II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )

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1 II : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) pdf F 6 69 C 7 79 B 8 89 A 9 1 S

2 II A4 A4 Cafe avid Cafe avid 12: 13:3 1 2 (1) C: (2) R: (3) Q: (4) Z: (5) N: (6) : (1) α: (2) β: (3) γ, Γ: (4) δ, : (5) ϵ: (6) ζ: (7) η: (8) θ, Θ: (9) ι: (1) κ: (11) λ, Λ: (12) µ: (13) ν: (14) ξ, Ξ: (15) o: (16) π, Π: (17) ρ: (18) σ, Σ: (19) τ: (2) υ, Υ: (21) ϕ, Φ: (22) χ: (23) ψ, Ψ: (24) ω, Ω: (1) x X x X x X (2) X {x X x } N = {n Z n > } (3) X Y X Y ( (4) A := B A B e := lim n. n n) (5) 1 : ( 2) 1 2 {a n } : M n a n M

3 II : October 2, 214 Version : 1.1 z = x 2y z = sin x sin y x 4, 4 y 4 (1) z = f(x, y) = x y (2) z = f(x, y) = x 2 + y 2 (3) z = f(x, y) = x 2 y 2 (4) z = f(x, y) = sin((x y)π) 1-2. xyz (a, b, c) (A, B, C) A x a A(x a) + B(y b) + C(z c) = B y b = C z c

4 II

5 II : October 9, 214 Version : (1/2) S L S = S(x, y) = xy, L = L(x, y) = 2(x + y) x y 2 V A V = V (x, y, z) = xyz, A = A(x, y, z) = 2(xy + yz + zx) x y z 3 n x 1, x 2,, x n y = f(x 1, x 2,, x n ) n 1: 2 z = f(x, y) xyz R z = xy z = (x 2 + 3y 2 )e 1 x2 y 2. 2 z = f(x, y) 2 z = xy z = (x 2 + 3y 2 )e 1 x2 y 2.

6 II : y = Ax + B 2 z = Ax + By + C 3 w = Ax + By + Cz + n y = A 1 x 1 + A 2 x A n x n 2 1 z = Ax + By + C p1 = ( a 1 b 1 ), p2 = ( a 2 b 2 )

7 II p1 p 2 := a 1 a 2 + b 1 b p1 p 2 = p1 p2 cos θ. p 1 = a b2 1 p 2 = a b2 2 θ p 1 p cos θ 1 p 1 p 2 p 1 p 2 p 1 p 2 2 q 1 q 2 p p q1 = p q 2. p p z = Ax + By 1 z = f(x, y) = Ax ( + ) By A = B = A z = B ( ) ( ) A x z = f(x, y) = Ax + By z = B y ( ) A B ( ) x 1 y 2 z = f(x, y) k E k = {(x, y) f(x, y) = k} 1 1 y = Ax

8 II z = f(x, y) = Ax + By E k E k = {(x, y) Ax + By = k} E k Ax + By = k ( ) ( ) A x Ax By = k = k B y ( ) ( ) A x E k k B y ( ) A 2 E k B 1 k 1, k 2, ( ) E k1 E k2... A B z = Ax + By z = Ax + By 3 z = Ax + By 3 xz yz z = Ax + By xz y = z = Ax A yz x = z = Bx B z = Ax + By 3

9 II z = Ax + By + C 1 ( ) ( ) A x z = + C B y z = Ax + By +C 3 z = Ax + By z +C x 2 y x 2 + 2y 2 sin xy (1) lim (x,y) (,) x 2 + y 2 (2) lim (x,y) (,) 2x 2 + y 2 (3) lim (x,y) (,) xy 2-2. x 3 + y 3 sin(x 2 + y 2 ) (1) x 2 + y 2 ((x, y) (, )) ((x, y) (, )) (2) x + y ((x, y) = (, )) ((x, y) = (, )) Hint: (2) x 2 + y 2 x + y

10 II : October 16, 214 Version : 1.1 (1/9) 2 3 xy R 2 1 [a, b] (a, b) 2 p = (a, b) r > R 2 ( p, r) := { x R 2 x p < r } p r ( p, r) := { x R 2 x p r } a b c d R 2 [a, b] [c, d] := { (x, y) R 2 a x b, c y d } 1 y = 1 x 2 1 x z = f(x, y) (x, y) f z = xy R 2 z = x/y R 2 {x } z = 1 x 2 y 2 (, 1) z = f(x, y) f {k R f(x, y) = k (x, y) } z = xy z = x/y R z = 1 x 2 y 2 [, 1] z = f(x, y) f : R, f : (x, y) z, (x, y) f z f A f : A z = f(x, y) = 1 x 2 y 2 f : R f : [, 1] 1

11 II (x, y) (a, b) (x, y) (a, b) (x, y) (a, b) ( ) ( ) x a y b = (x a) 2 + (y b) 2 (x, y) (a, b) ( ) ( ) x a y b (x, y) (a, b) 1 x a y a (x a) 2 + (y b) 2 (x, y) (a, b) x a y b (a, b) f(x, y) L (x, y) (a, b) f(x, y) L f(x, y) (x, y) (a, b) L L = L lim f(x, y) (x,y) (a,b) f(x, y) = e x+y (x, y) (1, 2) e x+y e 1+2 = e 3 f(x, y) = ex2 +y 2 1 x 2 + y 2 (, ) (x, y) (, ) f(x, y) 1 t = x 2 + y 2 (x, y) (, ) t f(x, y) = e t 1 1 t

12 II lim (x,y) (,) x 2 y 2 x 2 + y 2 y = x (x, y) (, ) x 2 y 2 lim (x,y) (,) x 2 + y 2 = lim x 2 x x 2 = 1. y = x (x, y) (, ) x 2 y 2 lim (x,y) (,) x 2 + y 2 = lim y 2 y y 2 = 1.. x = r cos θ y = r sin θ r > θ < 2π θ r x θ θ x 2 y 2 lim (x,y) (,) x 2 + y 2 = lim r 2 (cos 2 θ sin 2 θ) r r 2 (cos 2 θ + sin 2 = cos 2θ. θ) 1: 3 x2 y 2 x 2 + y 2 [ 1, 1] [ 1, 1] 2x3 y 3 x 2 + y 2 lim (x,y) (,) 2x 3 y 3 x 2 + y 2 r = x 2 + y 2 x r, y r 2x 3 y 3 x 2 + y 2 2x3 + y 3 x 2 + y 2 2r3 + r 3 r 2 = 3r. (x, y) (, ) r lim 2x 3 y 3 (x,y) (,) x 2 + y 2 = lim 3r =. r lim (x,y) (,) 2x 3 y 3 x 2 = + y2 2

13 II lim (x,y) (a,b) f(x, y) = α lim (1) lim {f(x, y) + g(x, y)} = α + β. (x,y) (a,b) (2) lim f(x, y)g(x, y) = αβ. (x,y) (a,b) f(x, y) (3) β lim (x,y) (a,b) g(x, y) = α β. (x,y) (a,b) g(x, y) = β 2 1 (a, b) f(x, y) (x, y) (a, b) f(x, y) f(a, b) lim f(x, y) = f(a, b) (x,y) (a,b) f(x, y) (a, b) f(x, y) (a, b) f(x, y) 1 1 z = f(x, y) = Ax + By + C xy R 2 (a, b) R 2 f(x, y) f(a, b) = A(x a) + B(y b) A x a + B y b. (x, y) (a, b) x a y b f(x, y) f(a, b) f(x, y) g(x, y) (a, b) f(x, y) + g(x, y) f(x, y)g(x, y) (a, b) g(a, b) f(x, y) (a, b) g(x, y) xy, 1 x 2 3xy 2 + y 3 x y x 2 + y2. f(x, y) = ex2 +y 2 1 x 2 + y 2

