5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h 0 g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)

5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h 0 f(a + h, b) f(a, b) h............................................................... ( ) f(x, y) (a, b) x A (a, b) x (a, b) (x, y) (x, y) f x, y f(x, y) x x (paritial derivative) y ( ) B = lim h 0 f(a, b + h) f(a, b) h.............................................................. ( ) x x x y (ordinary differentiation) z = f(x, y) Lagrange, Leibniz, Cauchy z x, f x, f x (x, y), z y, f y, f y (x, y) z, f (x) z x, f x, z y, f dy y dx, df dx D x z, D x f, D y z, D y f Df (a, b) f f x (a, b), x f x=a,y=b x f f y (a, b), y f x=a,y=b y (x,y)=(a,b) (x,y)=(a,b) D dx x rund f partial f (over) partial x x

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h 0 g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)......................................................................... ( ) g (a) a g(x) g(x) y = b f(x, y) f x (a, b) f(x, y) (a, b) x z f(a, b) = f x (a, b)(x a), y = b x a y = b + 0 t z f(a, b) f x (a, b) (a, b) y z f(a, b) = f y (a, b)(y b), x = a x a 0 y = b + s z f(a, b) f y (a, b) ( ) g (x) = f x (x, b) f x (x, b) x x b y f x (x, y) f(x, y) (x, y) x f y (x, y) x y ( ), ( ) (a, b) x y (a, b) x θ { x = a + h cos θ (h R) y = b + h sin θ (a, b) θ f(a + h cos θ, b + h sin θ) f(a, b) lim h 0 h θ x 0 y π θ θ

5.. ( ), ( ) (x, y) f(x + x, y) f(x, y) f x (x, y) = lim x 0 x x y x f(x, y) y x y (...) x x (...) y y f(x, y) = xy y x f x (x, y) = y (x) x = y x y f y (x, y) = x (y) y = x f(x, y) = x y 3xy + x y + f x (x, y) = y (x ) x 3y (x) x + (x ) x + ( y + ) x = xy 3y + x f y (x, y) = x (y ) y 3x(y ) y (y) y + (x + ) y = x y 6xy f(x, y) = sin(x + 3y) f x (x, y) = sin(x + 3y) f y (x, y) = 3 sin(x + 3y) f(x, y) = (sin x + e y )(cos y + e x ) f(x, y) = x y f x (x, y) = (sin x + e y ) x (cos y + e x ) + (sin x + e y )(cos y + e x ) x = cos x(cos y + e x ) + (sin x + e y )e x f y (x, y) = (sin x + e y ) y (cos y + e x ) + (sin x + e y )(cos y + e x ) y = e y (cos y + e x ) (sin x + e y ) sin y f x (x, y) = yx y f y (x, y) = x y log x f(x, y) f(x, y) = g(x) + h(y) f(x, y) x y f x (x, y) = g (x) f y (x, y) = h (y) f x x f y y h(y) 0 f(x, y) x f y (x, y) = h (y) = 0 y g(x) 0 h(y) c g(x) f(x, y) = g(x)h(y) x y f x (x, y) = g (x)h(y) f y (x, y) = g(x)h (y) 3

f(x, y) = f(y, x) x, y f x (x, y) x y f y (x, y) f(x, y) f(y, x) = x x x y f(x, y) y : f(x, y) = x y + xy f x (x, y) = xy + y x y xy + x f y (x, y) f(x, y) = g(x + y) f(x, y) x + y f x (x, y) = f y (x, y) = g (x + y) f(x, y) x + y = t f x f y f(x, y) = g(ax + by + c) f x (x, y) = ag (ax + by + c) f y (x, y) = bg (ax + by + c) f x : f y = a : b f(x, y) = g(h(x, y)) f x (x, y) = h x (x, y)g (h(x, y)) f y (x, y) = h y (x, y)g (h(x, y)) h(x, y) = t f(x, y) f(x, y) : f(x, y) = (xy) + (xy) h(x, y) = xy g(t) = t + t f(x, y) = g(h(x, y)) f x (x, y) = y (t + ) = y(xy + ) f y (x, y) = x (t + ) = x(xy + ) f(x, y) ( ), ( ) xy f(x, y) = x + y ((x, y) (0, 0)) 0 ((x, y) = (0, 0)) f x (x, y) = y(y x ) (x + y ) f y (x, y) = x(x y ) (x + y ) f(h, 0) f(0, 0) 0 0 f x (0, 0) = lim = lim = 0 h 0 h h 0 h f(0, h) f(0, 0) 0 0 f y (0, 0) = lim = lim = 0 h 0 h h 0 h (x, y) R x y 4

