[, + 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

1 865.. 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 * /3 8 4 : function; :the domin; : the rnge; : the imge. 3 set; 4 : rel numbers; : the set of rel numbers; R R R 5 : n element; : subset; X Y Y 6 Y X 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} 4 I f : I R {, f I } f..3,.4 f f 5. 5 cosine, sine, tngent. cot := cos, sec :=, csc := sin cos sin. : rtionl number; : n irrtionl number. 3 f 3 R 4 f : the grph of function f. 5 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.

3 = cos, π = Cos Π Π 4 Π 4 Π f = tn b f = c f 3.4 d f 4.4 e f ,.4 f 5 cotngent, secnt, cosecnt d d tn = + tn = sec, d d sec = sec tn, d tn d = log cos, d d cot = + cot, csc = csc cot, d cot d = log sin. - sec d = + sin log sin = log + tn tn,.3 csc d = cos tn log + cos = log. 6 cosec sec = cos cos 7 := 8 = Cos = cos π. = sin, π π = Sin = Sin = sin π π. = tn, π < < π = Tn = Tn = tn π < < π. Cos, Sin, Tn 9 Cos = π 6 + 3, Sin 4.7. Cos + Sin = π = 5 π, Tn = π 8. 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 9 rc cosine; rc sine; : rc tngent; inverse trigonometric functions. rccos, rcsin, rctn, cos, sin, tn

4 7 865 < β < α = Tn < Tn = π 4. < α + β < π α + β = Tn = π d d Cos =, d d Tn = +. d d Sin =, = Cos = cos d d = d/d = sin. π sin sin = cos =.7 Tn.5-3 Cos, Sin = ±.4 d.5 + = Tn, d = Sin Tn =, Sin =.6 Tn = dt + t, Sin = dt t.7 Mchin α C cosh = e + e, sinh = e e, tnh = sinh cosh = e e e + e = e e cosh sinh = 67 cosh + = cosh cosh + sinh sinh, sinh + = sinh cosh + cosh sinh, tnh + = tnh + tnh + tnh tnh. elementr functions. 3 n α { > } 4 hperbolic cosine; hperbolic sine; hperbolic tngent; hperbolic functions. 5 cosh t ht cos ht 6 cosh cosh 7 t, t = cosh t, sinh t =

5 d d cosh = sinh, sinh = cosh, d d d d tnh = tnh. cosh d = sinh, tnh d = log cosh. t, t sinh d = cosh, t + t 4 + N t N = t N+ = + t + t N+ t N+ + t t =.6 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 Tn = N N + N+ + R N N k k+ = + R N, k + k= N+3 N R N N+3 N lim RN = N.8 Tn = = k k+ k + k=..8 =, π.9 4 = R N 3 N π = N + 3 R N, N + 3 R 4 N N + 3 R N N 3 N α = /5, β = /39. 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 = = =5 8 TSUBAME Pet flops 5

6 865 4 = tn 5 = tn π, π - -3.,.4-4 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 + Sinh, Tnh Sinh = log + +, Tnh = log M =, N = -6 log = log log Cos, Sin, Tn -7 n I n = π/ I n = n n I n m π/ I n = cos n m m 3 m... π d = m m + m m... 3 cos n d n n = m, n = m + m sin n = sin θ u = + -9 f = + /f -, b / + b b =, / b > 3 b < / + u -.3 t = tn t cos = cos sin u = sin 3 = cosh u cos - / + d = log + + = Sinh / + = tn θ 3 = sinh u -4, 4, R R R

7 R = R. n n R n R n = {,..., n,..., n R}. R = {,, R} = {,, } R 3 = {,, z,, z } R R R 3 R n R n R n,..., n f,..., n f n f n. f R n R f : R..., R f, = + f R 3 f : R R f : R, f, = + R f, =, f, =, f, = * / : the number line; : the coordinte plne, the Crtesin plne; : the coordinte spce; : point. R n R,, f,.. f, f, f, =.3. f p, f p, 4 f : R R R 3 { },, f,, f f R 3 f : R c R {, f, = c} f c 4 f c z = c f c 3 n f : R n R c {,,..., n, f,..., n,..., n } R n+, {,,..., n f,..., n = c } R n f c.4. R f, = + [, + c, + {, f, = c} 4 : the contour, the level curve, the level set.