14 II f(, ) := R 2 2x 3 y 3. f(x, y) = x 2 + y 2 (x, y) (, ) R 2 (x, y) = (, ) x 2 y 2. f(x, y) = x 2 + y 2 (x, y) (, ) R 2 (x, y) = (, ) {(, )} f(, ) R (1) z = x 2 + y 2, (x, y) = (a, b) (2) z = xy, (x, y) = (1, 1) (3) z = (x + y) 3, (x, y) = (1, 2) (Hint: XY (X 2 + Y 2 )/2 X + Y 2 2(X 2 + Y 2 ) ) 3-2. z = f(x, y) = sin x + sin y (1) sin x sin y f(x, y) (a, b) ( ) f(x, y) = sin a + sin b + (x a) cos a + (y b) cos b + o (x a) 2 + (y b) 2 (2) [, 2π] [, 2π] z = sin x + sin y 3

15 II : October 23, 214 Version : 1.1 (1/16) y = f(x) x = a A f(x) f(a) x a A (x a) (1) 1 x a f(x) = f(a) + A(x a) + o( x a ) (2) x a f(x) 1 f(a) + A(x a) o( x a ) 1 f(a) + A(x a) f(x) x = a y = b + A(x a) y = Ax (a, b) z = Ax + By (a, b, c) z c = A(x a) + B(y b) z = c + A(x a) + B(y b) (2) 2

16 II z = f(x, y) (x, y) = (a, b) A, B (x, y) (a, b) f(x, y) = f(a, b) + A(x a) + B(y b) ( (x ) (3) +o a) 2 + (y b) 2 z = f(x, y) f(x, y) (x, y) (a, b) f(x, ( y) 1 f(a, b) + A(x a) + B(y b) (x ) o a) 2 + (y b) 2 f(x, y) (a, b) 1 z = f(a, b) + A(x a) + B(y b) ( (x ) 1 o a) 2 + (y b) f(x, y) 1 f(a, b) + A(x a) + B(y b) E(x, y) := f(x, y) {f(a, b) + A(x a) + B(y b)}

17 II (x, y) (a, b) E(x, y) (x a) 2 + (y b) 2 ( ) E(x, y) = o (x a) 2 + (y b) 2 ((x, y) (a, b)) E(x, y) (x a) 2 ( + (y b) 2 (x ) o a) 2 + (y b) 2 1 z = f(x, y) = Ax + By + C (a, b) f(x, y) f(a, b) = (Ax + By + C) (Aa + Bb + C) = A(x a) + B(y b) (3) 2 z = f(x, y) = x 2 + y 2 (1, 2) x = x 1, y = y 2 f(x, y) = ( x + 1) 2 + ( y + 2) 2 = x + 4 y + x 2 + y 2 f(x, y) = 5 + 2(x 1) + 4(y 2) + (x 1) 2 + (y 2) 2. f(1, 2) = 5 (x, y) (1, 2) (x 1) 2 + (y 2) 2 (x 1)2 + (y 2) 2 = (x 1) 2 + (y 2) 2 (3) 1 2 z = f(x, y) (a, b) (3) 1 z = f(a, b) + A(x a) + B(y b) (4) f(x, y) (a, b)

18 II : z = x 2 + y 2 [, 3] [, 3] [.8, 1.2] [1.8, 2.2] 1 f(x, y) = 5 + 2(x 1) + 4(y 2) [.8, 1.2] [1.8, 2.2] (4) f(x, y) (a, b) 1 z = f(a, b)+ A(x a) + B(y b) 2: z = f(x, y) (a, b). 1 z = f(x, y) = Ax + By + C (a, b) z = Ax + By + C = f(a, b) + A(x a) + B(y b).. 2 z = f(x, y) = x 2 + y 2 (1, 2) z = 5 + 2(x 1) + 4(y 2) (4) ( ) ( ) A x a f(x, y) = f(a, b) + + ( ) B y b ( ) ( ) A x a f(x, y) f(a, b) B y b (5)

19 II ( ) a b ( ) ( ) x x a := y y b f(a, b) z := f(x, y) f(a, b) (5) ( ) ( ) A x z B y ( x, y) z ( x, y) (A, B) ( x, y) (A, B) 1 (A, B) (4) (A, B) f(x, y) (a, b) f(a, b) grad f(a, b) (nabla) grad gradient 1 z = f(x, y) = Ax + By + C (a, b) (A, B) 2 z = f(x, y) = x 2 + y 2 (1, 2) (2, 4) n a = (a 1, a 2,..., a n ) b = (b 1, b 2,..., b n ) a b := a1 b 1 + a 2 b a n b n a := a a = a a a2 n x = (x 1, x 2,..., x n ) x a x a 1 (a, b) ( A, B)

20 II x = (x 1, x 2,..., x n ) y = f( x ) a = (a1, a 2,..., a n ) A = (A 1, A 2,..., A n ) x a f( x ) = f( a ) + A ( x a ) + o( x a ) (6) A = (A 1, A 2,..., A n ) a f( a ) n = 1 (6) (2) f x f y (1) f(x, y) = x 2 + y 2 (2) f(x, y) = x 2 y 4 + 2xy (3) f(x, y) = x2 y 2 (4) f(x, y) = e x+y (5) f(x, y) = 1 x 2 y 2 x 2 + y (1) f(x, y) = 3 x 2 y 2, (x, y) = (1, 1) (2) f(x, y) = sin(x + 2y), (x, y) = (, ) (3) f(x, y) = e x+y, (x, y) = (, ) Hint

21 II : October 3, 214 Version : 1.1 (1/23) z = f(x, y) (x, y) = (a, b) A, B (x, y) (a, b) f(x, y) = f(a, b) + A(x a) + B(y b) ( (x ) (1) +o a) 2 + (y b) 2 z = f(x, y) f(x, y) (1) (1) A B z = f(x, y) (a, b) (1) z = f(x, y) 3 (a, b, f(a, b)) z = f(a, b) + A(x a) + B(y b) z = f(x, y) 3 y = b x = a y = b z = f(x, b) x x = a z = f(a, y) y A B A z = f(x, b) (x, z) = (a, f(a, b)) B z = f(a, y) (y, z) = (b, f(a, b))

22 II A = lim x a f(x, b) f(a, b) x a B = lim y b f(a, y) f(a, b) y b z = f(x, y) z = f(x, y) (x, y) = (a, b) A, B (x, y) (a, b) A = lim x a f(x, b) f(a, b) x a B = lim y b f(a, y) f(a, b) y b (2) A B f(x, y) (a, b) A = f x (a, b), B = f y (a, b) z = f(x, y) f(x, y) A = f x (a, b) x B = f y (a, b) y z = f(x, y) (a, b) f x (a, b), (a, b) f y (a, b), f(x, y) z x = f x (x, y) z y = f y (x, y) z x = f z (x, y) x y = f (x, y) y x z = x f(x, y) y z = y f(x, y) z x = f x (x, y) x z y = f y (x, y) y f x (a, b) f (a, b), x xf(a, b), f x, (x,y)=(a,b) z x (x,y)=(a,b) z = f(x, y) = x 2 y 3 z x y x f(x, y) x z x = (2x)y 3 = 2x 2 y 3

23 II z y x y f(x, y) y z y = x 2 (3y 2 ) = 3x 2 y 2 (1) z = Ax + By + C (2) z = x 2 + y 2 (3) z = sin(x 2 + y 3 ) (1) z x = A z y = B. (2) z x = 2x z y = 2y (3) z x = 2x sin(x 2 + y 3 ) z y = 3y 2 sin(x 2 + y 3 ). vs = z = f(x, y) (a, b) (1) A = f x (a, b) B = f y (a, b). (a, b) (1) x a, y = b f(x, b) f(a, b) = A(x a) + B + o( (x a) 2 ) x a x a = A + f x (a, b) A f y (a, b) o( x a ) x a A α = z = f(x, y) f x (x, y) f y (x, y) (x, y) (a, b) f(x, y) = f(a, b) + f x (a, b)(x a) + f y (a, b)(y b) ( (x ) +o a) 2 + (y b) 2. (3). x := x a, y = y b, A = f x (a, b) B = f y (a, b) ( x, y) (, ) K(x, y) := f(x, y) {f(a, b) + A x + B y} x2 + y 2 y f(x, y) f(a, b) = f(x, y) f(a, y) + f(a, y) f(a, b) = f x (c 1, y)(x a) + f y (a, c 2 )(y b) = f x (c 1, y) x + f y (a, c 2 ) y c 1 x a c 2 y b 1 vs. vs.