5..3 ( ), ( ) f(x, y) x y { (x 0) f(x, y) = 0 (x < 0) y x y 0 y y y (x, y) f y (x, y) = 0 x x = 0 y y f(x, y) = x + y x x +b x = 0 f x (x, y) x = 0 y y + a y f y (x, y) (x, y) y f(x, y) x y z = x + y (x 0) z = x + y (x 0) y z = y x f(x, y) = 3 xg(y) g(y) y f y (x, y) = 3 xg (y) (x, y) f x f x (x, y) = g(y) 3 x g(y) 0 x = 0 0 g(y) = 0 y f(x, y) = 0 f x (x, y) = 0 (0, g(y)) g(y) 0 x f(x, y) = x g(y) x, y { 0 (x = 0 y = 0 xy = 0) f(x, y) = ( xy 0) z = x y f x (0, 0) = f y (0, 0) = 0 x, y x y x, y 5

x f y y f x xy f(x, y) = x + y ((x, y) (0, 0)) 0 ((x, y) = (0, 0)) (x, y) f x, f y y = mx (m 0) f(x, y) = xy x + y = mx ( + m )x = m + m 0 x y ((x, y) (0, 0)) f(x, y) = (x + y ) 3 0 ((x, y) = (0, 0)) (x, y) x y f(x, y) = = r cos θ sin θ (x + y ) 3 r 0 r cos θ sin θ 0 x, y cos θ 0, sin θ 0 r cos θ sin θ y = mx x y m x 4 f(x, y) = = (x + y ) 3 ( + m ) 3 x = m x 3 ( + m ) 3 x x = 0 0.5 0.8 0.7 0.6 0.5 0 0.4 0.3 0. 0. 0.5 0 0 0 0 0 5. 5.0 5.. y x = A + ε ( y = f(x 0 + x) f(x 0 )) x 0 ε 0 A = f (x 0 ) y = A x + ε x = f (x 0 ) x + ε x 6

ε x ( x, y) (x, y) (dx, dy) dy = f (x)dx dx x dy y 5.. z = A x + B y + ερ z = f(x 0 + x, y 0 + y) f(x 0, y 0 ) ρ = ( x) + ( y) ρ 0 ε 0 (x 0 + x, y 0 + y) (x 0, y 0 ) f(x, y) (x 0, y 0 ) (x 0, y 0, z 0 ) z 0 = f(x 0, y 0 ) (dx, dy, dz) dz = Adx + Bdy dx, dy, dz x, y, z (x, y, z) dx = x x 0 z z 0 = A(x x 0 ) + B(y y 0 ) f(x, y) (x 0, y 0, z 0 ) 5..3 f(x, y) (x 0, y 0 ) (x 0 + x, y 0 + y) x (x 0, y 0 ) y = 0, ρ = x z = A x + ε x x z x = A ± ε ± x x 0 ε 0 A (x 0, y 0 ) x (x 0, y 0 ) x A f x (x 0, y 0 ) B = f y (x 0, y 0 ) θ x = ρ cos θ y = ρ sin θ 7