8 f b f..4 c f c c f z. R f, = R c {, f, = c} c =. b I I R f f + h f. lim h h f. f f 7 I f f : I f R f f = f f = d d f f f f f f f = f n f n = dn d n f = f.,.3 f R f R 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 5 : sclr field. 6 7 : differentible; : the differentil coefficient; : the derivtive; f : f-prime dsh. 8 : the second derivtive; 3 : the third derivtive; n : the n-th derivtive. 9 domin 3

9 7 865 f, b f f, b, b f, b f f f :,, R f f f.5. d f = f, f = f f f, f,,..3 f f, f f, f, f, f, 4 f, f,, f, f = f = f, f = f = f, f = f = f, f = f = f f.6. f, = f, = 3 + 6, f, = 3 +. f = 6 + 6, f = 6, f = 6, f =..7. R = {, } f, = Tn.8 f, = + / = f, = + / = + / + / = = + +, f = +, f = f = + +, f = +..6,.7 f f f f 3-9 f f 3 f f, 3 3 f = f, 3 f = f, 3 f = f,... : prtill differentible; : the prtil derivtive with respect to. : the second prtil derivtives. : the commuttivit of prtil differentils.

10 f 3, 3 f, 3 f, 3 f 3 4 n f,..., n i i =,..., n f i f = f k, l n. k l l k ut... 3 ut.8. A t A ut ut du.4 dt = λu k.5 ut = ke λt λ m t t 5 k k > ρ d ρ > t dt 3 n ordinr differentil eqution : Hooke s Lw; Hooke, Robert, , En. 865 t.6 m d dt + ρ d dt + k = = t u = u,, 3 w = w,, z u = u + u, w = w + w + w z 7 u = w = u w 8 w = ρ ρ = ρ,, z,, z w = ρ ρ 9.. t ut, u u.7 t = c u c.8 u t, = πct ep 4ct {t, t > } R B t 6 prtil differentil eqution. 7 the Lplcin; Lplce, Pierre-Simon , F 8 hrmonic function. 9 the Poisson equiton; Poisson, Siméon enis 78 84, F. the het eqution.

11 u.5, u., u.5, u., u.5, u.3, u.35, u.4, u.45, u.5,. c = u., u., u.3, u.4, u.5,.. t ut, u. u t = c u c 3 ut, = F + ct + G ct F, G - 4 u tt = c u u.6, u.7, u.8, u.9, u.,.3.9 c = u t, ct ct u t, d = πct ep d = 4ct t lim ut, = t + { = t = t >. { f = >.9 ut, = u t, f d ut,.7 t f.3 - I R f R f , f,, F f, = F + f f f 7 3 the wve eqution. 4

12 ,, f, = +, =, -4 f, f + f = + f + f + f + f : f,, f,, f,, f, 3. f, 4, f3, 6, f4, 8. f, m m,. f 3 f -7 n -8 n m -9,, f, = +, =, f 5 f, = log + + +, g, = log cos cos, h, = Tn. -5 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, z = + i fz = z m m Re fz Im fz,, m =, 3, 4 m ut, = sin + qt + b sin qt, b, q., b, q - f, = log +, g, = Tn., 3-3 F t f,, z = F + + z 3 f F 5 : miniml surfce.

13 I R f I lim f = f f I f I {, f = =. lim f = lim + f f = lim f = f = f = { cos, =. n = nπ, n = n+π n =,, 3... { n}, { n } lim f n =, lim f n = n n lim f 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 =. * / : continuous; : continuous function. f f 3. f = f = f = 3. f = 3 R f fh f h = 3 + h h f sin f = + = R sin f = cos + = f 3 C k - I f I C - 4 I C - I f I C k - k > f k f k I C - k C k - 3 : the squeeze theorem. 4 C - : of clss C ; C r - : of clss C r ; C - : of clss C C-infinit.

14 f R C - 3. f C - n { > } 3. n n > R 5 R 6 {, R + < r }, {, R < < b, c < < d} r,, b, c, d r >, < b, c < d f A 3. lim f, = A f, A,,b,, b 7,, b f, A 8 + h, b + k, b h, k, 3. lim f, = lim f + h, b + k.,,b h,k, 3.6. 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, 5 : domin; 6 : n open disc; : n open rectngle rectngulr domin. 7,, b f, b the limit {h n }, {k n } 9 lim n f + h n, b + k n = A. 3. {f + h n, b + k n } A {h n }, {k n } 3.8. R,, f, = +, =, -6 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 3.7,, f, f,,, f, lim f, = lim = lim lim + f, =, lim f, = lim = lim lim f, =. + f, = / +,, lim lim f, =, lim lim f, =. 3 f, = / +,, r > θ = r cos θ, = r sin θ 9 : rbitrr; X P : P holds for n rbitrr X.