24 II K(x, y) = (f x(c 1, y) A) x + (f y (a, c 2 ) B) y x2 + y 2 x f x (c 1, y) A x2 + y + f y y(a, c 2 ) B 2 x2 + y 2 f x (c 1, y) f x (a, b) 1 + f y (a, c 2 ) f y (a, b) 1. ((x, y) (a, b)) (x, y) (a, b) (c 1, c 2 ) (a, b) C 1 C C 1 z = f(x, y) C 1 f x (x, y) f x (x, y) C 1 = z = f(x, y) C 1 C 1 C 2 C 3 C C 1 C z = f(x, y) (x, y) = (x(t), y(t)) dz dt (1) f(x, y) = x 2 + y 2, (x, y) = ( t, 2t) (2) f(x, y) = x + y, (x, y) = (cos t, sin t) (3) f(x, y) = sin xy, (x, y) = (t 2, t) 5-2. Φ : (u, v) (x, y) uv E xy Φ(E) (1) (x, y) = Φ(u, v) = (u + v, uv), E = {(u, v) u >, v > } (2) (x, y) = Φ(u, v) = (v cos u, v sin u), E = {(u, v) u π/2, 1 v 2} (3) (x, y) = Φ(u, v) = (2u + v, u v), E = {(u, v) 1 u 3, 2 v 4}

25 II : October 3, 214 Version : 1.1 (1/3) y = f(x) z = g(y) z = g(f(x)) {g(f(x))} = g (f(x)) f (x) dz dx = dz dy dy dx t xy ( ) ( ) x x(t) C : = y y(t) xy z = f(x, y) ( ) x(t) f t f(x(t), y(t)) = z y(t) t dz dt (t ) x = x(t) x = y(t) z = f(x, y) C 1 t = t (a, b) := (x(t ), y(t )) (a, b) f (A, B) := (f x (a, b), f y (a, b)) t = t t C ( ) ( ) ( ) x x(t + t) x(t ) := y y(t + t) y(t ) x(t) y(t) t v 1 := dx dt (t ), v 2 := dy dt (t )

26 II t x(t + t) x(t ) = v 1 t + o( t) y(t + t) y(t ) = v 2 t + o( t) ( ) ( x = y v 1 v 2 ) ( ) o( t) t + o( t) (1) z = f(x, y) (a, b) (x, y) (a, b) f(x, y) = f(a, b) + A(x a) + B(y b) + o( (x a) 2 + (y b) 2 ). ( x, y) (, ) z :=f(a + x, b + y) f(a, b) =A x + B y + o( ( ) ( ) x 2 + y 2 A x ) = + o( x 2 + y 2 ) B y (1) ( ) ( ) ( ) A v 1 o( t) z = t + B v 2 o( t) + o( x 2 + y 2 ) ( ) ( ) A v 1 = t + A o( t) + B o( t) + o( x 2 + y 2 ) B v 2 ( ) ( ) A v 1 = t + o( t). B v 2 o( x 2 + y 2 ) = o( t) 1 ( ) ( ) dz dt (t z ) = lim t t = A v 1 B v 2 1 o( p x 2 + y 2 ) t = ( f x (a, b) f y (a, b) ) dx dt (t ) dy dt (t ) = f x (a, b) dx dt (t ) + f y (a, b) dy dt (t ) = o(p x 2 + y 2 ) p x2 + y 2 p x2 + y 2 t = o(p x 2 + y 2 q ) p (v 2 x2 + y v2 2 ) + o(1).

27 II (x, y) = (x(t), y(t)) z = ( f(x, y) z(t) = f(x(t), y(t)) t z f (f x, f y ) = x, z ) ( dx (x, y) y dt, dy ) dt dz dt = z dx x dt + z y dy dt = z dx x z dt dy y dt (x, y) = (t, t 3 ) z = x 2 + y 2 dz dt t= (x, y) z (z x, z y ) = (2x, 2y) (x, y) t (1, 3t 2 ) t = 1 (x, y) = (1, 1) ( ) ( ) ( ) ( ) dz 2x dt = t=1 2y 3t 2 = = (x,y)=(1,1) t=1 z = x 2 + y 2 = t 2 + t 6 dz dt = 2t + 6t5 t = 1 dz dt = ( ) ( ) x u 6-1. z = f(x, y) = Φ y v z = F (u, v) := f(x(u, v), y(u, v)) = ( ) x(u, v) y(u, v) f f = (f x, f y )

28 II Φ = ( x u y u x v y v ) F F = (F u, F v ) ( ) x u x v (F u F v ) = (f x f y ) ( ) ( ) ( ) (1) z = f(x, y) = x 2 + y 2 x u u + v, = Φ = y v uv ( ) ( ) ( ) x u u cos v (2) z = f(x, y) = x + y, = Φ = y v u sin v ( ) ( ) ( ) x u 2u + v (3) z = f(x, y) = xy, = Φ = y v u v ( ) ( ) ( ) (4) z = f(x, y) = e x 2y x u u 2 v, = Φ = y v u y u y v 2 ( x, y) (, ) f(a + x, b + y) = f(a, b) + P x + Q y (A x2 + 2B x y + C y 2 ) + o( x 2 + y 2 ) P = f x (a, b) Q = f y (a, b) A = f xx (a, b) B = f xy (a, b) C = f yy (a, b) 7-1. f(x, y) (a, b) 2 (1) z = f(x, y) = x 2 + y 2, (a, b) = (α, β) (2) z = f(x, y) = e xy, (a, b) = (1, 1) (3) z = f(x, y) = 3 x 2 + y 2, (a, b) = (1, 1) (4) z = f(x, y) = 3 x 2 + y 2, (a, b) = (, )

29 II : November 2, 214 Version : 1.1 (11/6) (u, v) C 1 x = x(u, v), y = y(u, v) (u, v) (x, y) ( ) ( ) ( ) u x x(u, v) Φ : = v y y(u, v). ( ) ( ) ( ) r x r cos θ Φ : = θ y r sin θ (x, y) E := {(r, θ) r 1, θ π/2} Φ(E) = = { (x, y) x 2 + y 2 1, x, y } 1. ( ) a b c d ( ) ( ) ( ) ( ) ( ) u x a b u au + bv Φ : = = v y c d v cu + dv 1 1 (a, c) (b, d) ad bc ( ) ( ) ( ) u 1 = u + v, v 1 (1) (2) ( ) ( ) ( ) x a b = u + v y c d (1) uv xy u v u, v (2)

