4, the Lplce opertor; the Lplcin; Lplce, Pierre- Simon , F

24 = + f rr f = + f rr f rθ f rθ + r = + + f θθ + + f θθ f r f r 6.7 f = f + f = f rr + r f r + r f θθ + f θ. + f θ dg = df = df = = cos θ r r sin θ θ, f = cos θf rr r cos θ sin θf rθ cos θ sin θ r sin θ r cos θ = r r θ = sin θ r + r cos θ θ. + r sin θf θθ + r sin θf r + r sin θ cos θf θ f = sin θf rr + r cos θ sin θf rθ r cos θf θθ + r cos θf r r sin θ cos θf θ θ f, f = f + f = f -4 F t f, = F + f, = tn 6- c ξ = + ct, η = ct t, ξ, η C - ft, f t c f = 4c f ξ η f tt c f = C - f C - F, G ft, = F + ct + G ct f tt = c f f,, z f = f + f + f zz -5 = r cos θ cos φ, = r sin θ cos φ, z = r sin φ r >, π < θ < π, π < φ < π r r r z cos θ cos φ sin θ cos φ sin φ θ θ θ z = sin θ cos θ r cos φ r cos φ φ φ φ z cos θ sin φ sin θ sin φ cos φ r r r f = f rr + r f r +. r cos φ f θθ + r f φφ r tn φf φ 3 d Alembert, Jen Le Rond , F.

25 R F, 7. F, = F, = = φ. 7. = φ = φ 7.. F, = F, = = φ φ = = ψ = 3 5 F, = + F, = = F U := {, R > }, U F, = = < < U := {, R < } = F {, > } F, = = 3 F,..., m F,..., m = m m = φ,..., m F,..., m = φ * n implicit function. f, = 7.. R C r - F : R P =, F, = r =,,..., P F, P U, R I C r - φ: I R, U F, = I = φ I 7. F, φ =. P F, = 7. F P P F, = = ψ m R m C r - F : R P =,..., m F,..., m = r =,,..., P F m,..., m P U R m V C r - m - φ: V R,..., m U F,...,, m =,..., m V = φ,..., m,..., m V 7.3 F,..., m, φ,..., m =.

26 R 3 3 F,, z := + + z C - P =,, F P = F z P = U := {,, z R 3 z > } V := {, R + < } F,, z =,, z U z =, V F,, z = z {,, z R 3 F,, z = } R 3 z R C - F C = {, F, = } C P P R U C U C U C -C P 3 C 7.5. C = {, R + = } 4 P C U := {, > }, U := {, < }, U 3 := {, > }, U 4 := {, < } j =,, 3, 4 C U j C - < < f = 3 g = 3 = g = 3 C - f sphere; bll the Northern Hemisphere. 3 smooth curve. 4 circle; the circle centered t the origin with rdius R C - F C := {, F, = } P C df P, C G, = + C = {, R G, = } 7.5 C C P df P =, 7.8., b R F, := + b df, =, b df, =,, =, F, = C = {, R F, = } df, C C >, b > b < F, := + df, = 4, df, =,, =,,,,, F ±, F, = C = {, R F, = }, C 5 F df 5 33

27 = F, = = φ F dφ, d = F,. = φ. F, φ = = d d F, φ = F, φd d + F, φdφ d = F F, +, dφ d. 7. F 7.9 d d = F F. F, = = ψ F dψ d = F, F, = ψ d d = F F C P =, C F, = b 7. P C = φ = φ 7.9 dφ d = d d = F, F, = =. b b 6 n ellipse; hperbol; prbol; conics; the lemniscte = b + + b = + b. = F, 7.3 φ j,..., m = F,..., m j F,..., m m j =,..., m m = φ,..., m 7.3 m. R m k C r - F,..., F k k < m P =,..., m F j,..., m = j =,..., k F F P... P k det F k F k P... P k P U k+,..., m R m k V V k C r - φ,..., φ k,..., m U F,..., m = = F k,..., m = k+,..., m V = φ k+,..., m,..., k = φ k k+,..., m.

