D xy D (x, y) z = f(x, y) f D (2 ) (x, y, z) f R z = 1 x 2 y 2 {(x, y); x 2 +y 2 1} x 2 +y 2 +z 2 = 1 1 z (x, y) R 2 z = x 2 y

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1 D xy D (x, y z = f(x, y f D (2 (x, y, z f R z = x 2 y 2 {(x, y; x 2 +y 2 } x 2 +y 2 +z 2 = z 5.2. (x, y R 2 z = x 2 y + 3 (2,,, (, 3,, 3 (,, 5.3 (. (3 ( (a, b, c A : (x, y, z P : (x, y, x AP a(x x + b(y y + c(z z = ax + by + cz = d (a, b, c, d (

2 第5章 2 * x ^ 2 + x * y + 2 * y ^ 2, 8x, -2, 2<, 8y, -3, 3<D In[]:= In[6]:= 偏微分 ^ 3 * y + 4 * x * y ^ 3 - x * y, 8x, -.5,.5<, 8y, -.5,.5<D Out[]= Out[6]= 図 5.2: z = x3 y + 4xy 3 xy 図 5.: z = 4x4 + xy + y 2 In[3]:= Plot3D[Exp[-x^2 - y^2], {x, -2, 2}, {y, -2, 2}] In[7]:= Plot3D[Sin[Pi x] Sin[Pi y], {x, -2, 2}, {y, -2, 2}] Out[7]= Out[3]= 図 5.3: z = e x 2 y 2 図 5.4: z = sin(πx sin(πy

3 D R 2 D z = f(x, y (a, b D (a, b D {(x n, y n } n= f(x n, y n f(a, b (n 5.4. f(x, y = x2 x 2 + y 2 y (x, y (, f(, y = (y f(x, y x (x, y (, f(x, = (x f(x, y y = x (x, y (, f(x, x = 2 (x f(x, y 2 f(,

4 z = f(x, y xy D (a, b D f(a +, b f(a, b lim f (a, b x f x (a, b f x(a, b y = b x f(x, b x = a f(x, y = x 2 y 3 f (a + 2 b 3 a 2 b 3 (a, b x (a + 2 a 2 b 3 = 2ab 3 y f(a, y y y = b f(a, b + k f(b lim k k y b f(a, y f(a, b y b f (a, b y f y (a, b f y(a, b f D x ( y f D x ( y D (x, y f x (x, y = f x(x, y D (x, y f y (x, y = f y(x, y x ( y x y ( f x x x ( f y y y f f

5 f(x, y = x 2 y 3 f x (x, y = f f(x +, y f(x, y x(x, y (x + 2 x 2 y 3 = 2xy 3 f y (x, y = f f(x, y + k f(x, y y(x, y k x 2 (y + k3 y 3 k = 3x 2 y 2. (x + 2 y 3 x 2 y 3 x 2 (y + k 3 x 2 y ( f(x, y = e ax+by x, y f x, f y (2 f(x, y = e ax cos(by x, y (3 f(x, y = e ax sin(x + y x, y x, y ( ( ( 5.2. ( f(x, y = arctan y x x, y f x, f y (2 f(x, y = x y (x >, y R x, y f x, f y ( tan f(x, y = y x x cos 2 f(x, y f x(x, y = x( y = y x x 2 cos 2 f(x, y = + tan2 f(x, y = + y2 x = x2 + y 2 x2 + y 2 f 2 x 2 x 2 x (x, y = y x 2 f x (x, y = y x 2 + y 2 tan f(x, y = y x x cos 2 f(x, y f y(x, y = ( y = y x x cos 2 f(x, y = x2 + y 2 x2 + y 2 f x 2 x 2 y (x, y = x f y(x, y = (2 log(f(x, y = y log x x, y f x (x, y f(x, y = y x, f y (x, y f(x, y = log x f x, f y f x (x, y = y x xy = yx y, f y (x, y = x y log x. x x 2 + y z = f(x, y ( f(x, y = (x 2 + y 2 yf x = xf y ( x (2 f(x, y = xf x + yf y = y

6 y = f(x f (a y = f(x (a, f(a 3 z = f(x, y y = b z = f(x, b (a, b, f(a, b x a y = b + t z f(a, b f x (a, b z = f(x, y x = a z = f(a, y (a, b, f(a, b x a y = b + t z f(a, b f y (a, b, (a, b, f(a, b f x (a, b f y (a, b l m n l + nf x (a, b =, m + nf y (a, b = f x (a, b f y (a, b z = x a f(x, y (a, b, f(a, b (x, y, z y b z c f x (a, b(x a + f y (a, b(y b + ( (z f(a, b =, z = f(a, b + f x (a, b(x a + f y (a, b(y b 5.4. z = f(x, y = x2 y2 4 9 (, 3, 5 6

7 z = f(z, y f x = f x, f y = f y (f x x = x ( f, (f x y = x y ( f, (f y x = x x ( f, (f y y = y y ( f y f xx 2 f x 2, f xy 2 f y x, f yx 2 f x y, f yy 2 f y 2 f xxx, f xxy 5.5. ( f(x, y = e ax+by f xx, f xy, f yx, f yy f xy = f yx ( f(x, y = e ax cos(by f xx, f xy, f yx, f yy f xy = f yx 5.6. f(x, y 2 2 f xy, f yx, k δ = (f(x +, y + k f(x +, y (f(x, y + k f(x, y y, k x ϕ(x = f(x, y + k f(x, y δ = ϕ(x + ϕ(x δ = ϕ (c = {f x (c, y + k f x (c, y} c x x + y δ = f xy (c, c 2 k c 2 y y + k f xy lim,k δ k = f xy(x, y δ = (f(x +, y + k f(x, y + k (f(x +, y f(x, y y ψ(y = f(x +, y f(x, y