30 II x = x(u, v) y = y(u, v) C 1 ( ) ( ) ( ) u x x(u, v) Φ : = v y y(u, v) ( ) ( ) ( ) p a x(p, q) (p, q) Φ = = q b y(p, q) (u, v) (p, q) (x, y) ( ) ( ) ( ) u u p := v v q ( ) ( ) ( ) ( ) x u p x(u, v) x(p, q) := Φ Φ = y v q y(u, v) y(p, q) x = x(u, v) C 1 P 1, Q 1 (u, v) (p, q) ( u, v) (, ) x(u, v) = x(p, q) + P 1 (u p) + Q 1 (v q) + o( (u p) 2 + (v q) 2 ) (1) x = P 1 u + Q 1 v + o( u 2 + v 2 ) (2) y = y(u, v) C 1 P 2, Q 2 y = P 2 u + Q 2 v + o( u 2 + v 2 ) (3) ( ) ( ) ( x P 1 u + Q 1 v o( ) u = v 2 ) y P 2 u + Q 2 v o( u 2 + v 2 ) ( ) ( ) ( ) ( ) x P 1 u + Q 1 v P 1 P 2 u = y P 2 u + Q 2 v Q 1 Q 2 v (4) (p, q) u v (a, b) x y 1

31 II (2)(3) ( ) ( ) P 1 Q 1 x u (p, q) x v (p, q) = P 2 Q 2 y u (p, q) y v (p, q) 1 ( ) ( ) ( ) u x x(u, v) Φ : = v y y(u, v) ( x u y u x v y v ) (5) Φ Φ (u, v) = (p, q) ( ) x u (p, q) x v (p, q) y u (p, q) y v (p, q) Φ(p, q) (p, q) (5) Φ Φ (6) Φ(p, q) Φ (p, q) det Φ = x u y v x v y u (2)(3) (4) ( ) ( ) x r cos θ. = y r sin θ ( ) ( ) x r x θ cos θ r sin θ = sin θ r cos θ y r y θ (6)

32 II ( ) 2 (r, θ) = (2, π/2) 1 ( ) ( ) ( ) ( ) x a b u au + bv 1. 1 = = y c d v cu + dv ( x u y u x v y v ) = ( a c ) b d C 1 z = f(x, y) (x, y) (u, v) ( ) ( ) ( ) Φ : u x x(u, v) = v y y(u, v) ( ) ( ) u Φ x(u, v) f f(x(u, v), y(u, v)) = z v y(u, v) z = F (u, v) := f(x(u, v), y(u, v)) u v z = F (u, v) = f(x(u, v), y(u, v)) (p, q) F (p, q) = (F u (p, q), F v (p, q))

33 II xy( ) ( ) z = f(x, y) uv z = F (u, v) a p = Φ b q (x, y) = (a, b) z = f(x, y) f(a, b) = (f y (a, b), f y (a, b)) = (A, B) (u, v) = (p, q) z = F (u, v) (A, B ) := (F u (p, q), F v (p, q)) (A, B) ( u, v) ( x, y) z = F (p + u, q + v) F (p, q) = f(a + x, b + y) f(a, b) z = f(x, y) z = A x + B y + o( ( ) ( ) x 2 + y 2 A x ) B y ( x, y) (, ) ( u, v) (, ) z = A u + B v + o( ( ) ( ) u 2 + v 2 A u ) B v (7) (8) (A, B ) Φ (p, q) Φ(p, q) = ( ) ( ) ( ) x P 1 Q 1 u ( u, v) (, ) y P 2 Q 2 v ( P 1 Q 1 P 2 Q 2 ) (4) (7) ( ) ( ) ( ) A x x z = (A B) B y y ( ) ( ) P 1 Q 1 u (A B) P 2 Q 2 v ( ) u = (AP 1 + BP 2 AQ 1 + BQ 2 ) v ( ) ( ) AP 1 + BP 2 u = AQ 1 + BQ 2 v (8) { ( ) A = AP 1 + BP 2 B ( A B ) P 1 Q 1 = (A B) = AQ 1 + BQ 2 P 2 Q 2

34 II ( ) ( ) ( ) x u x(u, v) 6-1 = Φ = z = f(x, y) y v y(u, v) z = F (u, v) = f(x(u, v), y(u, v)) f f = (f x, f y ) ( ) x u x v Φ = y u y v F F = (F u, F v ) F = f Φ ( ) { x u x v F u = f x x u + f y y u (F u F v ) = (f x f y ) y u y v F v = f x x v + f y y v z z u = z x x u + z y y u z v = z x x v + z y y v. 5-1 F u v u (x(u, v), y(u, v)) f f(x(u, v), y(u, v)) = F (u, v) u u (x(u, v), y(u, v)) ( 5-1 F u (f x, f y ) x u, y ) u F u = ( fx f y ) ( xu y u ) z u = z x x z u y = z x y u x u + z y 6-1 ( ) ( ) x r cos θ = z = x 2 + y y r sin θ (z r z θ ) =(z x z y ) =(2x 2y) =(2r ). ( x r y r x θ y θ ( cos θ sin θ z r = 2r z θ = ) ) r sin θ = (2r cos θ 2r sin θ) r cos θ ( cos θ sin θ y u ) r sin θ r cos θ z = x 2 + y 2 = (r cos θ) 2 + (r sin θ) 2 = r 2 z r = 2r z θ =

35 II : November 2, 214 Version : 1.1 (11/13) = C < C 1 < C 2 < < C z = f(x, y) C 1 f x f y f x f y 1 1 f x f y (f x ) x, (f x ) y, (f y ) x, (f y ) y f 2 (f x ) x = ( ) f, (f x ) y = ( ) f x x y x f,... f xx = 2 f x 2, f xy = 2 f y x,... 2 f xxx = ((f x ) x ) x = 3 f x 3, f xyy = ((f x ) y ) y = 3 f y 2 x, (= 8) 3 n C n C n =, 1, 2, 3, f(x, y) C n n f(x, y) C n C n z = x 2 + y 2 z x = 2x, z y = 2y 2 z xx = 2, z xy =, z yx =, z yy = 2. 3 z = x 3 y 5 z x = 3x 2 y 5, z y = 5x 3 y 4 2 z xx = 6xy 5, z xy = 15x 2 y 4, z yx = 15x 2 y 4, z yy = 2x 3 y 3. z = sin xy 2 z x = y 2 cos xy 2, z y = 2xy cos xy 2 2 z xx = y 4 sin xy 2, z xy = 2xy 3 sin xy 2, z yx = 2xy 3 sin xy 2, z yy = 4x 2 y 2 sin xy

36 II z xy = z yx 7-1 z = f(x, y) C 2 f xy = f yx. C C f xxyy = f xyxy = f yxxy = n 2 n n + 1. (a, b) f (a, b) (x, y) x := x a, y = y b {f(x, y) f(x, b)} {f(a, y) f(a, b)} = {f(x, y) f(a, y)} {f(x, b) f(a, b)} (1) K(x, y) := f(x, y) f(x, b) (1) K(x, y) K(a, y) a x a K(x, y) K(a, y) = K x (a, y) x = {f x (a, y) f x (a, b)} x y f x (a, y) f x (a, y) f x (a, b) = f xy (a, b ) y b y b (1) f xy (a, b ) y x L(x, y) := f(x, y) f(a, y) a b x a y b ( ) f yx (a, b ) x y x, y (x, y) (a, b) f xy f yx (a, b ), (a, b ) (a, b) f xy (a, b) = f yx (a, b) 2 1 y = f(x) C f(x) = f(a) + f (a)(x a) + f (c) (x a) 2 2! c x a x a f(x) = f(a) + f (a)(x a) + f (a) (x a) 2 + o( x a 2 ) 2! x := x a f(a + x) = f(a) + f (a) x + f (a) x 2 + o( x 2 ) 2! 2 C