28 F j φ k+,..., m,..., φ k k+,..., m, k+,..., m = j =,..., k m k k m k,..., k 7.6 F F P... P m rnk = k F k F k P... P m rnk 7.6 {,..., m R m F j,..., m =, j =,..., k} P R m C r - 7. R m C r - f = f,..., f m : R m R m P f f P... P m det f m f m P... P m P U fp R m V V C r - g : V R m f g = V, g f = U f F = F,..., F k : R m R k f = F,..., F k, k+,..., m : R m R m F, = 3 F, = R C = {, R F, = } P =, F P C = φ d d = φ = FF F F F + F F F 3 F, φ 7-4 F, = +, C = {, R F, = } C = φ φ C 7-5 R 3 C 3 F,, z P =, b, c F P = F, b, c = P F, F, F z 7.3 F,, z =,, z P R 3 U, b, c c,,, b R V, V, V 3 C - ξ : V R, η : V R, ζ : V 3 R F,, z = = ξ, z, = ηz,, z = ζ,. ξ, z = F,, z = ξ, z. F,, z P F,, z = ξ, z η z z, ζ, z = z z = 7 submnifold.

29 ut... ut 8.. A t A ut ut du 8. dt = λu λ k 8. ut = ke λt ut 8. d e λt ut = λe λt λt du du ut + e t = eλt t + λut = dt dt dt e λt ut t = t A k kg ut 8.3 ut = k ut = k ep{ λt t } 3 * n ordinr differentil eqution. 3 e X epx ep the eponentil function , λ du 8.4 = λu u dt u = u, < u < ut 8.4, 8.5 t = I u, u 8.4 du λ = u u dt t = t = T T IU = ut λt = T du u u dt dt = = log { U U U du u u u = } u u [ u log = log { ut ut ] U u u } u I u, u T t ut ut = 8- u u + u e λt 8.3. m t t k k > ρ d dt 4 n initil condition; n initil vlue problem. 5 the logistic eqution. u ρ >

30 t t 8.7 m d dt + ρd dt + k = = t 8. t = t d 8.8 t =, dt t = v {t t > } ft 8.9 f t + p t f t =, f = α, f = β p, α, β ft = α + β t p p p ft = α + β log t p = { t p f t } = 8. 6 u = u,, 3 w = w,, z u = u + u, 6 prtil differentil eqution. w = w + w + w z C - u, w,, z u = w = u w 8 w = ρ ρ = ρ,, z,, z w = ρ ρ u = u, F u, = F + u 6., = r cos θ, r sin θ 8. u = u rr + r u r + r u θθ u u r θ : u = ur u u rr + r u r = 8.4 u = α + β log + α, β 8.6. w = w,, z w = F + + z 8-4 w = α + β r α, β 7 the Lplcin; Lplce, Pierre-Simon , F 8 hrmonic function. 9 the Poisson equiton; Poisson, Siméon enis 78 84, F. rottionll smmetric.

31 u.5, u., u.5, u., u.5, u., u., u.3, u.4, u.5, u.3, u.35, u.4, u.45, u.5, u.6, u.7, u.8, u.9, u., 8. c = c = t ut, u u 8. t = u c c - 8. u t, = πct ep 4ct {t, t > } R C t u t, ct ct u t, d = πct ep d = 4ct t lim u t, = t + = t = t > 8. the het eqution ut, = f = > u t, f d ut, 8. t f 3 8.,, t ut,, u u 8.4 t = c u c, u = u + u. u, u = ut, + = r cos θ, = r sin θ 8.4 u t = c u rr + r u r 8.5 ut, r = 4π ct ep r 4ct 3

32 u = ut,,, z u t = c u = + + z t ut, u 8.6 u t = c u c 4 ut, = F + ct + G ct F, G 6-5 u tt = c u Cs λ ut t + ut 8.4 u = u u > t = u < u = u γ = ρ/m, ω = k/m d dt d + γ dt + ω = =, ẋ = v γ ω > γ ω < t = e γt cosh µt + d sinh µt µ = γ ω, d = γ + v µ t = e γt cos µt + d sin µt µ = ω γ γ + v, d = µ γ ω = t = e γt + d t d = γ + v , θ e iθ = cos θ + i sin θ i z = + i, e z = e +i = e cos + i sin, e z Re e z Im e z, 8-7 z = + i fz = z m m Re fz Im fz,, m =, 3, 4 m 4 the wve eqution. 5

33 [, b] = {,,,..., N } = < < < N = b := m{,,..., N N } I = [, b] f I = {,,..., N } N N 9. S f := f j j, S f := f j j, j= j= f j := [ j, j ] f, f j := [ j, j ] f j = j j 9.. I f I 3 I S, S * 4 6 m{,..., N },..., N f [ j, j ] the superimum the infinimum 3 integrble; I f the integrl of f on the intervl I. f d = I I f 4 f d 9.. [, ] f = 5 = {,,..., N } [ j, j ] N N S f = j j = N =, S f = j j =, j= f [, ] 9.3. [, ] = f = [, ] = {,..., N } k = k [ k, k ] f [ k, k ] j= f S f = k k k = k, S f =. < k k S f f [, ] 9. f I = [, b] f d := f d, I b f d := f d I 4 < b 9. 5