15 ,, r = + f, = fr cos θ, r sin θ = r cos θ sin θcos θ sin θ = r sin θ cos θ = 4 r sin 4θ sin 4θ r R f, b lim f, = f, b,,b f f f, f, = f, = -9, 3.8 3,, f, = +, =,, R f,, b A, B h, k, 3.3 f + h, b + k f, b = Ah + Bk + εh, k h + k εh, k lim εh, k =. h,k, f,, b f, b 3.3 A, B A = f, b, B = f, b 3.3 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 R f,, b f, b 3.4 lim εh, k = h,k, εh, k := f + h, b + k f, b f, bh f, bk h + k 3.4. f, b, b 3.3 h, k, f, f f, f f 35

16 sin,, f, = +, =,, f, f 3.8. R f f, f, f = f C k - R f C - C - f, f C - f f, f, f, f C k - k f k C - k C k R f C C k m C m - C k - 4 C - f = f df P f P, f,, f, df f f 3.6 df =, f. 3.. φ, =, ψ, = dφ =,, dψ =, d =,, d =, df = f f d + d h, k, h + k h, k,, 3.8 f f f, b +, b, f := f +, b + f, b I t, t t, t I R γ : I t γt = t, t R. t, t γ 3.3 f, P =, b f f 3.5 df P =, b,, b, - 4 ; row vector column vector; totl differentil; differentil. curve; prmetric representtion of the curve. γ γt = t sin t, cos t ccloid C - t = nπ nπ, n =, ±, ±,...

17 γt = t, t γt = dγ dt t = ẋt, ẏt d d = t, dt dt t t, t 3 t b+v b+v b v P v O s coss,sins t O s t +t, t +t γt t γt O v +v +v γt t v = v, v P =, b γt = + tv, b + tv t = P v P v 3. s σs = cos s, sin s π < s < π, 4 sin s, cos s 3. 3, t σt := + t, t + t < t <. t = tn s 3. f, γt = t, t 3.9 F t = f t, t f, γt = t, t 3.9 df f d f d t = t, t t + t, t dt dt dt t. t δ hδ, kδ hδ := t + δ t, kδ := t + δ t 3 the velocit vector; the speed. 4 line; circle , δ hδ, kδ t + δ t ε δ := ẋt = hδ ẋt, ε δ := kδ ẏt δ δ δ t, t δ ε δ, ε δ 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 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

18 I = [, b], γt = t, t t b R γ γb γ,. R P, Q P Q γ : [, b] R γt t b. R P =, b ε U εp := {, R + b < ε } R R P ε. R 5 P U εp ε. F : R R {, R F, > } R. R f I I + h I h θ f + h f = f + θhh < θ <. 3.6, b h, k 3.4 εh, k k F h := f + h, b + k f, b + k f F h F h = f + h, b + k, F = F F h = F h F = F θhh = f + θh, b + kh < θ < θ Gk = f, b + k f, b k Gk = G δkk = f, b + δkk < δ < δ εh, k = F h + Gk f, bh f, bk h + k 5 : n open set; : connected set; : disc disk. 6 : the men vlue theorem = 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, 3.8, 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 V = k F θ h = k f + θ h, b + k f + θ h, b = k Fk F F t := f + θ h, b + t 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 , 3.8, 3., C - f, = +, 3-3 f, = e cos + sin f., f. γt t 3.9

19 n n n,,,,, 3, = = t,, t =,. t , = +,,, 3 3 = = t A * / vector, mtri, mtrices. sclr; row vector; column vector. 3 trnsposition. 4 squre mtri A = =, = = α α α =, α =,, = A A 4., ξ α A =, ξa = ξ, ξ. α A B α AB = b α b α A =, B = b α b α b α, b. A 4. AA = A A = E E := A A A A E 5 ξ, A 4.3 E =, ξe = ξ, EA = AE = A. 4. A 4.4 det A := det A A 6 A det A A 4.5 A =. det A 5 regulr mtri; the inverse mtri; the identit mtri. 6 : determinnt.

