f(x,y) (x,y) x (x,y), y (x,y) f(x,y) x y f x (x,y),f y (x,y) B p.1/14

B p.1/14

f(x,y) (x,y) x (x,y), y (x,y) f(x,y) x y f x (x,y),f y (x,y) f(x 1,...,x n ) (x 1 x 0,...,x n 0), (x 1,...,x n ) i x i f xi (x 1,...,x n ) B p.1/14

f(x,y) (x,y) x (x,y), y (x,y) f(x,y) x y f x (x,y),f y (x,y) f(x 1,...,x n ) (x 1 x 0,...,x n 0), (x 1,...,x n ) i x i f xi (x 1,...,x n ) [ ] x (x 0,y 0 ) y (x 0,y 0 ) f (x 0,y 0 ) ( ) B p.1/14

B p.2/14

[ ] f(x,y) x(t),y(t) f(x(t),y(t)) df dt = x dx dt + y dy dt B p.3/14

[ ] f(x,y) x(t),y(t) f(x(t),y(t)) df dt = x dx dt + y dy dt [ ] f(x(t),y(t)) = f(x(t 0 ),y(t 0 )) + f x (x(t 0 ),y(t 0 )) (x(t) x(t 0 )) +f y (x(t 0 ),y(t 0 )) (y(t) y(t 0 )) + R (x(t),y(t)) B p.3/14

[ ] f(x,y) x(t),y(t) f(x(t),y(t)) df dt = x dx dt + y dy dt [ ] f(x(t),y(t)) = f(x(t 0 ),y(t 0 )) + f x (x(t 0 ),y(t 0 )) (x(t) x(t 0 )) +f y (x(t 0 ),y(t 0 )) (y(t) y(t 0 )) + R (x(t),y(t)) f(x(t),y(t)) f(x(t 0 ),y(t 0 )) t t 0 =f x (x(t 0 ),y(t 0 )) x(t) x(t 0) t t 0 +f y (x(t 0 ),y(t 0 )) y(t) y(t 0) R(x(t),y(t)) R(x(t),y(t)) t t 0 = {x(t) x(t 2 0 )} 2 +{y(t) y(t 0 )} R(x(t),y(t)) {x(t) x(t 0 )} 2 +{y(t) y(t 0 )} 2 t t 0 + R(x(t),y(t)) t t 0 {x(t) x(t 0 )} 2 +{y(t) y(t 0 )} 2 t t 0 = {x (t 0 )(t t 0 )+R 1 (t)} 2 +{y (t 0 )(t t 0 )+R 2 (t)} 2 t t 0 B p.3/14

[ ] f(x 1,...,x n ) x 1 (t),...,x n (t) f(x 1 (t),...,x n (t)) f = f x1 x 1 + + f xn x n B p.4/14

[ ] f(x 1,...,x n ) x 1 (t),...,x n (t) f(x 1 (t),...,x n (t)) f = f x1 x 1 + + f xn x n [ ] f(x,y) x(u,v),y(u,v) f(x(u,v),y(u,v)) u = x x u + y y u, v = x x v + y y v B p.4/14

[ ] f(x 1,...,x n ) x 1 (t),...,x n (t) f(x 1 (t),...,x n (t)) f = f x1 x 1 + + f xn x n [ ] f(x,y) x(u,v),y(u,v) f(x(u,v),y(u,v)) u = x x u + y y u, v = x x v + y y v f(x 1,...,x m ) x 1 (y 1,...,y n ),...,x m (y 1,...,y n ) f(x 1 (y 1,...,y n ),...,x m (y 1,...,y n )) = x 1 + + x m y i x 1 y i x m y i B p.4/14

B p.5/14

[ ] f(x) x(u,v) f(x(u,v)) u = df dx x u, v = df dx x v B p.6/14

[ ] f(x) x(u,v) f(x(u,v)) u = df dx x u, v = df dx x v f(x) x(y 1,...,y n ) f(x(y 1,...,y n )) = df y i dx x y i B p.6/14