34 f I = [, b] f, g f g b f d g d S f S g, S f S g 9.5. f, g [, b] f + g, αf α [, b] b f + g d = f d + αf d = α f d. 7 g d, 9.6., b, c f f d = c f d + c f d 6 the definite integrl. 7 c [, b] [, c] [c, b] [, b], b, c < c < b I = [, b] f I I 8 I α 9.3 lim α f = fα α { n } lim n f n = fα I = [, b] f I I f fα, f fβ α, β I 9.8. I = [, b] f I f f 9.9. f I = [, b] f f d f d f f 9.8 f f f continuous, continuous function. 9

35 I = [, b] f F = ft dt b F I F = f I = [, b] + h h 9.6 F + h F = = +h ft dt ft dt + 9.5, 9.9 F + h F h f = h +h = h +h +h ft dt ft dt ft dt { ft f } dt h +h ft dt = +h +h ft dt. f dt ft f dt. {h n } + h n I t gt := ft f + h n I n gt n = ft n f gt t I n t n I n t n I n n t n 9.4 F + h F h f h +h gt n dt = h gtn = gtn. h n g gt n g = F + h F lim f = h h I f F, G G = F + d { } G F = G F = f f = d G F I I f 9.. I f I F = ft dt 9.. e f F F = f d e t dt 9.3. I f F I, b f d = F b F F = f F f the primitive +C

36 [, b] C - f + {f } d [, b] = {,,..., N }, f,, f,..., N, f N I = N fj f j + j j j j j= ξ j I = N j= f j f j j j = f ξ j j < ξ j < j + f ξ j j j F = + f N F j j, j= F j := [ j, j ] F, F j := [ j, j ] F, I N F j j j = j j F I F b 9.5. t γt = t, t t b C - d + dt d dt dt j= f F f d = F b F 9- E : + = > b > b E πb E π 4 k sin b t dt, k = km, km 43.5±.km 9-3 = P =, O =,, A =, OA, OP, AP t/ P, t 9-4 = 9-5 γt = t sin t, cos t t π 9-6 R r ρ = ρrkg/m 3 ρ ρ [, R]

37 [, b] [n] : = [n] < [n] < < [n] N n = b n =,,... lim n [n] = n j =,,..., N n ξ [n] j f [, b]. ξ [n] j [ [n] j, [n] j ] [n] j f d = lim n N n fξ [n] j [n] j ξ [n] j S n j= S [n]f S n S [n]f f [n] n f [, b] * [, b] [c, d] [, b] [c, d] = {, [, b], [c, d]} = {, b, c d} R [, b] [c, d] R [, b] [c, d]. = < < < m = b, c = < < < n = d I = [, b] [c, d] mn I = [, b] [c, d] = j =,..., m k =,..., n jk, jk = [ j, j ] [ k, k ] 3 := m{,,..., m m,,..., n n } R 3 4 f,..., f n { R f,..., f n } R R R I I R 5 the Crtesin product; rectngle. 3 t lest t most 4 closed set the comlement. 5 bounded set; : compct set. R n

38 m j= k= I = [, b] [c, d] f I m fξ jk, η jk j+ j k+ k ξ jk [ j, j ], η jk [ k, k ] ξ jk, η jk f I I f 6 f, d d 7 I R f I f,, f, =, I f f f, d d = I f, d d f 8 R f, =.3 := d d 9 6 double integrl; multiple integrl. 7 8 I 9 mesurble set; re R f Ω f f.. R f f, d d.3. I = [, b] [c, d] R I = b d c.4. [, b] φ, ψ φ ψ b := {, R φ ψ, b} R.4 f, d d = [ ] ψ f, d d φ.4.4 n iterted integrl. d ψ φ f, d

39 [, b] = < < < m = b [ j, j ] j := {, [ j, j ]} f [ ] ψj f j, d j j = j j φ j j.5. I = [, b] [c, d] R f, I f, d d = d d c f, d = d c d f, d.6. = {, + } f, = { =, },.4 [ ] d d = d d [ ] = d d = = d = 4 d = π 4. {, }, [ ] d d = d d [ ] = d d = 3 3 d = 4 3 d = π 3 4. R 3 R m.7. = = b ρ kg/m ρ d, ρ, kg/m ρ, d d,, z ρ,, z kg/m 3 ρ,, z d d dz