20 f, = t, f = f, 7 3. P = t, b v = t v, v γt γt = P + tv = t + v t, b + v t. 4.. R f P =, b v = t v, v F t := f + v t, b + v t = fp + tv t = F f P v 8 v v f P R f P =, b f P v 4.6 df P v = f, bv + f, bv v = t v, v. P f f grd f P :=, b = t dfp f, b f P df P v = grd f P v grd f P P f the directionl derivtive. 9 the grdient vector 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 4.3 f ξ = f ξ + f ξ, f η = f η + f η z = f, = f ξ, η, ξ, η = fξ, η z ξ = z ξ + z ξ, z η = z η + z η. R R R F : R, R F, F, R ξ, η ξ, η, F : R R the chin rule. ξ: i; η: et. ξ, η, ζ zet,, z

21 R 4.7 F : R, ξ,, η, R. ξ,, η, F ξ = F ξ = ξ, η, =, F = ξ, η: R R C k - 3 ξ, η C k R C - F = ξ, η: R n ξ ξ ξ ξ df = = η η F 4, U R F : R G: U R =, F U η η G F : R G F R G F : R R F G F, G C - dg F = dg df, dg F = dg F df R id : id = 6 R U R F : mp : components. 3 C k - 4 the differentil the Jcobin mtri Jcobi, Crl Gustv Jcob 84 85,. 5 the composition. 6 the identit mp id id U G F = id, F G = id U G: U G F G = F { = r, θ R r >, π < θ < π }, U = {, R > } F : r, θ F r, θ = r cos θ, r sin θ U, G: U, G, = +, Tn G = F, F = G, r, θ π < θ < π Tn tn θ = θ.6 r > G F r, θ = Gr cos θ, r sin θ = r cos θ + r sin θ, Tn r sin θ 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 = + = +. =, = id U,. 4.8., 4.7 r, θ = G,, r, θ, 8 7 the inverse mp; F : the inverse of F /F -inverse; 8 the polr coordinte sstem; the orthognonl cooridnte sstem; the Crtesin coordinte sstem; : escrtes, René Rentus Crtesius;

22 , = {r, θ r >, π < θ < π}, Ũ = {, > } h: Ũ R Tn h, := Tn + π Tn π 9 >, >, < F : r, θ F r, θ = r cos θ, r sin θ Ũ, G: Ũ, G, = +, h, F = G, G = F, r, θ = G,, 4.9. F : R U R G = F F F C - df = df df F = df E F F = id 4.6 df df = E, F F = id U 4.6 df df = E df df = r, θ = r cos θ, = r, θ = r sin θ. F : r, θ, 4.5 r θ cos θ r sin θ 4.9 df = =, sin θ r cos θ r θ 9 h,,, C Fortrn tn, G = F r r cos θ sin θ 4. dg = = df = θ θ r sin θ r cos θ 4. = cos θ r r sin θ θ, = sin θ r + r cos θ θ. C - f, 4. z = f = f + f. f, 4.8 r, θ f f r, θ 4. f = cos θf rr r cos θ sin θf rθ + 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 4.3 f = f + f = f rr + r f r + r f θθ. 4.4 R F, F, = F, = = φ. F, = = φ = φ 4.. F, = 3+5 F, = = 3 +5 = 3 5 F, = + F, = n implicit function.

23 F U := {, R > }, U F, = = < < U := {, R < } = {, > } F, = = f, = 4.. R C k - F : R F, =, F, P U R I C k - φ: I R, U F, = I = φ. I F, φ =. P F, = 4. F P P F, = 4.3. 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 sphere; bll the Northern Hemisphere R C - F C = {, F, = } P C P R U C U C - C P C 4.4. C = {, R + = } 3 P C U := {, > }, U := {, < }, U 3 := {, > }, U 4 := {, < } j =,, 3, 4 C U j C - < < R C - F C := {, F, = } P C df P, C 4.6. F, := + df, = 4, df, =,, =,,,,, F ±, F, = C = {, F, = }, = F, = = φ dφ d = F / F,, = φ. smooth curve. 3 circle; the circle centered t the origin with rdius. 4 the lemniscte.

