[ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b y, y d y d + a dy d + by = f = dy d y dy y d f f f.4 { + y = 5 y + y = 7 s6 3 s36
.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =, y = = [m], y y [kg/m ] [kg] 3 s366. y + y y = y = 4, y = 3 s367. 9y dy d + 4 = dy d + y = sin 3 s368. + y + z = + 3y z =.3 A =.4 A = [ 3 4 5 4 ] A 3 s369 3 s36 3 s36.5 f P P {,, } T : f fα + β 3 s36
.6 f = e sin y grad f v = 3zi + yj yz k div v i, j, k -, y-, z-.7 d log + d 3.8 f, y = log + y f + f y 3 s363 e d 5 s56 5 s56.9 = r cos θ, y = r sin θ r, θ, y f, y f f r θ r, f θ 5 s563.,,,,, D y ddy D. y =, y = y + 3y + y = 5 s564 5 s565. 3, 4 + y 3 = 45.3 A = 5 s566 P AP = D P D 5 s567.4 d.,. h r.5. D 3 ddy D : 6 s66 a + y a, b > b 6 s66.6. 3
dy d 3 dy = y d 3y = 5 6 s663.7., y, y, y. y y = y + 3y + y = 6 s664.8 yz π : + y + 3z 5 = g :. π. g. 3 = y 5 = z + 3,.9 A = 3 π g. 3 3 9 6 s665, AX = O, Y A = O 3 3 X, Y,..3. K 3 4 5 k.5..5..5 6 s666 f k.5.. 3. 4.4 k=5, p = a + b. G = f k p k, a, b. k= 6 s667.3. 4y 4y 3y = y = 4y + 7 s76.3., a >. I = e a d.33 z = y z =..34 A = 5 3, A A det A. 7 s76 7 s763 7 s764 4
.35 S 3 + y + z =, S P,, 3 S. 7 s765.36,. 7 s766.37 r = i + yj + zk, r = r,. r = r r, = i + j y + k z, i, j, k. 7 s767.38., = y = y. dy d = a by a, b 7 s768.39 e iπ + =., i =. 7 s769.4 f = a,, ε δ. 7 s76.4 f, g. fg, f g + fg.,,..4 A = 7 s76 R R A : A,. 4 ImA. KerA. 3 3 A =.,. 6 4 A =.,. 7 s76.43 + + 3 = R. + + 3 =, A A. A. 3 P, P AP T Λ., T 4,. P Λ. y y = P., y, y 5
5 4,,. 7 s763.44 a = 6,,, b = 3, 4,, c =,,,. a b, c., a, b, a b, a b, a b.. 7 s764.45 A. A = 7 s765.46 y = sin n. [ y y, y,, n. ] [ sin = sin + π ] 7 s766.47 y = sin 3 cos. 7 s767.48,, A, B, C.,.,,,,., A, B, C,,,,., A, B C. B C A B, C. A C B,,, A. A, A,,.. B.,,, B. B, B,,..,, C,,,, A, B..49 A = a a a a a 7 s768 A A = 6
8 s86.5 i i a + bi a, b 8 s86.5 dy d + e =.5 9 + 6 3 d + + sin 7 cos d 8 s863 8 s864.53 f, y = + y e y f + f y 8 s865.54 y = + 9 D + yddy D 8 s866.55 A = a,, B = b, 3 A, B 3 6 b > a > a, b A 3 B 8 s867.56 3 + y 4 = ABCD AB = a S D y C S a A O a B < a a.57 p p = + a < a a + a a a < a > pd 8 s868 pd 7
8 s869 3 5.58 A A = 7 3 4 A A A A 3 + y + 5z = 3 A A 7 + 3y + 4z = 3 + y + z = 3.59 4 + i4 i =.6 = y = dy d = y 3 8 s86 8 s86 8 s86.6 A = 3, B = det A, det B, det C 3 3, C = A 3 C rank C 8 s863.6, y f, y D f + f y = f, y D D = R f, y = 3 ay a f, y g, y = y f + f y g, y f, y 8 s864.63 cosπd cos πd.64 y = + sin, 8 s865.65 8 s866 4 + 8 s867 8
.66 a =,,, b =,, α α < 8 8 s868 5.67 y = m y = m m.68 d y d + 6 dy + 9y = 5e d.69 5 6 8 s869 8 s86 8 s86.7, y r r θ = y = n θ = π/n θ θ =, = r, y = θ i+ = θ i + θ i =,,,, n i+ = y i+ =, y,, y,, n, y n r, θ i, θ i+ 8 s86.7 f, y = loge y + e y.,. f f,..7 f y 9 s96 3 4 5 y 3 5 7 9 7 6 5 4 3 z =. u 3 5 7 9 v 9 s96.73 n A A 3 = O., O. A =., A A. a b n =. A =, A a + da = O. c d 9
3 n = A = O. 9 s963.74., D y,. + y ddy.75 dy = yy d D,. y = a., a. 9 s964 y, lim y. 9 s965.76 3 A,,, B 3,,, C,, 3. e 3.77 lim. [ : lim log e + =.].78 f = + 5 + +, 3.79, dz dt. fd. z = sin cosy, = e t, y = log e 3t 3.8 3. 4,..8 4. 8 6. 9 s966 9 s967 9 s968 9 s969 9 s96 4 + 8, 6. 4 8, 6...8 A. A = 3 9 s96 A. A P,, P AP P.