40 R I = [, b] [c, d] C - F I F d d = F b, d F, d F b, c + F, c d d, = {, +,, }, d d, = {,, 4}, d d, = {, π, sin }, d d, = {,,, + }, + + z d d dz, = {,, z,, z, + + z } R = {, R + } C - F, G G, F, d d = π F cos t, sin t sin t + Gcos t, sin t cos t dt t, t t b C F, G C F, d + G, d := F t, t d dt + G t, t d dt dt C F d + G d C G, F, d d = C F, d + G, d R 3 d d dz R 4, R Ω = Ω = {,, z {,, z } + b + z c } + b + z4 c 4, b, c., b, c.

41 R.. = {, } R := {, 3 + 3,, } = 4 = d d = 3π/ d d = [ 3 3 ] d = 3π 3. R f Ω f := {,, z, ; z f, } R 3 R 3 f, [, + ] [, + ] f, f, f, d d.. f, = b,b f, { =, } + b f, d d = b d d = 3 πb R 3 Ω Ω.3. = {,, z z 4, } = {, } = d d dz = d d dz = 4 d d = = 3 5. * 4 7 volume; R 3 R

42 z = f,, R f C - [, + ] [, + ] 3 P =,, f,, Q = +,, f +,, R =, +, f, + P Q, P R P Q P R =,, f +, f,,, f, + f, f +, f, f, + f, = + + f f +, +,. + f + f d d.4. f, = + = {,, }. + + d d = d log π d = lm Mkg m ρ := M/lkg/m ρ [, b] [, b] [, ] Mkg [, + ] M + M M + M kg/m M M ρ ρ d kg Am Mkg M/Akg/m,, ρ, ρ,, 3, [, + ] [, + ] line densit surfce densit volume densit, densit. 3

43 ρ, ρ, d d kg R 3 Ω ρ,, zkg/m 3 ρ,, z d d dz kg Ω.5. [, ] kg/m d = 3 kg = {,,, + }, kg/m d d = d d = 4 kg R B = {,, z + + z R } r = + + z ρrkg/m 3 ρ + + z d d dz kg B 4π R ρr dr C = {, = } t P t, C Q = t C OP Q A t A t t - A t A t lim t + log t Ω := {,, z +, + z } R 3

44 R > S R = {,, z + + z = R } N =,, R S R P P N P N P r N r N r C r C r C r L r r 3 C r S R N A r r < r < πr/ 4 lim r + L r πr, lim A r r + πr. -4 {, > } π d d -6 C C : γt = t, t t b t, t t C - [, b] t > C π C L d t + dt d dt dt d d b d t + dt, t + dt dt dt d d L = + dt dt d dt dt dt d d, d d = d d -5 = f b π f + f d [, b] f >

45 [, b] f [α, β] C - φ φα =, φβ = b. f d = β α f φu φ u du.. = u = φu. j j = φu j φu j = j j φ η j u j u j uj u j φ u du φ η j u j u j,. φ η j u j u j j j φ η j u j u j u j u j j j : φ φ [u j, u j ] gu := f φu φ u, ξ j = φη j, ξ j = φη j gη j u j u j = fξ j φ η j u j u j fξ j j j gη j u j u j = fξ j φ η j u j u j fξ j j j. β α f u d du du N S g fξ j j j, j= N S g fξ j j j j=.. [α, β] : α = u < u < < u N = β j = φu j j =,,..., N φ : < < < N [, b] [u j, u j ] φ φ η j, η j [u j, u j ] 9.4 * integrtion b substitution. φ C -.. f, g g, f. φ [, b] [α, β] φ. R L A : R X = A R A

46 9 473 A det A L A L A A L A.3. L A R L A P, Q R l P, Q p, q l L A l L A l = { tp + tq t R} L A tp + tq = tap + taq l = { t p + t q t R} p = Ap, q = Aq OP = p, OQ = q P, Q P Q l P, Q P = Q l P det A L A.4. L A R L A.5. l P, Q l R, S l det P R, P Q det P S, P Q R det t, b = P Q n = t b, det P Q, v = v, n R n l l R l n P R n P R, n > L A R R L A L A P QRS p = OP, q = OQ P Q { tp + tq t } P, Q P, Q L A P, Q P QRS det, b = P Q, b = P R, b θ.3 b sin θ = b b cos θ = b, b., b, b = t,, b = t b, b.3.8. L A det A = P QRS p, q, r, s = P Q = q p, b = P R = r p P, Q, R L A P, Q, R P Q = Aq Ap = Aq p = A, P R = Ab = deta, Ab = det A, b = det A det, b = det A.