24 F, φ = = d d F, φ = F d, φ d + F = F F dφ, +, d., φ dφ d 4. F 4.7 d d = F. F F, = = ψ F dψ d = F, F, 4-4. = ψ 4 d d = F F. 4-, f, = P =, b f P df P, f P v df P v v grd f P v 4-4 P =, b f P df P, P f γt = t, t γ = P t = γ γ grd f P 4-5 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 -3 = 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 4-7 F, = 3 F, = R C = {, R F, = } P =, F P C = φ d d = φ = F F F F F + F F. F, φ 4-9 F, = +, C = {, R F, = } C = φ φ C 4- R 3 C 3 F,, z F,, z = P =, b, c F P = F, b, c = P F, F, F z P F,, z =,, z F,, z = = ξ, z, = ηz,, z = ζ,. P F,, z = ξ, z η z z, ζ 5 d Alembert, Jen Le Rond , F. F 3 z, z = z =

25 [, b] = {,,,..., N } = < < < N = b := m{,,..., N N }. I = [, b] f I = {,,..., N } N N 5. S f := f j j, S f := f j j, j= j= j = j j f j := [ j, j ] f, f j := [ j, j ] f. 5.. I f I 3 I S f, S f I f 4 b f d = f d. I * / m{,..., N },..., N f [ j, j ] the superimum the infinimum integrble; I f the integrl of f on the intervl I. 4 < b [, ] f = 5 = {,,..., N } [ j, j ] N N S f = j j = N =, S f = j j =, j= j= f [, ] 5.3. [, ] = f = [, ] = {,..., N } k = k [ k, k ] f [ k, k ] f S f = k k k = k, S f =. < k k S f f [, ] f I = [, b] 5. f d := f d, f d := f d I b I f I = [, b] f, g f g b f d 5 6 the definite integrl. g d.

26 f, g [, b] f + g, αf α [, b] b f + g d = f d + αf d = α f d. 7 g d, 5.6., b, c f f d = c f d + c f d [, b] [n] : = [n] < [n] < < [n] N n = b n =,,... lim n [n] = n j =,,..., N n [n] j ξ [n] j f [, b] 5.3 f d = lim n N n j= ξ [n] j [ [n] j, [n] j fξ [n] j [n] j [n] j S n S [n]f S n S [n]f ]. f [n] n f [, b] I = [, b] f I I 8 I α lim α f = fα 5.4 α { n } lim n f n = fα I = [, b] f I I f fα, f fβ α, β I 5.9. I = [, b] f I f f 5.. f I = [, b] f f d f d. f f 5.9 f f f I = [, b] f F = ft dt b F I F = f 8 continuous, continuous function. 7 c [, b] [, c] [c, b] [, b], b, c < c < b 5. 9

27 [, b] + h [, b] h 5.6 F + h F = = +h ft dt ft dt + +h ft dt ft dt 5.5, 5. F + h F f h = +h h ft dt = +h { } h ft f dt h +h ft dt = +h +h ft dt. f dt ft f dt. {h n } + h n [, b] t gt := ft f I n := [, + h n ] gt n = ft n f gt t I n t n I n t n I n n t n 5.4 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 F = f F f I f F, G G = F + { G F } = G F = f f = G F I I f 5.. I f I F = 5.3. e the primitive ft dt e t dt f F F = f d 5.4. I f F I, b f d = F b F [, b] C - f + {f } d [, b] = {,,..., N }, f,, f,..., N, f N N fj f j I = + j j j j j= f j f j j j = f ξ j j < ξ j < j ξ j N I = + f ξ j j j j = j j j= f C - + f 5.7 I 5.6. t γt = t, t t b C - d + dt +C d dt dt

28 [, b] [c, d] [, b] [c, d] = {, [, b], [c, d]} = {, b, c d} R [, b] [c, d] 3 R [, b] [c, d] 5.5 = < < < m = b, c = < < < n = d I = [, b] [c, d] mn I = [, b] [c, d] = jk, jk = [ j, j ] [ k, k ] j =,..., m k =,..., n 4 := m{,,..., m m,,..., n n } R 3 5 f,..., f n { R f,..., f n } R R R I I R I = [, b] [c, d] f I 5.5 m m fξ jk, η jk j+ j k+ k j= k= ξ jk [ j, j ], η jk [ k, k ] ξ jk, η jk f I I f 7 8 f, d d. I R f I f,, f, =, I f f f, d d = f, d d I f 9 R f, = 5.6 := d d 3 the Crtesin product; rectngle. 4 t lest t most 5 closed set the comlement. 6 bounded set; : compct set. R n 7 double integrl; multiple integrl. 8 9 I mesurble set; re.