8 8 5 δ = ψ(y + k ψ(y δ = ψ (d k = {f y (x +, d f y (x, d }k d y y + k x δ = f yx (d 2, d k d 2 x x + f yx lim,k δ k = f yx(x, y 2 f xy (x, y = f yx (x, y 5.6. f xx + f yy = ( ( f(x, y = log(x 2 + y 2, (2 f(x, y = x x 2 + y 2, y2 (3 f(x, y = ex2 sin(2xy 5.7. a R (t, x, t >, x R f(t, x f(t, x = e (x a2 2t t f t = 2 f xx

9 5.5. ( ( d dt g(ϕ(t = (g(ϕ(t = g (pi(tpi (t 5.7. ( d ( (pt + q 3 (αt + β 4 = 3p(pt + q 2 (αt + β 4 + 4α(pt + q 3 (αt + β 3. dt (2 ϕ(t, ψ(t F (t = ϕ(t 3 ψ(t 2 F (t = (ϕ(t 3 ψ(t 2 + ϕ(t 3 (ψ(t 2 = 3ϕ(t 2 ϕ (tψ(t 2 + ϕ(t 3 2ψ(tψ (t f(x, y = x 3 y 2 F (t = f(ϕ(t, ψ(t f x = 3x 2 y 2, f y = x 3 2y F (t = d dt f(ϕ(t, ψ(t = f x(ϕ(t, ψ(t ϕ (t + f y (ϕ(t, ψ(t ψ (t 5.8 (. f(x, y R 2 ϕ(t, ψ(t t R F (t = f(ϕ(t, ψ(t t d dt F (t = d dt f(ϕ(t, ψ(t = f x(ϕ(t, ψ(t ϕ (t + f y (ϕ(t, ψ(t ψ (t.. F (t + F (t = f(ϕ(t +, ψ(t + f(ϕ(t, ψ(t 34 = f(ϕ(t +, ψ(t + f(ϕ(t, ψ(t f(ϕ(t +, ψ(t + f(ϕ(t, ψ(t + + f(ϕ(t, ψ(t + f(ϕ(t, ψ(t

10 5 k = ϕ(t + ϕ(t, k 2 = ψ(t + ψ(t f(ϕ(t +, ψ(t + f(ϕ(t, ψ(t = f(ϕ(t + k, ψ(t + f(ϕ(t, ψ(t + ϕ(t + ϕ(t k + f(ϕ(t, ψ(t + k 2 f(ϕ(t, ψ(t k 2 f x (ϕ(t, ψ(t ϕ (t + f y (ϕ(t, ψ(t ψ (t ψ(t + ψ(t 5.8. f(x, y = y 2 e x F (t = f(cos t, sin t t F (t = e cos t sin 2 t 5.9. f(x, y 2 ϕ(t, ψ(t t 2 F (t = f(ϕ(t, ψ(t F (t =f xx (ϕ(t, ψ(t (ϕ (t 2 + f x (ϕ(t, ψ(tϕ (t + 2f xy (ϕ(t, ψ(t ϕ (tψ (t + f yy (ϕ(t, ψ(t (ψ (t 2 + f y (ϕ(t, ψ(tψ (t. ϕ(t + ϕ(t = ψ(t + ψ(t =

11 5.5. ( x, y 2 2 ϕ, ψ x = ϕ(u, v, y = ψ(u, v. f(ϕ(u, v, ψ(u, v u, v u v v 5.9 (. f(x, y R 2 ϕ(u, v, ψ(u, v u, v R F (u, v = f(ϕ(u, v, ψ(u, v u, v u f(ϕ(u, v, ψ(u, v = f x(ϕ(u, v, ψ(u, v ϕ u (u, v + f y(ϕ(u, v, ψ(u, v ψ (u, v, u v f(ϕ(u, v, ψ(u, v = f x(ϕ(u, v, ψ(u, v ϕ v (u, v + f y(ϕ(u, v, ψ(u, v ψ (u, v. v 5. (. x = r cos θ, y = r sin θ (r, θ < 2π f(x, y f(r cos θ, r sin θ r f(r cos θ, r sin θ = f x(r cos θ, r sin θ cos θ + f y (r cos θ, r sin θ sin θ, θ f(r cos θ, r sin θ = f x(r cos θ, r sin θr sin θ + f y (r cos θ, r sin θr cos θ. 2 x f y f = 2 2 r f + 2 r r f + 2 r 2 θ f. 2

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x [ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),

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. sinh x sinh x) = e x e x = ex e x = sinh x 3) y = cosh x, y = sinh x y = e x, y = e x 6 sinhx) coshx) 4 y-axis x-axis : y = cosh x, y = s . 00 3 9 [] sinh x = ex e x, cosh x = ex + e x ) sinh cosh 4 hyperbolic) hyperbola) = 3 cosh x cosh x) = e x + e x = cosh x ) . sinh x sinh x) = e x e x = ex e x = sinh x 3) y = cosh x, y = sinh x y =

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