47 R C - F : R u, v F u, v = u, v, u, v R , b,, b,, b + u, b u,, b + u, b u,, b + v, b v,, b + v, b v,, b + u, b u + v, b v,, b + u, b u + v, b v F + h, b + k u, b v, b h = F, b + + h + k εh, k u, b v, b k εh, k h, k, t h, k F df 6.4 h, k.4 Φh, k := F + h, b + k F, b., u, v = det u v 4.4 u u, b v, b h v u, b v, b.9. u, v uv-, b, + u, b,, b + v, + u, b + v F u, v = u, v, u, v k, u, v u v.9.. R C - u, v u, v, u, v uv E f f, d d = f u, v, u, v, u, v du dv E.. d d + + := {, +, }.5, = r cos θ, r sin θ u, v, u, v, ε, ε ε = t ε, ε 4 the Jcobin. E := { r, θ r, π θ π }

48 r, θ, r, θ = det r θ cos θ r sin θ = det = r sin θ r cos θ. r θ d d + + = r dr dθ π/ E + r = π/... [ ] r dr + r dθ = π log 3 := {, + }, := {, + } R n E f... f,..., n d... d n =... f u,..., u n,..., n u,..., u n J du du... du n E J := u... un,..., n u,..., u n = det..... n u... n un.5 E := {r, θ r, π θ π}, E := {r, θ r, π θ π}, j d d + + = r dr dθ E j + r.3. R n C - u,..., u n u,..., u n,..., n u,..., u n - - = r cos θ, = r sin θ. = uv, = v. 3 = u, = v sin u. 4 = r cos θ, = r sin θ. 5 = r cos θ cos φ, = r sin θ cos φ, z = r sin φ ,, z = r cos θ cos φ, r sin θ cos φ, r sin φ.7-3 C - φ φ = φ = φ u du u = t t φ = ψ ψ

49 , b] f lim f d ε + +ε f [, b [, f f d f d M lim f d f d M + f d 3.. ε, ε d = [ ] ε = ε ε +, ] ε, ε d =. d = [log ] ε = log log ε = log ε + ε +, ] d. * improper integrl 3 M M e d = [ e ] M = e M M + [, + 4 M M e d =. d = [log ]M = log M + M + [, + d α α d α > β β d β < 3 e d > 3.3. k, k 3. d

50 ε, ε k d = sin ε k sin t dt = sin t [, π ] ε + k π d = k sin t dt ε, ε, ε = ε = ε + ε +ε d = [ ] ε log + log + = +ε d = f, b ε +ε f d ε, ε, f d = lim ε,ε, ε +ε f d ε, ε, ε +ε [ d = ] ε log +ε [ ] ε = log + log + +ε = log ε + log ε log ε log ε ε + d I =, b] f, g I f, g f g I, f d g d m m m!e. m f m = m!e m m f m e = e e e e =. f k f k f k+ = kf k f k+ 5 > f k+ f k+ = m! > 4 5 < f f f

51 p p Me M = [p] +! [p] p > p [p]+ m = [p] p lim + p e =. p p p e e, +. p 3.6 / p 3. p [p] +!e / p e p [p] +!e / p p e d p e e / > [, p p e d. [, ] [, + e e p e p e [, p = e d = e d + e d = e d 6 4 π k u E, k := u du, k 7 E, k = π/ k sin d k k < 8 E, k 9. [, ], k, 3.. s > 3.3 e s d 6 the Gussin integrl; Guss Guß, Crl Friedlich , G. 7 k 8 the elliptic integrl of the second kind, the complete elliptic integrl of the second kind. n n =,, 3

52 Γ s = e s d s > 3.3. p, q Bp, q = p q d 3 ft = t k k * s > ˆfs = k!/s k+ 4 ft = e t * s > ˆfs = /s 5 ft = cos ωt ω * s > ˆfs = s/s + ω 6 ft = sin ωt ω * s > ˆfs = ω/s + ω , ] [, s Γ s + = sγ s n Γ n = n! [, + ft * ˆfs := e st ft dt s ˆf f ft = * s > ˆfs = /s ft = t * s > ˆfs = /s 9 B b β s

53 e d = π, e d = π r, θ = r cos θ, = r sin θ R { R := r, θ r R, θ π } [ = [, R], π ], / r, θ = r π 4.4 J R = e r R r dr dθ = re r dr dθ R R dr = π M e r dθ = π e R 3., 3. e 4. M M M E M 4. M M I M := M M M 4. I M = e d = = M R 4.3 J R := d d, * R e e M e d M e d e d d = E M := [, M] [, M] e d E M e d d R := {, R + R,, } 4. M M E M M 4. d d e EM d d M e M 4., 4.3, 4.4 π e M I M π e M, 4 4 M + e d = lim M + I M 4. = π 4. e d d 3.