29 R f Ω f f 5.7. R f 5.8. I = [, b] [c, d] R I = b d c 5.9. [, b] φ, ψ φ ψ b := {, R φ ψ, b} R 5.7 f, d d = [ ] ψ f, d d. φ d ψ φ f, d. 5.9 [, b] = < < < m = b [ j, j ] j := {, [ j, j]} f [ ] ψj f j, d j j = j j φ j j n iterted integrl [, b] [c, d] R f, I f, d d = d d c f, d = d c d f, d. 5.. = {, + } f, = = {,, } 5.9 d d = = = [ d ] d = 4 [ ] d = d d {,, d d = = [ d 3 ] d = 3 d = 4 d = π 4. } [ ] d d 3 d = π 3 4. R 3 R m [, b] ρ b kg/m ρ d kg, ρ, kg/m ρ, d d kg,, z ρ,, z kg/m 3 ρ,, z d d dz kg.

30 f F f d = F b F 5- E : + = > b > b E πb E π 4 k sin b t dt, k = km, km 5-3 = P =, O =,, A =, OA, OP, AP t/ P, t 5-4 = 5-5 γt = t sin t, cos t t π { } + +, d d, =,,, d d, = {,, 4}, { } π, d d, =, sin { },, d d, =,, + { } + + z,, z d d dz, =,, z. + + z R I = [, b] [c, d] C - F F d d = F b, d F, d F b, c + F, c I 5-9 R 3 d d dz R 4, R , b, c {,, z } { + b + z c,,, z } + b + z4 c 4 5- R = {, R + } C - F, G G, F, d d = π F cos t, sin t sin t + Gcos t, sin t cos t dt C : t, t t b F, G F, d + G, d C := F t, t d dt + G t, t d dt dt C F d + G d C G, F, d d = F, d + G, d 5. C 5-7 R r ρ = ρrkg/m 3 ρ ρ [, R]

31 R 6.. = {, } 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, * 8 5 /5 f, d d. 6.. f, = b,b f, { =, } + b f, d d = b d d = 3 πb R 3 Ω Ω 6.3. = {,, z z 4, } = {, } = d d dz = d d dz = 4 d d = = 3 5. z = f,, R f C - [, + ] [, + ] 3 P =,, f,, Q = +,, f +,, R =, +, f, + P Q, P R volume; R 3 R

32 P Q P R =,, f +, f,,, f, + f, f +, f, f, + f, = + + f f +, +, 6. + f + f d d 6.4. f, = + = {,, } d d = d + + d log π 8 6. = [, b] f [α, β] C - φ φα =, φβ = b 6. 3 f d = β α f φu φ u du integrtion b substitution. 3 φ C = u = φu 6. β α f u d du du [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 ] 5.4 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, 6.3 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. S g N fξ j j j, j= S g N fξ j j j 6.3 f, g g, f j=

33 φ [, b] [α, β] φ 6.3 R 4. R R L A : R X = A R A R 4 A 38 L A 6.7. L A L A p = L A q Ap = Aq A p = q 6.8. L A R L A P, Q R l P, Q p, q l 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 L A L A R L A 6.. l P, Q l R, S l det P R, P Q det P S, P Q R det 38 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 > 6.. 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 6.4 b sin θ = b b cos θ = b, b., b, b = t,, b = t b, b liner trnsformtion.

34 L A det A = P QRS P, Q, R p, q, r = 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. R C - F : R u, v F u, v = u, v, u, v R 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 4.5 h, k u, b v, b h 6.5 Φh, k := F + h, b + k F, b u, b v, b k 5 3. u, v, u, v, ε, ε ε = t ε, ε , u, v = det u v u, v uv-, b, + u, b,, b + v, + u, b + v F u, v = u, v, u, v u, 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, u, v u v v R C - u, v u, v, u, v uv E f f, d d = f u, v, u, v, u, v du dv 6 the Jcobin. E

35 d d + + := {, +, } 5 6.6, = r cos θ, r sin θ E := { r, θ r, π θ π } r, θ, r, θ = det r θ cos θ r sin θ = det = r sin θ r cos θ 6.5 d d + + = E r θ [ r dr dθ π/ ] + r = r dr π/ + r dθ = π log 3 := {, + }, := {, + } 6.6 E := {r, θ r, π θ π}, E := {r, θ r, π θ π}, d d + + = j E j r dr dθ + r j =, R n C - u,..., u n u,..., u n,..., n u,..., u n 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 6-6 Ω := {,, z +, + z } R 3 6- 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/ L r 4 lim r + πr, lim A r r + πr.