z = Aρ cos θ + Bρ sin θ + ερ ρ z ρ = A cos θ + B sin θ + ε ρ 0 A cos θ + B sin θ = f x (x 0, y 0 ) cos θ + f y (x 0, y 0 ) sin θ θ f x, f y : y = A x + ε x y x = A + ε x 0 A = f (x 0 ) z = A x + B y + ερ ρ z ρ = A x ρ + B y ρ + ε ρ 0 0 z A x B y ρ 0 z, x. y dx, dy, dz : 5..4 5..4, 5..5 (x 0, y 0, z 0 = f(x 0, y 0 )) z z 0 = f x (x 0, y 0 )(x x 0 ) + f y (x 0, y 0 )(y y 0 ) (x 0, y 0, z 0 ) x y x x 0 y = + 0 t z x y z = y 0 z 0 x 0 y 0 z 0 f x (x 0, y 0 ) 0 + s f y (x 0, y 0 ) 8

x x 0 0 y = + 0 t + s z y 0 z 0 f x (x 0, y 0 ) f y (x 0, y 0 ) t = x x 0, s = y y 0 x, y x = x, y = y z x, y θ θ θ : f(x, y) = xy (, ) θ = π/4 y = x f x = y, f y = x π/4 cos π 4 + sin π 4 = + = x = t cos π 4 = t y = t sin π 4 = t (, ) t = z t z = (t ) z = t (x, y, z) x 0 t/ y = 0 + t/ = z t (, ) z = x + y 0 0 + y = x x = y = t/ x = y = t x 0 y = 0 + t z t t 5..5 ax + by + cz + d = 0 9

t (a b c) (x, y, z ), (x, y, z ) ax + by + cz + d = 0 ax + by + cz + d = 0 a(x x ) + b(y y ) + c(z z ) = 0 a b c x x y y z z 0 z z 0 = f x (x 0, y 0 )(x x 0 ) + f y (x 0, y 0 )(y y 0 ) f x (x 0, y 0 ) f y (x 0, y 0 ) f x (x 0, y 0 ) f y (x 0, y 0 ) xy z ( ) f x (x 0, y 0 ) f y (x 0, y 0 ) t ( f x (x 0, y 0 ) f y (x 0, y 0 )) f(x, y) (0 0) 5.3. ( ) f y (x 0, y 0 ) f x (x 0, y 0 ) z x x 0 = f y (x 0, y 0 )t y y 0 = f x (x 0, y 0 )t z = z 0 + f x (x 0, y 0 )(x x 0 ) + f y (x 0, y 0 )(y y 0 ) = z 0 + f x (x 0, y 0 )f y (x 0, y 0 )t f x (x 0, y 0 )f y (x 0, y 0 )t = z 0 z f(x, y) : k 0

f(x, y) = k (x, y) y x f(x, y), f x (x, y), f y (x, y) k (f y, f x ) f(x, y) = x + y f x (x, y) = x f y (x, y) = y (x, y) (x y) (y x) 5..6 () (x, y) = (r cos θ, r sin θ) f(x, y) = x sin θ ( x sin tan y ) (x 0) f(x, y) = x 0 (x = 0) θ.5 0.5 0 0.5.5 0 0 :

x = 0 y = 0 f(x, y) = 0 f(x, y) x, y x x y y xy z = 0 xy f(x, y) 5..7 () f x, f y C f x, f y f x, f y x y xy f(x, y) = x + y ((x, y) (0, 0)) 0 ((x, y) = (0, 0)) (x, y) x, y f x y(y x ) f x (x, y) = (x + y ) ((x, y) (0, 0)) 0 ((x, y) = (0, 0)) (x, y) (0, 0) y(y x ) (x + y ) = r sin θ(r sin θ r cos θ) r 4 = sin θ(sin θ cos θ) r (x, y) (0, 0) r 0 θ 0 f x (x, y) f y : 5.3 dx, dy, dz dz = f f dx + x y dy f x, f y