3. d dt t t = A t t, =, =.83 log = a, Taylor., a. 9 s96 9 s963.84 lim sin =. 9 s964.85 logsin t + t..86 9 s965 e e + e d. 9 s966.87,. dy + y =, = y =. d 9 s967.88 F, P Q, W, W = F P Q. F = 4i 3j + k, P 3,,, Q,, 4 W., i =,,, j =,,, k =,,. 9 s968.89 e, =,., e 3,. 9 s969.9 φ, y, z, A, y, z = A i + A y j + A z k,.. rot grad φ = :. div rot A =, i, j, k 3, y, z 9 s96.9., C. sin sin 3d sin sin 3, cos,.. 9 s96
.9. 9 s96.93. 3 + d.94 y dy d = e.95 A = 3.96 A A = A λ. e log d 9 s963 y =.. 4 λ X. 9 s964 9 s965,. s6.97 f, y f =, f, y. f, y. = / + / y. f = + y.98.,, y, f = e sin y y = y + y 3 y + 6y + y = s6.99 A = 3.. s63 λ λ. A. s64. + y = + z =. s65. a, a, a 3. a =, a =, a 3 = s66
. A 3. A. A = A. 3 A P P AP..3 lim + log e =. s67 log e d. s68.4 X = { : π, R}, Y = {y : y, y R}. f,., R. f : X Y, sin. f. f..5 A = 3. s69 A. A. 3 A. 4 A..6, 7. s6 d dt = a d. y =, y t d dt + b dt. 3. dy, dt = dy d d, y 4 dt. 5.,, ν[m/s] 37.5 ν + 5 [m/s ]., ν = [m/s].75[m/s ],. 75[m/s]= 7[km/h] 6 [s], 7 [m]..7 f. f = e + f = + sin 3 s6 3
s6.8,., y, z, u. y, = y =,. u, y y z,., y, z, u. u, y, z.,,,,. s63.9. π 4 cos d y e y dy d s64. a,. + y + z + w = + y + 4z + w = 4 3 + y + 3z + w = y + w = a s65. y,. y +y = y 7y +y = 6 +5+8 3 y 4y +4y = cos s66.. A = 3 4 5 9 5 s67.3. s68.4 -y. f = 3 + 3y, OABC, O,, A,, B,, C,. OABC 4, OAB 3 f f ds OAB f ds OABC,. ds. s6 4
.5 A =, P a AP = P 5 3 b a, b. s6.6 n. a, b, c, d ad bc, c, f = a + b c + d n. D = {, y y 4, },. + y ddy D s63.7 3y 5z = a y 4z = b 3 + y + 3z =, a, b., a, b,. s64.8 3 A,, 3,., E 3. 3A E A A s65.9 y.,. dy + y cos = sin cos d y =. y π.. n,. n d π 6. + y 3z =, 3 y z = 4. s66 s67 s68. 3 CPU. CPU.8,.,. 5
3 CPU CPU,., 3 CPU,. 3, CPU CPU,,..3. 5 5 a 5 a 3 a. 3 a 4 4 3 3 5 6 3 7. 6 9 5.4 A = a c b d s69 s6 A a + da + ad bce =.. E. A = A A. 3 A =, A n. 4 s6.5 3. m n d π cos n d 3 e d s6.6,, α α. t ft., f = β. s63.7. tan θ = /3. sin θ + cos θ. + tan α/ tan α = + 3, cos α. s64.8 y,, 3,., y + =, + y 4 =, 3 + 3y + 3 =. 6
