- Page 3 and 4: R R n
- Page 5: 2 × 2 SO(
- Page 10 and 11: R A R a ∈ A δa > 0
- Page 13 and 14: R n X τ X τ X
- Page 15: R n {xk}k∈N R
- Page 18 and 19: lim x→+∞ x∈D f(x) − ℓ = 0
- Page 20 and 21: f : D → Rm D ⊆ Rn c D f
- Page 23 and 24: α > 0 ⎧ | sin(xy) − xy|
- Page 25 and 26: y 2 y = 0 +∞
- Page 27 and 28: D ⊆ RN {fn}n∈N fn : D →
- Page 29 and 30: x ∈ [0, +∞[ fn f(x) =
- Page 31: f ′ n(x) = ne −n2 x 2 (1 − 2n
- Page 34 and 35: x = 0 1/n x > 0 e
- Page 37 and 38: a ∈ N a ≥ 2 ∞ an
- Page 39 and 40: π 4 /96 3π 2 = 9 4 π2 + 72 π2
- Page 41: ∞ π 2 1 = , 4 π (2n + 1) 2
- Page 44 and 45: u(x) = 4 π ∞ sup n2 n
- Page 46 and 47: v = (v1, ..., vm) df(p)(v) = Jac
- Page 48 and 49: y ∂xf(x, y) = 2 |xy| sign(xy)
- Page 50 and 51: |f(ρ cos θ, ρ sin θ)| = arctan
- Page 52 and 53: ¯x = 0 y = 0
- Page 54 and 55: H h, k ∈ R n H(h, k)
- Page 56 and 57:
f(x, y) = f(y, x) f ∂x
- Page 59 and 60:
λ, µ ∈ R f(x, y) = x 3 + x
- Page 61 and 62:
fn g : R 2 → R g(x, y) =
- Page 63:
(1, 1, 1) g(1, 1, 1) = 4 g(
- Page 66 and 67:
f(x, y) = x 3 − 6xy + 3y 2 + 3x
- Page 69 and 70:
f, ϕ : Ω → R ¯x f
- Page 71 and 72:
Zhk h, k ∈ Z g(x, y) = 0
- Page 73 and 74:
λ = 0 yz = 0 x = 0
- Page 75 and 76:
⎧ ⎪⎨ 2x + λ2x/9 = 0, 2y + 2
- Page 77 and 78:
g1(x, y, z) = x 2 +y 2 −1 g2(x,
- Page 79:
√ 2 (0, 1/2) f f(0, 1
- Page 82 and 83:
df(x, y) = ∂xf(x, y) dx + ∂yf(x
- Page 84 and 85:
I g(m1, m2, m3) = 0 g(m1, m2, m
- Page 87 and 88:
Γ = {(x, y) ∈ R 2 : x 3 − 3xy
- Page 89 and 90:
Γ ∂yf(x, y) = 0 f(x, y) =
- Page 91 and 92:
x 4 − 4xy + y 4 = 0
- Page 93 and 94:
∂xz(0, 0) = ∂yz(0, 0) = 0 ∂x
- Page 95 and 96:
q = lim x→∞ ϕ(x) − m∗ x =
- Page 97 and 98:
f(x1, ..., xn) dx1 ... dxn, D D
- Page 99 and 100:
y = 4 − x A C y = 4 − 3x A
- Page 101 and 102:
V = π b a f 2 (x) z = f(x,
- Page 103 and 104:
α + t 2 dt = t α + t 2 − α
- Page 105 and 106:
V = 2π 2π r s(s cos φ +
- Page 107 and 108:
|y| x 2 + y 2 D √ 2
- Page 109:
x y = sin x 0 ≤ x ≤ π A
- Page 112 and 113:
P1 P2 P3 √ 1 3 df(−1, 0) =
- Page 114 and 115:
f ∈ C ∞ C 1 −1 < k
- Page 116 and 117:
[0, 1] fn = nxe−nx2 N N
- Page 118 and 119:
x = k |k| > 4 x = 4
- Page 120 and 121:
4(x 2 + y 2 − x) 3 = 27(x 2 + y 2
- Page 122 and 123:
(x 2 + y 2 ) 3 = 4x 2 y 2
- Page 125:
(x 2 + y 2 ) 3 = 4x 2 y 2
- Page 128 and 129:
P Q A Q(x, y) = x
- Page 131 and 132:
I := x 2 dxdy. D = {(x, y) ∈ R 2
- Page 133 and 134:
ω 1 F 1 F ω 1
- Page 135 and 136:
⎛ det ⎝ n1 n2 n3 ∂xψ1 ∂xψ
- Page 137 and 138:
√ √ 2π 12 Area(L) := 12 :=
- Page 139 and 140:
S rotF · ˆn dσ = +∂S
- Page 141 and 142:
S − θ ∈ [−π/4, π/4
- Page 143:
(1, 0, 0) ϕ(0, 0) ⎛ det
- Page 146 and 147:
a ∈ R f(x) = e−x /x [a, +
- Page 148 and 149:
k → +∞ ⌊
- Page 150 and 151:
[−1, 0] log |x|χS(x) [−1,
- Page 152 and 153:
α, β : [a, b] → D α β D
- Page 154 and 155:
ω = ωx dx + ωy dy ∂yωx