37 II / z = f(x, y) (a, b) C ( x, y) (, ) f(a + x, b + y) = f(a, b)+p x + Q y (A x2 + 2B x y + C y 2 ) 2 +o( x 2 + y 2 ) P = f x (a, b) Q = f y (a, b) A = f xx (a, b) B = f xy (a, b) C = f yy (a, b). z = x 2 + y 2 (a, b) = (1, 2) ( x, y) (, ) (1 + x) 2 + (2 + y) 2 =5 + 2 x + 4 y (2 x2 + 2 x y + 2 y 2 ) + o( x 2 + y 2 ). =5 + 2 x + 4 y + x 2 + y 2 + o( x 2 + y 2 ). z = e x+y (a, b) = (, ) z x = z y = e x+y z xx = z xy = z yy = e x+y (x, y) = ( x, y) (, ) e x+y =1 + x + y (x2 + 2xy + y 2 ) + o(x 2 + y 2 ). =1 + (x + y) + (x + y)2 2! + o(x 2 + y 2 ). 1 e t = 1 + t + t 2 /2 + o(t 2 ) (t ) t = x + y x y (a, b) (a + x, b + y) f t [, 1] ( ) ( ) ( ) x(t) a x = + t y(t) b y F (t) := f(x(t), y(t)) F (t) = d ( ) ( ) dt F (t) = fx (x(t), y(t)) x f y (x(t), y(t)) y = f x (x(t), y(t)) x + f y (x(t), y(t)) y ( ) F (t) = d dt {F (t)} = d dt {f x(x(t), y(t)) x + f y (x(t), y(t)) y} = d dt {f x(x(t), y(t))} x + d dt {f y(x(t), y(t))} y {( ) ( )} {( ) fxx (x(t), y(t)) x fyx (x(t), y(t)) = x + f xy (x(t), y(t)) y f yy (x(t), y(t)) ( )} x y y = f xx (x(t), y(t)) x 2 + 2f xy (x(t), y(t)) y y + f yy (x(t), y(t)) y 2 ( ) 2 o(t 2 ) = o((x + y) 2 ) o(x 2 + y 2 ) (x + y) 2 2(x 2 + y 2 )

38 II f(x, y) C 2 ( ) F (t) t F (t) C 2 1 c (, t) F (t) = F () + F ()t + F (c) t 2 2! t = 1 c (, 1) F (1) = F () + F () + F (c) 2! (x(), y()) = (a, b) (x(1), y(1)) = (a + x, b + y) ( ) f(a + x, b + y) = f(a, b) + f x (a, b) x + f y (a, b) y + F (c). ( ) 2! ( ) F (c) F () ={f xx (x(c), y(c)) f xx (a, b)} x 2 + {f xy (x(c), y(c)) f xy (a, b)} 2 x y + {f yy (x(c), y(c)) f yy (a, b)} y 2. 2 x y x 2 + y 2 F (c) F () f xx (x(c), y(c)) f xx (a, b) x 2 + f xy (x(c), y(c)) f xy (a, b) 2 x y + f yy (x(c), y(c)) f yy (a, b) y 2 f xx (x(c), y(c)) f xx (a, b) x 2 + y 2 + f xy (x(c), y(c)) f xy (a, b) x 2 + y 2 + f yy (x(c), y(c)) f yy (a, b) x 2 + y 2. f C 2 ( x, y) (, ) (x(c), y(c)) (a, b) f (x(c), y(c)) f (a, b) ( = xx, xy, yy) F (c) F () x 2 + y 2 F (c) F () = o( x 2 + y 2 ). ( ) F (c) = F () + o( x 2 + y 2 ). 7-1 f C 2 ( ) z = f(x, y) (a, b) C 2 (a + x, b + y) c (, 1) f(a + x, b + y) = f(a, b) + P x + Q y (A x 2 + 2B x y + C y 2 ) (a, b ) := (a + c x, b + c y) P = f x (a, b) Q = f y (a, b) A = f xx (a, b ) B = f xy (a, b ) C = f yy (a, b ). 3 2

39 II z = f(x, y) (a, b) C n (a + x, b + y) c (, 1) f(a + x, b + y) = f(a, b)+ 1 1! ( x x + y y )f(a, b) + 1 2! ( x x + y y ) 2 f(a, b) (n 1)! ( x x + y y ) n`1 f(a, b) + 1 n! ( x x + y y ) n f(a + c x, b + c y) x := x, y := y ( x x + y y ) j ( x x + y y ) 2 f(a, b) = ( x 2 2 x + 2 x y x y + y 2 2 y)f(a, b) = x 2 f xx (a, b) + 2 x y f xy (a, b) + y 2 f yy (a, b) ( x, y) (, ) f(a + x, b + y) = f(a, b)+ 1 1! ( x x + y y )f(a, b) + 1 2! ( x x + y y ) 2 f(a, b) (a, b) = (, ) n! ( x x + y y ) n f(a, b) + o( x 2 + y 2 n=2 ) A i,j := i+j f x i y j (a, b) = f x xy y(a, b) x i y j f(a + x, b + y) = f(a, b) + A 1, x + A,1 y + 1 A2, x 2 + 2A 1,1 x y + A,2 y 2 2! + 1 A3, x 3 + 3A 2,1 x 2 y + 3A 1,2 x y 2 + A,3 y 3 3! + 1 A4, x 4 + 4A 3,1 x 3 y + 6A 2,2 x 2 y 2 + 4A 1,3 x y 3 + A,4 y 4 4! nc A n, x n + n C 1 A n 1,1 x n 1 y + + n C n A,n y n n! +o( x 2 + y 2 n/2 ) (x, y ) l : ax + by + c = d d = ax + by + c a 2 + b 2. 2

40 II (x, y ) l (p, q) ( x, y) = (x p, y q) z = f(x, y) = ax + by + c l z f(p, q) = f(x, y ) f(p, q) = a(x p) + b(y q) ( ) ( ) a x f(x, y ) =. b y f ( ) ( ) a x θ π = b y cos θ = ± a 2 + b 2 d f(x, y ) = ax + by + c f(x, y) (1) z = f(x, y) = x 2 + 3y 2 (2) z = f(x, y) = xy (3) z = f(x, y) = x 3 y 3 3x + 12y (4) z = f(x, y) = x 2 y 2

41 II : November 27, 214 Version : 1.1 (11/2) y = f(x) x = a f (a) = x = a f (a) < f (a) < f (a) = z = f(x, y) (a, b) [ ] (a, b) (x, y) (a, b) f(x, y) < f(a, b) [f(x, y) > f(a, b)] f(a, b) [ ] f(x, y) C 1 f (a) = 8-1 z = f(x, y) (a, b) f x (a, b) = f y (a, b) =. f x (a, b) = f y (a, b) =. z = f(x, y) = x 2 f x (, ) = f y (, ) = 2 (, ) 2 z = f(x, y) = x 2 y 2 f x (, ) = f y (, ) = y = f(x, y) = x 2 x = f(x, y) = y 2

42 II (1) z = (x 2 + 3y 2 ) (2) z = x 2 y 2 (3) z = x 2 (4) z = (x 2 + y 4 ) x = a y = b (a, b) 8-1 f x (a, b) = f y (a, b) = 2 f(x, y) f(a, b) A x 2 + 2B x y + C y 2 A = f xx (a, b), B = f xy (a, b), C = f yy (a, b) A > AC B 2 > ( A x 2 + 2B x y + C y 2 = A x + B y ) 2 + A AC B2 y 2 > A ( x, y) (, ) (a, b) f A < AC B 2 > f (a, b) A > AC B 2 < At 2 + 2Bt + C = α, β A x 2 + 2B x y + C y 2 = A( x α y)( x β y) ( x, y) (, ) (a, b) f