36 {, > } π d d d d, d d = d d 6-4 = f b π f + f d [, b] f > 6-5 C C : γt = t, t t b t, t t C - [, b] t > C = 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 6-8 C - φ φ = φ = φ u du u = t t φ = ψ ψ 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

37 , b] f lim f d ε + +ε f [, b [, f M lim f d M + f d f d f d f d 7.. ε, ε d = [ ] ε = ε ε +, ] ε, ε d =. d = [log ] ε = log log ε = log ε + ε + * 8 6 5/6 improper integrl, ] 3 M M d. 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 > 7.3.

38 k, k 7. d ε, ε k sin ε d = k sin t dt = sin t [, π ] ε + k d = π k sin t dt. f, b [ + ε, b ε ] ε, ε +, + f d = lim ε,ε, ε +ε f d 7.4. ε, ε, ε [ +ε d = ] ε log +ε [ ] ε = log + log + +ε = log ε + log ε log ε log ε ε + d ε, ε, ε = ε = ε + ε +ε d = 5. [ ] ε log + log + = +ε d = 7.5. 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 4 e e =. f k f k f k+ = kf k f k+ f k+ f k+ = m! 7.7. p p Me M = [p] +! [p] p > p [p]+ m = [p] < f f f

39 p lim + p e =. p p p e e, +. p 7.6 / p 7. p [p] +!e / p p e [p] +!e / p p e d p e e / > [, p p e d. [, ] [, + e e p e p e 7.9 [, p = e d = e d + e d = e d 5 7. π 5 the Gussin integrl; Guss Guß, Crl Friedlich , G s > Γ s = e s d e s d s > 7.3. p, q Bp, q = p q d e d = π, e d = π. 7., 7. e M I M := M M M 7.5 I M = e d = R e = M e M e d M e d e d d = e d E M e d d E M := [, M] [, M] R { } 7.6 J R := d d, R :=, R + R,, 6 B b β

40 M M M M E M Π Θ M M E M 7.6. µ σ µ e σ d =, πσ µ e σ d = µ, πσ µ µ e σ d = σ. πσ M 7. r, θ = r cos θ, = r sin θ R { R := r, θ r R, θ π } [ = [, R], π ], / r, θ = r π 7.7 J R = e r R r dr dθ = re r dr dθ R R dr = π e r M M dθ = π e R M M E M M 7. π e M = J M I M J 4 M = π e M, 4 e d = lim M + I M = π r u = µ/ σ M, M M µ e σ d = e u du α j := Mj + µ, j =, M πσ π σ M j + j + j =, M, M M M M µ e σ d = πσ µ M πσ = = σ π σ π µ e σ σu + µ e u du π ue u µ du + π e u du e e + µ π e u d = σ u e u π [ = σ ue u] + e u π du du µ, +. σ du = π u e u du Γ = π. 7.3 u Γ = e d = e u du = π. 7., 7.3 Γ pγ q 7.7. p, q > Bp, q = Γ p + q.

41 v 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, Ε Ε M u ε +, M + Γ p + qbp, 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 lim ε + M + ε Iε, M = Γ pγ q. = uv, = u v ε,m uv { := u, v ε u v M u, M u v ε } u 7., / u, v = u u > Iε, M = e u u p+q v p v q du dv := [ ε, M ε ] [ ε, [ ε], := [ε, M] 7. ε M + ε, ] M M + ε X X = k C k / X j p j > j p j = µ := p j j, σ := p j j µ µ X σ σ 7, b X b P,b P,b = ρ d ρ ρ P,b 7. ρ d =, ρ 7 stochstic vrible, rndom vrible, probbilit distribution, the binomil distribution, the men, the vrince, the stndrd devition, probbilit densit function.