- Page 156 and 157:
U(1, 1) = 2 ∂xU(x, y) = y/x+2
- Page 158 and 159:
x = x(y) Γc dx y) = −M
- Page 160 and 161:
R2 \ {y = 0} h(x, y)ω(x, y) = y
- Page 162 and 163:
2e −x/2 (e x − √ y) = d, d
- Page 165 and 166:
˙x = e t−x /x x(α) = 1 α
- Page 167 and 168:
g(t) = 0 f(x) = −1/x x = 0
- Page 169 and 170:
y ′ + y = sin x y IV − 16y = 1
- Page 171 and 172:
u(0) = 0 2 = c k (ek − 1) + 1
- Page 173:
A A −1 ⎛
- Page 176 and 177:
Ω (0, 0) (x0, z0) V (
- Page 178 and 179:
y = ˙x + 2x − 4t2 ˙y = 3x + 2y
- Page 180 and 181:
⎧ ⎪⎨ y = f(p, C), ∂pf(p, C
- Page 182 and 183:
F (x, y, p) = 0 c = 0 dy = p dx
- Page 185 and 186:
α ∈ R x ≥ 0 y ′ (x) =
- Page 187 and 188:
˙y = 1 − x 2 y 2 y(0) = α α
- Page 189:
y ′ = y 4 − x2 y(1) = 0 (1 + |
- Page 192 and 193:
µ 2 + 2µ + n 2 = 0 ∆ = 4(1
- Page 194 and 195:
u(t, x) = T (t)X(x) ˙ T (t)X(x
- Page 197 and 198:
[0, π] ut − 5uxx = 0, 0 ≤
- Page 199 and 200:
λ ∈ R X(x) λ ∈ R 2
- Page 201 and 202:
a0 = 1 π an = 2 π π 0 π 0
- Page 203 and 204:
f : R 2 → R D := {(x, y)
- Page 205 and 206:
α < 2 N n=1 fn(x, y, α)
- Page 207 and 208:
cos(3θ) = cos θ(1 − 4 sin 2 θ)
- Page 209 and 210:
div F (x, y, z) = ∂xF1 + ∂yF2
- Page 211 and 212:
2π 0 cos θ sin θ dθ = 1 2 2π
- Page 213 and 214:
a (−∞, 0] a
- Page 215 and 216:
0 < log 1 − a 1 + a < 2 |1 −
- Page 217:
ω = n ∈ Z ω λ < 0 λ
- Page 220 and 221:
f(x, y, z) = x 2 +y 2 +z 2 x+y+z
- Page 222 and 223:
1 Jac ϕ(θ) 1 ds =
- Page 224 and 225:
f f(x, y) > 0 f |(x, y)| →
- Page 226 and 227:
1 A = √ 1−y2 √ 2/2 − √ 1
- Page 228 and 229:
Jac ϕ(θ, y) = ⎛ ⎜ ⎝ − y
- Page 230 and 231:
ω2 = det 2 B1 + det 2 B2 + det 2 B
- Page 232 and 233:
ω1(x, y) = yx y−1 dx + x y log x
- Page 234 and 235:
λω = ( 1 x2 − y) dx + (y − x)
- Page 236 and 237:
f Γ f(x,
- Page 238 and 239:
˜ F A A A (
- Page 240 and 241:
dσ = det 2 B1 + det 2 B2 + det 2 B
- Page 242 and 243:
p, q ∈ N q 2π 0 cos p θ sin
- Page 244 and 245:
t ∈ I LK > 0 f(t, y1) −
- Page 246 and 247:
ϕ ∈ C1 (I, Y ) ϕ ′ (t) ≤
- Page 248 and 249:
C R 2 x = ξ, y = ξη 1 N(
- Page 250 and 251:
F Fc := {(x, y) ∈ A : F (x, y
- Page 253 and 254:
K Y K = R C Y = R n
- Page 255 and 256:
a, b ∈ C 0 (I, K) I R A(t) ∈
- Page 257 and 258:
y1 y2 c ′ 1y ′ 1 + c
- Page 259 and 260:
uj n y(t) = e (t
- Page 261 and 262:
(t) = a(t)e αt α ∈ R a(t)
- Page 263 and 264:
y ′ = f(x, y) A
- Page 265 and 266:
t > 0 t = es y(es ) = u(s)
- Page 267:
Ω ⊆ R n F : Ω → R n C 1
- Page 270 and 271:
2 × 2 b = 0 ˙x y
- Page 273 and 274:
a∂ttu(t, x) + b∂tu(t, x) + c∂
- Page 275 and 276:
λ λ λn = − d2 + 4c 2 n 2
- Page 277:
u(t, x) = 1 π f(s) ds π 0 e −
- Page 280 and 281:
SO(3) λ1 = 1 e 2α = 1 α = 0
- Page 283 and 284:
R 2 γ 1 x 2 +y 2 = 1
- Page 285 and 286:
d −1 1 sinh x = √ dx x2 + 1 d
- Page 287 and 288:
z ∈ C e z ∞ z = Re(z)(cos(Im(z)
- Page 289:
z 1 + cos z cos = ± 2 2 z 1