54 Γ = π. u Γ = e d = e u du µ σ µ e σ d =, πσ µ e σ d = µ, πσ µ µ e σ d = σ. πσ u = µ/ σ M, M M πσ M µ e σ d = e u du π 4.8 = M + µ σ, = M µ σ M j + j + j =, M, M M M µ e σ d = πσ = = σ π σ π σu + µ e u du π ue u µ du + π e u du e e + µ π e u du µ, M µ M πσ µ e σ d = σ u e u π [ = σ ue u] + e u π du σ du = π u e u du X X = k C k / X j p j > j pj = µ := p j j, σ := p j j µ µ X σ σ, b X b P,b P,b = ρ d ρ stochstic vrible, rndom vrible, probbilit distribution, the binomil distribution, the men, the vrince, the stndrd devition, probbilit densit function.

55 ρ P,b 4.9 ρ d =, ρ ρ µ := ρ d, σ := µ σ ρ := πσ µ e σ µ ρ d ρ 4.9 ρ , 4.7 µ, σ 3 3., Bp, q = Γ pγ q Γ p + q p >, q >. p, q ε < /4 M > Iε, M := e p e q d d ε,m M M = e p d e q d ε,m = [ε, M] [ε, M] ε ε v Ε Ε lim ε + M M Iε, M = Γ pγ q = uv, = u v ε,m uv { := u, v ε u v M u, M u v ε } u 4., u, v = u u > Iε, M = e u u p+q v p v q du dv [ ε, M := ε [ := [ε, M] ] [ ε, ε] ε M + ε, 4. ] M M + ε u the norml distribution.

56 473 4 Iε, M e u u p+q v p v q du dv = M ε e u u p+q du M M+ε ε M+ε v p v q dv Iε, M e u u p+q v p v q du dv M ε ε = e u u p+q du v p v q dv ε ε +, M + Γ p + qbp, q ε -3 n R R n R 3 B n R := {,..., n R n n R } R n V n R :=... B n R d d... d n, f,..., n := e n e n n d d... d n = e dt t = π n. R n r = + + n f = e r r r + r f fr r r + r = fr V n r + r V n r = frα n r + r n r n = frα n nr n r + r , f R n r r 4... f,..., n d d... d n = nα n e r r n dr R n r = u nα n e u u n du = n n n α nγ = α n Γ j = R j j =,..., n V n R = R n α n α n := V n α n α = π, α 3 = 4 3 π α n = π n Γ n +. 3 bll. sphere 4 r = r = + e r r

57 8 - Yes; No; 3 No; 4 Yes, = No; 5 No; 6 Yes ,, 4/5, 3/5; 4/5, 4/5, 4/5; 3 m/ + m -7 ; z z = F z -8 c f, = c nn -8 : n -9 : n m nh m = n+m! m!n! nm I f = k I h = 3. f f = f + h f = hf +θh < θ < θ +θh h I f + θh = f = f = k I f I k 3-4 I, < =, h = > f 3. f f = hf + θh < θ < f + θh >, h > f > f - fh, f, f, = lim = lim h h h h = -4 3 : f, = +,,, =,, f, = +,,, =, cos Π Π sin Π tn f, = b c + d + p + q + m., b, c, d, p, q, m -5 f f,, z = r F r, r = + + z f,, z = r F r + r F r r 3 f + f + f zz = F r r + F r. F F r = /r + b, b -6 f {, + > }, g {, cos cos > } Π Π Π Π 4 Π 4 = cos = sin = tn sec Π Π Π Π csc Π Π Π Π Π cot = sec = csc = cot Π Π