,, 3 y.,, 3 y. 3 A, 3 B, 3 C, A, B, C. 4 A, B, C ABC. s65.9. y + y = y 6y + 8y = s66.3 y = + sin π V. y O π.3 A = 3 3.. s67 det A. A. 3 A y z = 3. s68.3 f = e, 4 fd. -y, = at sin t, y = a cos t, a >, t π. sin.33 lim 3..34 A = a b c. bc ca ab s69 s6 s6 7
.35 f = log >,., log. f = f. y = f. 3 f log. 4 y = f y = log S..36 A ranka. A = 3 4 s6 a,. a a y a a z = 3 4 w a.37 A =. s6 B = A 4A 4E, BA., E 3. 3 yz P =, y, z, XY Z Q = X, Y, Z T A. T : X Y = y Z yz P, T XY Z H Q. H. z s63.38 yz C l,., P, Q, P Q. = cos θ C : y = sin θ θ π l : z = cos θ + sin θ = t y = 3t z = t P C, Q l, P Q θ t. < t < + P C, Q l, P Q,, P Q..39 y, A,. s64 y tan y = e sin, π/ < < π/ A 8
y; y y = y A, y; y., y. y; y, lim y; y y. π/, y lim y; y.,, π/. s65.4 a. tan = a +,., tan, tan =. 3 s36.4 I = log + y ddy, D :, + y 4 D., log. 3 s36.4 R 3 a =, a = 3, a 3 =,. a, a, a 3 3 y z = α.,. a, a, a 3. a 3 a a..43 A = 3 a a b = a + ya a, y. 3 3 b = A,. b A. 3 A 5 b. 3 3 7 5 4 α 3 s363,. 3 s364 9
.44 f, y. f,, f y,. f h, f, h lim h h 3 s365.45 F, F = f. I = b a f d F., a, b., yf + y J = + y ddy, D :, y, p + y q D F., p, q..46 y 3 s366 y + y + ay =., a. a =,. a >, y lim y =. 3 s367.47 f = e e e + e. f = Taylor 3. 4 s46.48 a, b. I. ab I = a b b a y ddy, D : a + y b < D 4 s46.49 D. D = 3 5 7 4 5 3 4 8 4 s463.5 A,. A = 3 3 5 4 s464
.5 Γt.., Γt = e t d, t >. t > lim e t =. t > Γt + = tγt. n Γn + = n!. 3 Γ, e d e y y dy, y. 4, y r, θ o, y r, θ. o { = r cos θ, y = r sin θ 5 4 3 Γ.5 A. 4 4 A = P AP P. 3 {a n } a =, a = 5, a 3 =,. a n+3 = a n+ + 4a n+ 4a n n a n+3 a n+ = B. a n+ a n+ 6 s66 3 B. 4 a n. a n+ a n 6 s66.53 m.. Im = m + d I, I. r, N,. N r + = n r n + N+ r N+ r + n=
3. n n= n = + 3 4 + 7 s76.54 α. A = α 3 4 4 α. A A. A α. 3 A α., A..55 a, b >. y 7 s76 y a + = C. b,, y C A.. = a cos 4 t. y = b sin 4 t., y a + b., d t dt. C y = y, y = dy d. 3 A. 4 A..56 k. + +3 3 4 = + 3 + 4 = 3k + E 4 +k 3 + 4 = 8 3 + 3 k 4 =.. E. E k.., lim a y 3 E k,. lim + y 8 s86 8 s86 a a a 3.57 yz R 3 f : R 3 R 3 A = a a a 3., A f = A, a 3 a 3 a 33 R 3..
f, A. f, A. 3, + y + z = 5 y + 7z = f A.58 y = y, y = dy. d E y + y cos = sin cos.. y + y cos =. E. 8 s863 3 E, y =. y 4 3 y, lim. 8 s864 3