B p.7/14

[ ] n = α β (x,y) f(x,y) f(x + tα,y + tβ) f(x,y) D n f(x,y) = lim t 0 t B p.8/14

[ ] n = α β (x,y) f(x,y) f(x + tα,y + tβ) f(x,y) D n f(x,y) = lim t 0 t = α x + β y = (f x,f y ) α β B p.8/14

[ ] n = α β (x,y) f(x,y) f(x + tα,y + tβ) f(x,y) D n f(x,y) = lim t 0 t = α x + β y = (f x,f y ) α β f(x,y) n n B p.8/14

[ ] n = α β (x,y) f(x,y) f(x + tα,y + tβ) f(x,y) D n f(x,y) = lim t 0 t = α x + β y = (f x,f y ) α β f(x,y) n n (f x,f y ) gradf B p.8/14

[ ] n = α β (x,y) f(x,y) f(x + tα,y + tβ) f(x,y) D n f(x,y) = lim t 0 t = α x + β y = (f x,f y ) α β f(x,y) n n (f x,f y ) gradf D n f(x,y) f(x,y) n gradf B p.8/14

Dnf(x,y) (x,y) n B p.9/14

B p.10/14

[ ] f(x,y) f x (x,y),f y (x,y) f 2 f x = 2 x x, 2 f y x = y x, 2 f x y = x y, 2 f y = 2 y y B p.11/14

[ ] f(x,y) f x (x,y),f y (x,y) f 2 f x = 2 x x, 2 f y x = y x, 2 f x y = x y, 2 f y = 2 y f xx,f xy,f yx,f yy ( f xy,f yx x,y ) y B p.11/14

[ ] f(x,y) f x (x,y),f y (x,y) f 2 f x = 2 x x, 2 f y x = y x, 2 f x y = x y, 2 f y = 2 y f xx,f xy,f yx,f yy ( f xy,f yx x,y ) [ ] f xy f yx f xy = f yx y B p.11/14

[ ] f(x,y) f x (x,y),f y (x,y) f 2 f x = 2 x x, 2 f y x = y x, 2 f x y = x y, 2 f y = 2 y f xx,f xy,f yx,f yy ( f xy,f yx x,y ) [ ] f xy f yx f xy = f yx f(x 1,...,x n ) k f (= f xi1 x x ik x ik ) ( ) i1 y B p.11/14

B p.12/14

[ ] x = r cosθ,y = r sin θ ( ) f(x,y) f = 2 f x + 2 f 2 y = 2 f 2 r + 1 2 r r + 1 2 f r 2 θ 2 B p.13/14

[ ] x = r cosθ,y = r sin θ ( ) f(x,y) f = 2 f x + 2 f 2 y = 2 f 2 r + 1 2 r r + 1 2 f r 2 θ 2 x = r cosθ,y = r sin θ,z = z ( ) f(x,y,z) f = 2 f x + 2 f 2 y + 2 f 2 z = 2 f 2 r + 1 2 r r + 1 2 f r 2 θ + 2 f 2 z 2 B p.13/14

[ ] x = r cosθ,y = r sin θ ( ) f(x,y) f = 2 f x + 2 f 2 y = 2 f 2 r + 1 2 r r + 1 2 f r 2 θ 2 x = r cosθ,y = r sin θ,z = z ( ) f(x,y,z) f = 2 f x + 2 f 2 y + 2 f 2 z = 2 f 2 r + 1 2 r r + 1 2 f r 2 θ + 2 f 2 z 2 x = r sin θ cos ϕ,y = r sin θ sin ϕ,z = r cosθ ( ) f(x,y,z) f = 1 ( r 2 ) + 1 ( sin θ ) + 1 2 f r 2 r r r 2 sin θ θ θ r 2 sin 2 θ ϕ 2 B p.13/14

195 198 206 ( A) B p.14/14