58 4- cos, sin 4-3 cosh tnh tnh = e e e + e = e e + = e + = + e e +. f = f f f = f 3 cosh sinh tnh = cosh = sinh = tnh 4, 5 6 cosh u = +t t, sinh u = t t, tnh u = t +t. 7 A > B ± A B cosh + α, α = tnh B/A A A < B B A sinh + α, α = tnh A/B. A = B 8 = cosh Y = e Y 4-4 α = tn 5, β = tn 39 tn4α + β = 4α + β = π 4 + nπ n tn < 4α + β < 4 tn + tn = π n = sin = sin 4-6 cos n = sin cos n sin log + 4 log + 5 log /+, +b = α, β α β log α β 3 tn A A. A = b. + b = + b = A A + /A. 4-3 tn + 4 log +, 6 log tn log + 3, 3 3 log ++log + + tn + tn t 5-3 γt = t, t = + v t, b + v t ẋ = v, ẏ = v 5-4 γs = cos s, sin s F s := f γs = + sin s F s = cos s s π/4, π/4, 3π/4, π, π, 3π/4 3π/4, π/4, π/4, 3π/4 5-5 df P v = grd f P v v = grd f P df P v grd f P v = kgrd f P k > 5-6 γt P f f f t, t = t v > t < v /v ftv, tv =, v < t > v /v ftv, tv =, v = ftv, tv = ftv, tv t = df O v = d/dt t= ftv, tv = n = /n, n = /n n, n, f n, n = lim fn, n = f, f n f, = F r f θ f = F rr + r F r = log F r r = /r F r = log + + b, b tn / = θ 6- fξ, η = f ξ+η, ξ η c f ξη = f ξ η f ξ ξ, η = Hξ ξ fξ, η = Hξ dξ + Gη Gη η 6-3 r, θ, φ,, z r θ φ cos θ cos φ r cos φ sin θ r cos θ sin φ r θ φ = cos φ sin θ r cos θ cos φ r sin θ sin φ z r z θ z φ sin φ r cos φ,, z r θ φ

59 7 56,, z = rcos θ sin φ, sin θ sin φ, cos φ 7- P C df P, F P F P F P 7. P R U I I C - φ C U φ C P F P = F P P C = ψ = ψ P C 7- df =, 3 C := {, F, = }, C C > < = 3 = /3 b, b b, b, b, b ; > C = φ φ = + + φ = = + = φ = 3 = =, ± +3 3 < < 4 = C, b C = φ φ = φ [, b φ = φ = F, φ /F, φ 7.9 d d F, φ = F, φ + F, φ φ φ F, = 4 + C = C = {,,, } < { {, [ b, b ] [b, b ]} < < C {, [ b, b ] = 8 63 ξ = F F, η z = F z F, ζ = F F z. 8- T λ = log /T = t + b = +, b =. u C = ±b, ±b > = F = C = φ F = ψ C u u = u u u e λt

60 w = F r w rr + r w r = Re e z = e cos, Im e z = e sin 8-7 m = Re fz =, Im fz =, m = 3 Re fz = 3 3, Im fz = 3 3, m = 3 Re fz = , Im fz = m = b / b d = 4b [ ] u du = 4b u u + sin u., = cos t, b sin t π t π π/ π/ 4 sin t + b cos t dt = 4 π/ = 4 cos t + b cos t dt π/ π/ = 4 k cos t dt = 4 t = π u 3 k π/ 4 k sin t dt 4 π/ sin t + b cos t dt k sin u du. k π sin t dt = 4 k 8 π., b k = km 9-3 = + + u du + = log + + = sinh = sinh t, = + = cosh t. P = cosh t, sinh t log π, 8 = f π π d = d dt = 3π. dt = t = t sin t R 9-6 4πr ρr dr. [, R] = r < r < < r N = R [r j, r j ] r j r j 4 3 πr3 j r3 j ρr j 4 3 πρr j r j r j r j + r j r j + rj 4 3 πρr j3r j r j r j = 4πr j ρr j r j r j = r j r j. [, R] ρ m [, R] : = r < r < < r N = R I j := [r j, r j ] r j r j ρ ρ j ρ j = ρξ j, r j ξ j r j ξ j I j M j M j 4 3 π ρ jr 3 j r3 j = 4 3 πρξ jr j r j r j + r jr j + r j = 4 3 πρξ j3ξ j r j r j + µ j µ j := 4 3 πr j r j ρξ j r j + r jr j + r j 3ξ j µ j 4 3 πr j r j Mr j ξ j 4 3 πr j r j Mr j r j = 4πm 3 r j r j r j + r j 8πmR r j r j 3 8πmR r j r j 3 n µ j j= 8πmR n r j r j = 8πmR. 3 3 j=