42 ρ µ := ρ d, σ := µ σ ρ := πσ µ e σ µ ρ d ρ 7. ρ µ, σ 5-9 n R R n R 9 B n R := {,..., n R n n R } R n V nr := d d... d n B n R α n := V n j = R j j =,..., n V n R = R n α n α = π, α 3 = 4 π 3 n π 7.8. α n = Γ n +. f,..., n := e n n 7.3 n d d... d n = e dt t = π n. R n e 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... 8 the norml distribution. 9 bll. sphere , 5 f R n r r 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 , ] [, 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 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 r = + s

43 - Yes; No; 3 Yes, = No; 4 No; 5 Yes. - cos Π Π Π Π sin Π Π tn Π 4 Π 4 = Cos = Sin = Tn sec Π Π Π Π csc Π Π Π Π Π cot = sec = csc = cot -3 cos, sin -4 cosh tnh tnh = e e e + e = e e + = e + = + e e +. Π Π 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 -5 α = Tn 5, β = Tn tn4α β = 39 4α β = π 4 + nπ n Tn < 4α β < 4 Tn = 5 π n = Sin = Sin -7 cos n = sin cos n -8 + 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 Tn + 4 log +, 6 log Tn log + 3, 3 3 log ++log + + Tn + Tn mm. R + R Tn R+. R 4 R 8 /πm Tn.8 N =

44 ; z z = F z -6,, 4/5, 3/5; 4/5, 4/5, 4/5; m/ + m. m m m/ + m 3 fh, f, f, = lim = lim h h h h = f, = +,,, =,, f, = +,,, =,. nn+ -7 : n -8 : n m n H m = n+m! m!n! nm -9 f, = f, = f, = f, = f, = f, = ,,, =,, ,,, =,, ,, 3, =,, ,, 3, =,, { ,,, =,, + 3,,, =,. { : f, = b c + d + p + q + m., b, c, d, p, q, m -3 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 -4 f {, + > }, g {, cos cos > }, h {, > } -5 Re e z = e cos, Im e z = e sin m = Re fz =, Im fz =, m = 3 Re fz = 3 3, Im fz = 3 3, m = 3 Re fz = , Im fz = m 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 4-3 df P v = grd f P v v = grd f P df P v grd f P v = kgrd f P k > 4-4 γt P f f f t, t = t 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 - q = ±c;, b

45 4-6 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,, z = rcos θ sin φ, sin θ sin φ, cos φ 4-7 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 d d F, φ = F, φ + F, φ φ φ F, = 4 + C = C = {,,, } < { {, [ b, b ] [b, b ]} < < C {, [ b, b ] b = +, b =. C = ±b, ±b > = F = C = φ F = ψ C 4-3 = 5 59 ξ = F F, η z = F z F, ζ = F F z = b / b [ ] d = 4b u du = 4b u u + sin u., = cos t, b sin t π t π

46 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 5-3 = + + u du + = log + + = sinh 5-4 = sinh t, = + = cosh t. P = cosh t, sinh t log π, 8 = f π π d = d dt = 3π. dt = t = t sin t 5-6 /6, 8 log 6, R 5-7 4πr ρr dr. 4 ππ 3, /45, / [, 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= 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 r M 4πr ρr dr.

47 5-8 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 /4π, π /, 8π /5, /3bcπ, 8/5bcπ. 6- π r R πr sin r R. 3 4πR sin r R. f, = R + πr sinr/r 6. 4 r r 6-3,, +, + + π + π = π + π 5- G, d d = d G, d = [G, ] = d = = G, d G, d =, d d m,..., m n n,..., n = 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 [ ] d dz d = 4 d = 6 3, [ z ] d d dz z [ z ] = 8 d d dz = 6 3. 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 -

48 6-4 + z = f b z > z = F, = f =, := {, b, f } 6. S S = F F d d = [ b f = f + f f = π f + f d. f + f f ] d d f d d 6-5 [, b] C = f ẋ = C I = [, b ] C = f 6-4 π b 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 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. 6-7 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 6-8 φ = φ u du = φ t dt = φ t dt

49 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 + > 7- [, ] e e s s. s > 7., ] 7.5 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 7-4, ] q M, M = { q q < q < p q M p <. p > 7., /] 7.5 / p q d t p q d / s > ˆfs = e st dt = lim e sm = M + s s. 7.9 = + 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 s < ˆfs = e st dt Γ n = n Γ n = = n n... Γ = n!.

50 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 6 5 s + ω ω e st cos ωt dt = s ω 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.