43 II z = f(x, y) (a, b) f x (a, b) = f y (a, b) = = (a, b) := f xx (a, b)f yy (a, b) {f xy (a, b)} 2 f (a, b) (i) > f xx (a, b) > (a, b) (ii) > f xx (a, b) < (a, b) (iii) < (iv) =. (1) (i) (2) (iii) (3) (iv) (4) (iv) f(x, y) = 1 + 2x 3y 2x 2 xy y 2 { fx (x, y) = 2 4x y = f y (x, y) = 3 x 2y = (x, y) = (1, 2) f xx (x, y) = 4 f xy (x, y) = 1 f yy (x, y) = 2 (1, 2) = 4 ( 2) ( 1) 2 = 7 >. 8-2 (ii) (1, 2) f(1, 2) = 5 f(x, y) = 2x 2 y 2 { fx (x, y) = 4x = f y (x, y) = 2y = (x, y) = (, ) f xx (x, y) = 4 f xy (x, y) = f yy (x, y) = 2 (1, 2) = 4 ( 2) 2 = 7 <. 8-2 (iii) (, ) f f(x, y) = x 2 + y 4 { fx (x, y) = 2x = f y (x, y) = 4y 3 = (x, y) = (, ) f xx (x, y) = 2 f xy (x, y) = f yy (x, y) = 12y 2 (x, y) = 2 12y 2 2 = 24y 2. (, ) = 8-2 (iv) (, ) (x, y) (, ) f(x.y) > = f(, ) f (, )

44 II A = f xx (a, b), B = f xy (a, b) C = f yy (a, b), = AC B (a + x, b + y) (a, b) c (, 1) f(a + x, b + y) = f(a, b) (A x 2 + 2B x y + C y 2 ) (a, b ) = (a+c x, b+c y) A = f xx (a, b ) B = f xy (a, b ) C = f yy (a, b ) ( x, y) (, ) (a, b ) (a, b) A A, B B, C C 1 (i) > f xx (a, b) = A > ( x, y) (, ) A C (B ) 2 > A > ( ) 2 A x 2 + 2B x y + C y 2 = A x + B y A + A C (B ) 2 A y 2 > (a, b) f (ii) (iii) F (t) = At 2 + 2Bt + C = F (t ) > t ( x, y) x = t y A x 2 + 2B x y + C y 2 = y 2 F (t ) > y y = t y f(a + x, b + y) f(a, b) = 1 2 (A x 2 + 2B x y + C y 2 ) + o( x 2 + y 2 ) ( ) = y 2 F (t ) + o( y 2 ) = y 2 F (t ) + o( y2 ) y 2 y o( y2 ) y 2 F (t ) > y f(x, y) (a, b) f(a, b) F (t ) < t f(x, y) (a, b) f(a, b) f(a, b) [ ]p 14 [ ] p181 x 2 + 2y 2 = 1 xy C : g(x, y) = z = f(x, y) z = g(x, y) C (a, b) C f g C (a, b) f (a, b) C f g 1 f C 2 C

45 II f g α R {} ( ) ( ) f x (a, b) g x (a, b) = α f y (a, b) g y (a, b) a, b, c g(a, b) =, f x (a, b) α g x (a, b) =, f y (a, b) α g y (a, b) =. 3 F (x, y, λ) = f(x, y) λ g(x, y) F λ (a, b, α) = F x (a, b, α) = F y (a, b, α) =.

46 II : November 27, 214 Version : 1.1 (11/27) 1 z = f(x, y) xy z = 3 (x, y, z) w = f(x, y, z) K K (x, y) z = f(x, y) R [a, b] [c, d] = {(x, y) a x b, c y d} = [a, b] [c, d] z = f(x, y) [a, b] m a = x < x 1 < < x m 1 < x m = b [c, d] n c = y < y 1 < < y n 1 < y n = d = [a, b] [c, d] mn ij = {(x, y) x i 1 x x i, y j 1 y y j } 1 i m 1 j n

47 II ij (x ij, y ij ) ij Σ := n j=1 j=1 n f(x ij, y ij) (x i x i 1 ) (y j y j 1 ) f(x ij, y ij ) ij (x i x i 1 ) (y j y j 1 ) ij Σ {x i 1 i m} {y j 1 j n} {(x ij, y ij ) 1 i m, 1 j n } z = f(x, y) Σ max{ x 2 x 1,..., x m x m 1, y 2 y 1,..., y n y n 1 } {(x ij, y ij ) 1 i m, 1 j n } I I = f(x, y) dxdy f(x, y) 1 2 f(x, y) = x 2 + y 2 = [, 1] [, 1] [, 1] 5 1 2

48 II f : R f : R { f(x, y) (x, y) f(x, y) := (x, y) / f f(x, y) dxdy := f(x, y) dxdy Area() Area() := dxdy. f(x, y) = 1 1 dxdy f : R f { (x, y) x 2 + y 2 1 } { (x, y) x 2 + y 2 = 1 } { (x, y) x 2 + y 2 < 1 } 1 1

49 II (1) Area( 1 2 ) = f(x, y) dxdy = 1 2 f(x, y) dxdy + 1 f(x, y) dxdy. 2 (2) f(x, y) g(x, y) f(x, y) dxdy (3) α, β {αf(x, y) + βg(x, y)} dxdy = α g(x, y) dxdy. f(x, y) dxdy + β g(x, y) dxdy (1) I = (3) I = 1 1 dx dy 2x x 2 2y (x 2 + y 2 + 1) dy (2) I = y 1 + x 3 dx Hint f(x, y) I = (1) = {(x, y) x 1, y x}, f(x, y) = x 2 y (2) = {(x, y) 1 x 1, y } 1 x 2, f(x, y) = 1 x 2 (3) = { (x, y) x 2 + y 2 1 }, f(x, y) = x 2 y 2 1 y 2 x dy y dx f(x, y) dxdy

50 II : ecember 8, 214 Version : 1.1 (12/4) 1 = { (x, y) R 2 a x b, ϕ 1 (x) y ϕ 2 (x) } ϕ 1 (x) ϕ 2 (x) [a, b] 1 = { (x, y) R 2 c y d, ψ 1 (y) x ψ 2 (y) } 1-1 z = f(x, y) ( b ) ϕ2 (x) f(x, y) dxdy = f(x, y) dy dx (1) a ϕ 1 (x) b a dx ϕ2 (x) ϕ 1 (x) f(x, y) dy (1) ϕ2 (x) ϕ 1 (x) f(x, y) dy x y x x 1

51 II (1) z = f(x, y) 3 xy K K x = x S(x ) ϕ2 (x ) ϕ 1 (x ) f(x, y) dy (1) K b a S(x) dx z = f(x, y) = xy = {(x, y) x 1, y x} f(x, y) = xy 1-1 ϕ 1 (x) = ϕ 2 (x) = x xy dxdy = = 1 1 ( x x 3 2 dx = ) xy dy dx = [ x 4 8 ] 1 1 [ xy 2 2 ] x dx = 1 8. = {(x, y) a x b, ϕ 1 (x) y ϕ 2 (x)} b a {ϕ 2 (x) ϕ 1 (x)} dx f(x, y) = ( b ) ϕ2 (x) Area() = 1 dxdy = 1 dy dx = = a b a ϕ 1(x) b a [ x ] ϕ2(x) ϕ 1 (x) {ϕ 2 (x) ϕ 1 (x)} dx dx I = xe y2 dxdy, = { (x, y) x 1, x 2 y 1 }