61 fr := 4πr ρr f j I j f 4 3 πρξ j3ξ j r j r j f j r j r j N N M j S f + µ j j= j= S f + 8πmR. 3 f R fr dr M r M 4πr ρr dr. M j ρ 79 - r M 4πr ρr dr. d [ F, d d = d F, d = F, ] =d =c d c = F, d d F, c d = [ F, d ] =b = [ F, c ] =b = = F b, d F, d F b, c + F, c. - /6, 8 log 6, 4 ππ 3, /45, / -3 3/4π, π /, 8π /5, /3bcπ, 8/5bcπ. -5 G, d d = d G, d = [G, ] = d = = G, d G, d = = sin t π t π, = sin u π u 3π π/ 3π/ = Gcos t, sin t dt + Gcos u, sin u du π/ π/ 3π/ π = Gcos t, sin t dt = Gcos t, sin t dt. π/ π F 86 - = { t + t + logt + + t }, = t + t [ ] d dz d = 4 d = 6 3, [ z ] d d dz z [ z ] = 8 d d dz = π r R πr sin r R. 3 4πR sin r R. f, = R + πr sinr/r. 4 r r -4,, +, + + π + π = π + π, d d

62 m,..., m n n,..., n m + + m n n m + + m n k, l, k + k, l + l m kl := k l kl := k, l m kl = k l d d =, k,l k,l m kl kl = k k l, l k l k,l kl kl d d, d d z = f b z > z = F, = f =, := {, b, f }. S S = F F d d = [ b f = f + f f = π f + f d. f + f f ] d d f d d -6 [, b] C = f ẋ = C I = [, b ] C = f -5 π f + df d d = π t ẋ + ẏ dt = d/dt = t t = f t [, b ] ẋ = ẏ π b = π = π d dt dt = π ẏ dt = π ẏ dt ẋ + ẏ dt ẋ =, > ẏ C [, b] m j, j m j = t j, j = t j, t j π/ /cos θ+sin θ r 3 dr dθ = dθ r 3 dr = 6, { = r, θ rcos θ + sin θ, θ π }. 4 u uv du dv = u du v dv 4 u 4 u 4 4 u + u du v dv + u du v dv = 8 log 6, u { = u, v v, u v 4 u, v } u. π 3 u v sin u du dv = u sin u du v dv = 4 ππ 3, = {u, v u π, v }. π/ cos θ sin θr cos θ+ sin θ 4 dr dθ = cos θ sin θ dθ r dr { } = 45, = r, θ θ π, r cos θ + sin θ

63 5 r 4 cos φ dr dθ dφ π π cos θ cos φ+sin θ cos φ+sin φ = dθ dφ r 4 cos φ dr =, { = r, θ, φ θ, φ π }, rcos θ cos φ + sin θ cos φ + sin φ. - r cos φ, { = r, θ, φ r R, π θ π, π φ π } ρr d d dz = ρrr cos φ dr dθ dφ π π R R = dθ cos φ dφ r ρr dr = π r ρr dr. π π -3 φ = φ u du = φ t dt = φ t dt ψ := φ t dt ψ ε α d = ε { α+ ε α+ α log ε α = ε + α + > M M β d = { β+ M β+ β log M β = M + β + < 3 M { M e d = e M M = M + > 3- [, ] e e s s. s > 3., ] 3.5 e s d [, Γ = M M M e s d = e s d = [ e s] M M + s e s d M = e M M s + s e s d M Γ s+ = sγ s n Γ n = n Γ n = = n n... Γ = n!. 3-4, ] q M, M = { q q < q < p q M p <. p > 3., /] 3.5 / p q d t p q d / s > ˆfs = e st dt = lim e sm = M + s s.

64 3.9 = + s s. 3 ˆfs = te st dt = [ s ] t= te st + e st dt t= s ˆfs = te st dt = [ s ] t= tk e st + k t k e st dt t= s = k t k e st dt s 6 5 M M cos ωt e st sin ωt dt = e st dt ω [ ] = ω e st cos ωt s e st cos ωt dt ω = ω s ωs + ω = ω s + ω ˆfs = e st dt s < 5 ω = ω M M M sin ωt e st cos ωt dt = e st dt ω [ ] M = ω e st sin ω t + s M e st sin ωt dt ω = ω e sm sin ωm s M cos ωt e st dt ω ω = ω e sm sin ωm s ω e st cos ωm s M ω e st cos ωt dt. s + ω M ω e st cos ωt dt = ω e sm sin ωm s ω e st cos ωm e sm e sm sin ω e sm, e sm e sm cos ω e sm s > M + e sm cos ωm, e sm sin ωm s + ω ω e st cos ωt dt = s ω