5.4 f x, f y x, y ( ) f x ( x ) f y ( y ) f y ( x ) f x y = f x = f xx(x, y) = f y = f yy(x, y) = f x y = f xy(x, y) = f y x = f yx(x, y) : f xx f x : ( ) f f y x x y f x, y y x f xy f yx p.8: 5..3 (a, b) f x, f y f xy, f yx f xy (a, b) = f yx (a, b) f xy, f yx D f xy = f yx x y f xy = f yx f(x, y) = x f y (x, y) = 0 f yx = 0 f x (x, y) x = 0 x = 0 f xy (x, y) f yx f x, f y { x (x 0) f(x, y) = 0 (x < 0) f yx f xy x = 0 f xy, f yx xy x y f(x, y) = x + y ((x, y) (0, 0)) 0 ((x, y) = (0, 0)) p.58 59 f xy (0, 0) = f yx (0, 0) = f xy = f yx f xx, f yy n n n f r x s y (r + s = n) n + 3

f(x, y) n f(x, y) C n C n n C 5.5 5..7 5..0 5.5. x = x(t) y = y(t) x, y t f(x, y) t z = f(x, y) = f(x(t), y(t)) = f (t) dz dt = z x dx dt + z y dy dt f t (x(t), y(t)) xy f(x, y) t x = t, y = b b z = f(x, y) dz dt = z x f(x, y) y = b f x (x, b) y x = at + a 0, y = bt + b 0 (a b) dz dt = a z x + b z y (a b) a + b a x = a cos t, y = a sin t ( dz dt = a sin t z ) z + cos t x y z = xy dz dt = a(x cos t y sin t) = a (cos t sin t) = a cos t f (t) = a sin t cos t dz dt = a (sin t cos t) = a cos t y = x dz dx = z dx x dx + z dy y dx = z z + x x y z = xy 4

dz dx = y + x y = x z = xy = x 3 y = x y + x = x + x = 3x = (x 3 ) x dz dx z x 5.5. x, y xy (x, y) (u, v) x = x(u, v) y = y(u, v) z = f(x, y) = f(x(u, v), y(u, v)) = f (u, v) z u = z x x u + z y y u z v = z x x v + z y y v Lagrange z u = z x x u + z y y u z v = z x x v + z y y v z x y z u z = u x u y x z v v v y ( z u z v ) = ( x u x v y u y v ) ( z x z y ) (Jacobian) J : : 3: (u, v) (x, y) x = au + bv y = cu + dv x u = a, x v = b, y u = c, y v = d z u = az x + cz y z v = bz x + dz y x = u + v y = u v z = f(x, y) = x y 5

z x = x, z y = y z u = z x x u + z y y u = (x y) z v = z x x v + z y y v = (x + y) u = x + y v = x y z u = 4v z v = 4u f(x, y) = x y = (x + y)(x y) = 4uv z u, z v π/4 z = x y z = xy x = r cos θ y = r sin θ ( ) cos θ sin θ J = r sin θ r cos θ z r = z x cos θ + z y sin θ z θ = z x r sin θ + z y r cos θ J J cos θ sin θ = r cos θ sin θ r (r, θ) (x, y) z x = z r cos θ z θ sin θ r z y = z r sin θ + z θ cos θ r f(x, y) = g(x + y ) f(x, y) = g(x, y) h(x, y) g(x, y), h(x, y) 6

z = x + y z x, z y z r, z θ f(x, y) f (u, v) f (r, θ) z = x + y = r z r = z θ = 0 J z x, z y z = x xy y x + y z x, z y z = r cos θ r cos θ sin θ r sin θ r cos θ + r sin = cos θ cos θ sin θ sin θ = cos θ sin θ θ z r = 0 z θ = (sin θ + cos θ) z x, z y (r, θ) (x, y) J z x = sin θ (sin θ + cos θ) = sin θ ( cos θ sin θ + cos θ sin θ) r r = r 4 r sin θ((r cos θ)(r sin θ) + r cos θ r sin θ) = (x + y ) y(xy + x y ) r cos θ, r sin θ f r, f θ f x, f y x y f r, f θ r θ f(r, θ) = r g(r, θ) = sin θ g g(r, θ) = θ f r (r, θ) = f θ (r, θ) = 0 g r (r, θ) = 0 g θ (r, θ) = cos θ f r θ x g f r (r, θ) f θ f r, f θ f x, f y f r, f θ (r, θ) (x, y) r (x y) θ (y x) (dimension) x, y cm km r θ 7