52 II ( 1 ) 1 ( 1 ) I = xe y 2 dy dx = x e y 2 dy dx x 2 x 2 e y2 = {(x, y) y 1, x y} 1-1 ( 1 ) y 1 [ ] x 2 y I = = xe y2 dx dy = e y2 dy 2 [ ] 1 1 = e y2 y 2 dx = e y2 = (e 1 1) = 1 1/e. 4 = {(x, y) y 1, y 1 x 1 y} f(x, y) f(x, y) dxdy = 1 1 = {(x, y) 1 x, y x + 1} 2 = {(x, y) x 1, y x + 1} 1 y dy f(x, y) dx y f(x, y) dxdy = f(x, y) dxdy + 1 f(x, y) dxdy 2 = dx x+1 f(x, y) dy + 1 dx 1 x+1 f(x, y) dy (1) I = sin(x 2 + y 2 ) dxdy, = { (x, y) π/2 x 2 + y 2 π } (2) I = (3) I = (x + y)e x y dxdy, = {(x, y) x y 1, x + y 1} xy dxdy, = } {(x, y) x2 4 + y2 1, y

53 II : ecember 8, 214 Version : 1.1 (12/18) E ( ) ( ) ( ) x u x(u, v) Φ : E, = Φ = y v y(u, v) E x = x(u, v) y = y(u, v) C 1 ( ) x u x v Φ = y u y v det Φ = x u y v x v y u det Φ E Φ E f f(x, y)dxdy = f(x(u, v), y(u, v)) det Φ dudv. (1) E E E Φ 1 1 Φ 1 ( ) ( ) ( ) u a b u 1 ( v ( c d v a b c) d) ( 1 ) ( 1 ) ad bc 1 1 ad bc ( ) ( ) ( ) ( u a b u 1 uv ad bc = a v c d v det c ) b d Φ E E ij (1 i m, 1 j n) E ij (u ij, v ij ) ij := Φ(E ij ) J ij := Φ(u ij, v ij ) (x ij, y ij ) := (x(u ij, v ij ), y(u ij, v ij )) Area( ij ) det J ij Area(E ij ) 1 (, ), (a, c), (b, d) ad bc /2

54 II ij (1) f(x, y) dxdy f(x ij, y ij )Area( ij ) i,j i, j 1 i m, 1 j n mn ij f(x ij, y ij )Area( ij ) f(x ij, y ij ) det J ij Area(E ij ) i,j i,j = f(x(u ij, v ij ), y(u ij, v ij )) det J ij Area(E ij ) i,j f(x(u, v), y(u, v)) det Φ dudv. E E E ij (1) = {(x, y) x + 2y 1, x y 1} I = (x y) 2 dxdy 1 u = x + 2y, v = x y uv E = {(u, v) u 1, v 1} x y x = u + 2v y = u v 3 3 ( ) ( ) x u = Φ := 1 ( ) u + 2v y v 3 u v ( ) 1/3 2/3 Φ = det Φ = 1 (1) 1/3 1/3 3 I = = E 1 1 v ( 1 3 dudv = 1 1 [ ] v du = v 2 ) 3 dv du 2 9 du = 4 9.

55 II R > R = { (x, y) x 2 + y 2 R 2} I = ( ) x = Φ y e x2 y 2 dxdy ( ) r := θ ( ) r cos θ r sin θ rθ E = {(r, θ) r R, θ 2π} ( ) cos θ r sin θ Φ = det Φ = r sin θ r cos θ (1) ( 2π ) R 2π R I = e r2 r dudv = re r2 dr dθ = dθ re r2 dr E 2 θ [ ] R ( I = 2π e r2 = 2π ( e R ) ) ( = π 1 e R2) e x2 dx = π Q R := [ R, R] [ R, R] R Q R 2R y 2 dxdy R e x π(1 e R2 ) Q R e x Q R e x 2 y 2 dxdy 2R 2 y 2 dxdy π(1 e 2R2 ) R 2 y 2 dxdy = π. Q R Q R e x 2 y 2 dxdy = R R lim R Q R e x e x2 y 2 dxdy. R R ( R 2 R dy e x2 y 2 dx = e y2 dy e x2 dx = e dx) x2. R R R R 2 I = lim π. R R R e x2 dx =

56 II (1) I = sin(x 2 + y 2 ) dxdy, = { (x, y) π/2 x 2 + y 2 π } (2) I = (3) I = (x + y)e x y dxdy, = {(x, y) x y 1, x + y 1} xy dxdy, = } {(x, y) x2 4 + y2 1, y

57 II : January 15, 214 Version : 1.1 (12/18) E ( ) ( ) ( ) x u x(u, v) Φ : E, = Φ = y v y(u, v) E x = x(u, v) y = y(u, v) C 1 ( ) x u x v Φ = y u y v det Φ = x u y v x v y u det Φ E Φ E f f(x, y)dxdy = f(x(u, v), y(u, v)) det Φ dudv. (1) E E E

58 II Φ 1 1 Φ 1 ( ( ( ) ( ( u a b u 1 1 ( ) v) ( ) c d) v ) 1) a b ad bc c d 1 1 ad bc ( ) ( ) ( ) ( ) u a b u 1 uv ad bc = a b v c d v det c d Φ E E ij (1 i m, 1 j n) E ij (u ij, v ij ) ij := Φ(E ij ) J ij := Φ(u ij, v ij ) (x ij, y ij ) := (x(u ij, v ij ), y(u ij, v ij )) Area( ij ) det J ij Area(E ij ) ij (1) f(x, y) dxdy f(x ij, y ij )Area( ij ) i,j i, j 1 i m, 1 j n mn ij f(x ij, y ij )Area( ij ) f(x ij, y ij ) det J ij Area(E ij ) i,j i,j = f(x(u ij, v ij ), y(u ij, v ij )) det J ij Area(E ij ) i,j f(x(u, v), y(u, v)) det Φ dudv. E E E ij (1) (, ), (a, c), (b, d) ad bc /2

59 II = {(x, y) x + 2y 1, x y 1} I = (x y) 2 dxdy 1 u = x + 2y, v = x y uv E = {(u, v) u 1, v 1} x y x = u + 2v 3 y = u v 3 ( ) ( ) x u = Φ := 1 ( ) u + 2v y v 3 u v ( ) 1/3 2/3 Φ = 1/3 1/3 I = v ( 1 E 3 dudv = [ ] v = du = det Φ = 1 (1) 3 v 2 ) 3 dv du 2 9 du = 4 9. R > R = { (x, y) x 2 + y 2 R 2} I = ( ) ( x r = Φ := y θ) e x2 y 2 dxdy ( ) r cos θ r sin θ rθ E = {(r, θ) r R, θ 2π} ( ) cos θ r sin θ Φ = det Φ = r sin θ r cos θ (1) ( 2π ) R 2π R I = e r2 r dudv = re r2 dr dθ = dθ re r2 dr E

60 II θ [ ] R ( I = 2π e r2 = 2π ( e R ) ) ( = π 1 e R2) e x2 dx = π Q R := [ R, R] [ R, R] R Q R 2R 9-2 e x2 y 2 > 2 y 2 dxdy 2 y 2 dxdy e x2 y 2 dxdy. R e x Q R e x π(1 e R2 ) Q R e x 2R 2 y 2 dxdy π(1 e 2R2 ) R 2 y 2 dxdy = π. Q R Q R e x 2 y 2 dxdy = R R lim R Q R e x R R ( R 2 R dy e x2 y 2 dx = e y2 dy e x2 dx = e dx) x2. R R R R 2 I = lim π R R R e x2 dx = f(x, y) 3 (1) f(x, y) = x 2 + y 2, = { (x, y) x 2 + y 2 R 2} (2) f(x, y) = xy, = { (x, y) x 2 + y 2 R 2} E : a 2 + y2 b 2 + z2 c 2 a = b E x 2 = 1 2

61 II : January 22, 215 Version : 1.1 (1/22) A A A z = f(x, y) C 1 p = (x, y) z = f( p ) p = (x, y) V f ( p ) := f(x, y) = (f x (x, y), f y (x, y)) f f(x, y) = xy [ 2, 2] [ 2, 2] f( p ) f( p ) z = f(x, y) C 1 1 A A q = (a, b) q = (a, b ) A A C 1 C { C : p = p (t) = (x(t), y(t)) ( t 1) p () = q, p (1) = q [, 1] N = t < t 1 < t 2 < < t N = 1 1

62 II C N pk = (x k, y k ) := p (t k ) ( k N) { p k := p k+1 p k = (x k+1 x k, y k+1 y k ) f k := f( p k+1 ) f( p k ) = f(x k+1, y k+1 ) f(x k, y k ) f ( ) ( ) f x (x k, y k ) x k+1 x k f k = f y (x k, y k ) y k+1 y k Vf ( p k ) p k. + o( (x k+1 x k ) 2 + (y k+1 y k ) 2 ). q = q + p + p p N 1 f( q ) = f( q ) + f + f f N 1 f( q ) f( N 1 q ) = f k k= N 1 V f ( p k ) p k k= : A A ( ). C N max { p k k < N} ( ) C V f ( p ) d p Vf ( p ) C

63 II f( q ) f( q ) f Vf ( p ) = f( p ) 13-1 z = f(x, y) C 1 f( p ) q q C 1 p = p (t) t 1 p () = q, p (1) = q f( q ) f( q ) = C f( p ) d p. q q C x y = p = (x, y) ) V ( (u(x, y) p ) = v(x, y) u, v (x, y) C 1 C p k = (x k, y k ) ( ) ( ) x k x k+1 x k p k = := y k y k+1 y k N 1 k= V ( pk ) p k = = N 1 k= N 1 k= ( ) u(x k, y k ) v(x k, y k ) ( ) x k y k u(x k, y k ) x k + v(x k, y k ) y k C N ( ) max { p k k < N} ( ) C V ( p ) d p = C u(x, y) dx + v(x, y) dy Vf ( p ) C C V ( p ) = f( p ) = (f x (x, y), f y (x, y)) f( p ) d p = f x (x, y) dx + f y (x, y) dy C C

64 II C : p (t) = (x(t), y(t)) (α t β) C u(x, y) dx + v(x, y) dy = β α { u(x(t), y(t))x (t) + v(x(t), y(t))y (t) } dt. V ( p ) = (2x, 2y) α > C = C α : p (t) = (t, t α ) ( t 1) x dx+2y dy = 1 C ( 2t 1 + 2tα αt α 1) [ dt = ] t 2 + t 2α 1 = 2. α f(x, y) = x 2 + y 2 V ( p ) = f( p ) 13-2 α V ( p ) = ( y, x) r > C = C r : p (t) = (r cos t, r sin t) ( t 2π) 13-2 y dx + x dy = 2π C { r sin t ( r sin t) + r cos t r cos t} dt 2π = r 2 dt = 2πr 2. V ( p ) f 13-3 f( p (1)) f( p ()) = 2πr 2 = r > V ( p )

65 II C 1 C 1 C : p = p (t) ( t 1) p () = p (1) t < s < 1 p (t) p (s) C t p (t) p (1 t) 13-3 u = u(x, y), v = v(x, y) C C 1 u dx + v dy = (v x u y ) dxdy. (1) C f(x, y) C 2 u = f x v = f y C 2 v x u y = (f y ) x (f x ) y =. C f x (x, y) dx + f y (x, y) dy =. C 13-1 (u, v) = (, x) (u, v) = ( y, ) v x u y = Area() = C x dy = C y dx. a b πab C : p (t) = (a cos t, b sin t) ( t 2π) 13-4 Area() = C x dy = 2π 2π 1 + cos 2t a cos t b cos t dt = ab dt = πab C = C 1 + C 2 + C 3

66 II C 2 y = ϕ(x) u = u(x, y) u dx = u dx + u dx + u dx C C 1 C 2 C 3 = b = = a b u(x, c) dx + a b a = a b u(x, ϕ(x)) dx + {u(x, ϕ(x)) u(x, c)} dx + { } ϕ(x) u y (x, y) dy dx c u y dx dy a a u(a, y) dx x v dy = v x dx dy C ( ) V ( u(x, y) p ) = v(x, y) rot V ( p ) := u y (x, y) + v x (x, y) R V ( p ) V ( p ) d p = rot V ( p ) dxdy C

67 II : January 22, 215 Version : 1.1 (1/15) C 1 z = f(x, y) 3 xy z = P z = Ax + By + C P P E P (x, y, z) E P (x, y, ) := {(x, y, ) (x, y, z) E} P P θ θ < π/2 Area(E) cos θ = Area() Area(E) = Area() 1 cos θ (1) E cos θ A B P n = ±(A, B, 1) P n = (,, 1) 1 n ± n n θ cos θ = n n n n = ±(A + B + ( 1) 1) A 2 + B 2 + ( 1) = 2 ±( 1) A 2 + B θ [, π/2) cos θ > (1) Area(E) = Area() A 2 + B (2) 1 P Ax + By z + C = A < x B A : ; = 1 z C

68 II z = f(x, y) C 1 Area(K) := K = {(x, y, f(x, y)) (x, y) } (f x ) 2 + (f y ) dxdy K r > r xy 1, 2,..., N i (1 i N) (x i, y i ) i = [x i, x i + r] [y i, y i + r] K i K i K i K (x i, y i, f(x i, y i )) P i f C 1 P i i E i K i E i A i := f x (x i, y i ) B i := f y (x i, y i ) P i z = f(x i, y i ) + A i (x x i ) + B i (y y i ) z = A i x + B i y + f(x i, y i ) A i x i B i y i (2) Area(E i ) = Area( i ) A 2 i + B2 i + 1. K K i E i N N Area(E i ) = A 2 i + B2 i + 1 Area( i) i=1 i=1

69 II r N N (f x ) 2 + (f y ) dxdy = lim A 2 i + B2 i + 1 Area( i) r + N i=1 K 1: [, 1] [, 1] f(x, y) = 1 (x 2 + y 2 )/2 R 4πR 2 R x 2 + y 2 + z 2 = R 2 z = ± R 2 x 2 y 2 z = f(x, y) = R 2 x 2 y 2 2 f x = x R2 x 2 y, f y 2 y = R2 x 2 y 2 = { (x, y) x 2 + y 2 R 2} ( ) = 2 = 2R (f x ) 2 + (f y ) 2 ( x) + 1 dxdy = ( y) 2 + (R 2 x 2 y 2 ) R 2 x 2 y 2 dxdy 1 R2 x 2 y dxdy 2 x = r cos θ y = r sin θ rθ E = [, R] [, 2π] ( ) = 2R E 1 2π R2 r rdrdθ = 2R 2 R dθ r R2 r 2 dr [ = 2R 2π ] R R 2 r 2 = 4πR R = 4πR 2.

70 II R { (x, y, z) x 2 + y 2 + z 2 R 2, z } { (x, y, z) ( x R 2 ) } 2 + y 2 R2 4 z = f(x, y) = R 2 x 2 y 2 { ( = (x, y) x R ) } 2 + y 2 R2 2 4, y ( ) = 2 (f x ) 2 + (f y ) dxdy = 2R R2 x 2 y dxdy. 2 x = r cos θ y = r sin θ rθ E = {(r, θ) θ π/2, r R cos θ} 1 2π ( ) = 2R E R2 r rdrdθ = 2R 2 2π [ = 2R ] R cos θ R 2 r 2 dθ 2π ( R cos θ r R2 r 2 dr )dθ = 2R ( R sin θ + R)dθ ( π ) = 2R 2 1 = R 2 (π 2).

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