y x y y y = x y x f θ r f r f(x, y) = x + y = r f r = 5.6 5.7 5.. p.77 5.. f(a + x, b + y) (a, b) x, y 5..4 f(x, y) C (a, b) f(a + x, b + y) = f(a, b) + f x (a, b)x + f y (a, b)y + ( fxx (a, b)x + f xy (a, b)xy + f yy (a, b)y )! + ( fxxx (a, b)x 3 + 3f xxy (a, b)x y + 3f xyy (a, b)xy + f yyy (a, b)y 3) 3! + 0... n n n nc k f xk y n k(a, b)xk y n k k=0 f (...) (a, b) (a, b) (a+θx, b+θy) θ θ < f(a+x, b+y) 5.8 4, 5 5. 5.0 5.9 3 5.9. (a, b) D (x, y) D (x, y) (a, b) (x, y) f(x, y) < f(a, b) (a, b) f(a, b) f(x, y) > f(a, b) (a, b) f(a, b) 8

<, >, (a, b) 5.9. f(x, y) f x (a, b) = f y (a, b) = 0 (a, b) f(x, y) xy z = k x f x (a, b) = 0 (a, b) (saddle point) : (a, b) (a, b) (a, b) (a, b) 5.9.3 f x (a, b) = f y (a, b) = 0 f(a + x, b + y) f(a, b) = +! ( fxx (a, b)x + f xy (a, b)xy + f yy (a, b)y ) + (a, b) (a + x, b + y) x, y lim x 0 x n = 0 (n > ) x x, y f(a+x, b+y) f(a, b) f xx (a, b) = A f yy (a, b) = B f xy (a, b) = H f xx (a, b)x + f xy (a, b)xy + f yy (a, b)y = Ax + Hxy + By = P 9

P A = B = 0 H = 0 P 0 H 0 P = Hxy P x, y y = x P = Hx y = x P = Hx (x, y) = (0, 0) H (a, b) A, B 0 A 0 [ ( P = A x + H ) ] A y AB H + A y ( x + H A y ) 0 H AB p.84 5.3.4 = AB H = f xx f yy f xy > 0 < 0 > 0 A y 0 y = 0 0 x + H A y = 0 y = 0 x = y = 0 P x = y = 0 0 P A : = AB H > 0 AB > 0 B A B 0 A < 0 : P < 0 f(a+x, b+y) f(a, b) < 0 (x, y) (0, 0) (x, y) f(a, b) A > 0 : f(a, b) < 0 A < 0 A = C [ ( P = A x + H ) ] [ ( ) ] [ ( ) ] H H A y C y = A x + A + C y x + A C y P x, y P ( ) H x + A + C y = 0 ( ) H x + A C y = 0 (a, b) A = B = 0, H 0 = AB H = H < 0 0

= 0 P = A (x + HA ) y x + H A y = 0 0 P = 0 f(a + x, b + y) f(a, b) A = B = H = 0 x, y x, y 0 0 0 ( ) ( ) P = Ax + Hxy + By A H x = (x y) H B y x, y n n 5.9.4 z = x + y (x, y) (0, 0) x + y > 0 (0, 0) (0, 0) f x (x, y) = 0 f y (x, y) = 0 (x, y) (a, b) f xx (a, b) = A f yy (a, b) = B f xy (a, b) = H = AB H > 0: A < 0 A > 0 < 0: (a, b) = 0:

5.9.5 f(x, y) = x + y x + y f x (x, y) = x f y (x, y) = y f x = 0 x = 0 f y = 0 y = 0 (0, 0) f xx (x, y) = f yy (x, y) = f xy (x, y) = 0 = 0 = 4 > 0 A = > 0 x, y 0 + 0 = 0 f(x, y) = x 4 + y 4 f x (x, y) = 4x 3 f y (x, y) = 4y 3 (0, 0) f xx (x, y) = x f yy (x, y) = y f xy (x, y) = 0 (0, 0) A = B = H = 0 f xxx (x, y) = 4x f yyy (x, y) = 4y 0 (0, 0) 0 f x 4(x, y) = f y 4(x, y) = 4 0 (0, 0) f(x, y) = x y xy + 4xy 3y f x (x, y) = xy y + 4y = y(x y + ) f y (x, y) = x 4xy + 4x 3 f x = 0 y = 0 x y + = 0 y = 0 f y f y = x + 4x 3 = 0 x = ± 7 ( ± 7, 0) x y + = 0 y = x +

f y = x 4x(x + ) + 4x 3 = 0 3x + 4x + 3 = 0 f xx (x, y) = y f yy (x, y) = 4x f xy (x, y) = x 4y + 4 ( ± 7, 0) f xx ( ± 7, 0) = 0 f yy ( ± 7, 0) = 8 4 7 f xy ( ± 7, 0) = ± 7 = AB H = 0 (8 4 7) (± 7) = 7 < 0... f(x, y) = (x y)(y x) f x (x, y) = xy 3x + y f y (x, y) = x y 3y + x xy 3x + y = 0 x y 3y + x = 0 (x, y) = (0, 0) 0 0 x 0, y 0 x( 0) y( 0) x y 3x 3 + xy = 0 x y 3y 3 + xy = 0 3(y 3 x 3 ) = 0 y = x x 3 3x + x = 0 x(x 3x + ) = x(x )(x ) = 0 y = x (x, y) = (0, 0), (, ), (/, /) (0, 0) f xx (x, y) = y 6x f yy (x, y) = x 6y f xy (x, y) = 4xy + (0, 0) A = B = 0 H = 0 (, ) A = B = 4 H = 5 = ( 4) ( 4) 5 = 9 > 0 0 (/, /) A = B = 5/ H = 3

= ( 5/) ( 5/) = 9/4 > 0 /6 : y = x x = y f(x, y) = 0 (x y) (y x) f(x, y) +.5 + 0.5 + 0 0.5 + 0.5 0 0.5.5 f(x, y) = (x y) f x (x, y) = x y f y (x, y) = y x x = y f xx (x, y) = f yy (x, y) = f xy (x, y) = = ( ) = 0 0 x y = t f(x, y) = t t = 0 0 y = x : f(x, y) = x y f(x, y) = (y x )(y x ) = 0 f x (x, y) = x(4x 3y) f y (x, y) = y 3x f x = 0 x = 0 y = (4/3)x f y = 0 y = 0 f y = 8 3 x 3x = 3 x = 0 x = 0 (0, 0) f xx (x, y) = 6(4x y) f yy (x, y) = f xy (x, y) = 6x 4

(0, 0) = 0 0 = 0 x = 0 f(x, y) f(0, y) = y y = mx f(x, y) = (mx x )(mx x ) = x (m x)(m x) x x < m x (m x) (m x) x 0 f(x, y) > 0 f(0, 0) = 0 (0, 0) f(x, y) (0, 0) x < y < x f(x, y) < 0 y < x, y > x f(x, y) > 0 (0, 0) (0, 0) (0, 0) y = ( + α)x (0 < α < ) f(x, y) f(x, y) = (y x )(y x ) = αx (α )x = α(α )x 4 α(α ) < 0 x = 0 0 (0, 0) (0, 0) 3.5.5 + 0.5 0 0.5 +.5 0.5 0 0.5.5 f(x, y) = sin x cos y f x (x, y) = cos x cos y f y (x, y) = sin x sin y f x = 0 cos x = 0 cos y = 0 x = π + nπ y = π + mπ n, m f y = 0 sin x = 0 sin y = 0 x = nπ y = mπ ( π ) (x, y) = nπ, + mπ ( π ) (x, y) = + nπ, mπ... (A)... (B) f xx (x, y) = f yy (x, y) = sin x cos y f xy (x, y) = cos x sin y = ( sin x cos y) ( cos x sin y) = (cos x sin y + sin x cos y)(sin x cos y cos x sin y) = sin(x + y) sin(x y) 5

( sin (n + m)π + π ) ( sin (n m)π π ) sin ( (n + m)π + π ) ( sin (n m)π + π )... (A )... (B ) sin ± n + m, n m (A ) sin + = < 0 (B ) sin + = > 0 m + n 6

5.0 5.0.4 5.0. a b = b a 5.0. y = f(x) a x dx y dy dy = f (a)dx........................................................................... ( ) f dy df dx, dy xy (a, f(a)) ( ) y = f(x) dx, dy f (a) ( ) dx dy dx = f (a) ( ) dy = dy dx dx dx, dy dy dy, dx dx x = a a f(x) f (a), f (x) dy df(a) dx, x=a dx 7

: d d f(x) f(x) dx dx ( ) n dn f d n f(x) dxn dx z = g(y), y = f(x), h(x) = g(f(x)) h (x) = g (f(x))f (x) dz dx = dz dy dy dx y = f(x) dy 0 dz = g (y)dy dy = f (x)dx dy dz = g (y)f (x)dx = g (f(x))f (x)dx dx z y y x z x z = x dx dy dy dx = dx dx = dx dy = dy dx dy = f (x)dx f (x) y x x y dx = f (x) dy 5.0.3 z = f(x, y) (x, y) dz = f x (x, y)dx + f y (x, y)dy = z z dx + x y dy........................................... ( ) 8

df = f f dx + dy x y dx dz dx = z x + z dy y dx dx dx = dy x y dx dy dx = 0 dz dx = z x y z x z x y x y = ax + b dz dx = z x + a z y y = ax + b x z x z x y a z y ( ) (f x, f y ) (dx, dy) x = (x, y) f (x) = (f x, f y ) dx = (dx, dy) x, y x y ( ) dz = f (x) dx ( ) ( ) dx dz dx = f (x) dz = dz dx dx f (x) f gradf grad (gradient) 5.0.4 dz = f x (x, y)dx + f y (x, y)dy = z z dx + x y dy........................................... ( ) dx, dy, dz + a 9

dx =... dx dz = f x dx + f y dy, dz = f u du + f v dv dz f x dx + f y dy = f u du + f v dv z dz z z = f(x, y) dz df, z = x + y dz d(x + y ) 0 dx, dy, dz 0 dx = 0 d(f(x, y)) 0 dz dx = dz dx dz dx z, x dz dx = 0 z x dz dx z x z x x 5.0.3 ( ) dy = 0 dz = z x dx dx dz dx = z dx dy=0 x dx = z x x z y ( ) f x (a, b) = f y (a, b) = 0 dz = 0 (a, b9 ( ) dz = 0 0 = f x dx + f y dy........................................................................ ( ) 30

dz = 0 f(x, y) = k ( ) z x, y f y 0 dx dx 0 = f x dx + f dy y dx = f dy x + f y dx dy dx = f x f y z = f(x, y) x, y t x = x(t), y = y(t) z = f(x, y) = f(x(t), y(t)) = f (t) z t dx = x (t)dt, dy = y (t)dt ( ) dz = f x (x, y)x (t)dt+f y (x, y)y (t)dt = {f x (x, y)x (t)+f y (x, y)y (t)}dt = dz = dz dt dt dz dt = df (t) = z dt x dx dt + z y dy dt (5..8) ( ) dt z = f(x, y) = xy dz = d(xy) = ydx + xdy x = x(t), y = y(t) dt d(x(t)y(t)) dt = y(t) dx(t) dt + x(t) dy(t) dt ( z x dx dt + z y dy ) dt dt x = x(u, v), y = y(u, v) dx = x u du + x v dv dy = y u du + y v dv ( ) dz = f x dx + f y dy = f x (x u du + x v dv) + f y (y u du + y v dv) = (f x x u + f y y u )du + (f x x v + f y y v )dv 3

dz = f u du + f v dv ( ) du z u = f x x u + f y y u 3