5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h 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 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 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 8

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h 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 + 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 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 h θ x y π θ θ 9

5.. ( ), ( ) (x, y) f(x + x, y) f(x, y) f x (x, y) = lim x 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) f(x, y) x f y (x, y) = h (y) = y g(x) 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) (, )) ((x, y) = (, )) f x (x, y) = y(y x ) (x + y ) f y (x, y) = x(x y ) (x + y ) f(h, ) f(, ) f x (, ) = lim = lim = h h h h f(, h) f(, ) f y (, ) = lim = lim = h h h h 3

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

{ (x = y = xy = ) f(x, y) = ( xy ) z = x y f x (, ) = f y (, ) = x, y x y x, y x f y y f x xy f(x, y) = x + y ((x, y) (, )) ((x, y) = (, )) (x, y) f x, f y y = mx (m ) f(x, y) = xy x + y = mx ( + m )x = m + m x y ((x, y) (, )) f(x, y) = (x + y ) 3 ((x, y) = (, )) (x, y) x y f(x, y) = = r cos θ sin θ (x + y ) 3 r r cos θ sin θ x, y cos θ, sin θ 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 =.5.8.7.6.5.4.3...5 5. 5.. 33

y x = A + ε ( y = f(x + x) f(x )) x ε A = f (x ) y = A x + ε x = f (x ) x + ε x 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 + x, y + y) f(x, y ) ρ = ( x) + ( y) ρ ε (x + x, y + y) (x, y ) f(x, y) (x, y ) (x, y, z ) z = f(x, y ) (dx, dy, dz) dz = Adx + Bdy dx, dy, dz x, y, z (x, y, z) dx = x x z z = A(x x ) + B(y y ) f(x, y) (x, y, z ) 5..3 f(x, y) (x, y ) (x + x, y + y) x (x, y ) y =, ρ = x z = A x + ε x x z x = A ± ε ± x x ε A (x, y ) x 34

(x, y ) x A f x (x, y ) B = f y (x, y ) θ x = ρ cos θ y = ρ sin θ z = Aρ cos θ + Bρ sin θ + ερ ρ z ρ = A cos θ + B sin θ + ε ρ A cos θ + B sin θ = f x (x, y ) cos θ + f y (x, y ) sin θ θ f x, f y : y = A x + ε x y x = A + ε x A = f (x ) z = A x + B y + ερ ρ z ρ = A x ρ + B y ρ + ε ρ z A x B y ρ z, x. y dx, dy, dz : 35

5..4 5..4, 5..5 (x, y, z = f(x, y )) z z = f x (x, y )(x x ) + f y (x, y )(y y ) (x, y, z ) x y x y z z x y = = x y z x y z + f y (x, y ) t f x (x, y ) + s x x y = + t + s z y z f x (x, y ) f y (x, y ) t = x x, s = y y 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 t/ y = + t/ = z t (, ) z = x + y + t 36

y = x x = y = t/ x = y = t x y = + t z t 5..5 ax + by + cz + d = t (a b c) (x, y, z ), (x, y, z ) ax + by + cz + d = ax + by + cz + d = a(x x ) + b(y y ) + c(z z ) = a b c x x y y z z z z = f x (x, y )(x x ) + f y (x, y )(y y ) f x (x, y ) f y (x, y ) f x (x, y ) f y (x, y ) xy z ( ) f x (x, y ) f y (x, y ) t ( f x (x, y ) f y (x, y )) f(x, y) ( ) 5.3. ( ) f y (x, y ) f x (x, y ) 37

z x x = f y (x, y )t y y = f x (x, y )t z = z + f x (x, y )(x x ) + f y (x, y )(y y ) = z + f x (x, y )f y (x, y )t f x (x, y )f y (x, y )t = z z f(x, y) : k 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 ) f(x, y) = x (x = ) θ 38

.5.5.5.5 : x = y = f(x, y) = f(x, y) x, y x x y y xy z = 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) (, )) ((x, y) = (, )) (x, y) x, y f x y(y x ) f x (x, y) = (x + y ) ((x, y) (, )) ((x, y) = (, )) (x, y) (, ) 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) (, ) r θ f x (x, y) f y : 39

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) = f yx = f x (x, y) x = x = f xy (x, y) f yx f x, f y { x (x ) f(x, y) = (x < ) 4

f yx f xy x = f xy, f yx xy x y f(x, y) = x + y ((x, y) (, )) ((x, y) = (, )) p.58 59 f xy (, ) = f yx (, ) = f xy = f yx f xx, f yy n n n f r x s y (r + s = n) n + f(x, y) n f(x, y) C n C n n C 5.5 5..7 5.. 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, y = bt + b (a b) dz dt = a z x + b z y 4

(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 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) 4

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 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 43

J cos θ = sin θ sin θ r cos 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) 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 = 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 = 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 44

f(r, θ) = r g(r, θ) = sin θ g g(r, θ) = θ f r (r, θ) = f θ (r, θ) = g r (r, θ) = 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 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! +... n n 45

n nc k f xk y n k(a, b)xk y n k k= f (...) (a, b) (a, b) (a + θx, b + θy) θ θ < f(a + x, b + y) 5.8 4 5. 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) <, >, (a, b) 5.9. f(x, y) f x (a, b) = f y (a, b) = (a, b) f(x, y) xy z = k x f x (a, b) = (a, b) (saddle point) : (a, b) (a, b) (a, b) (a, b) 46

5.9.3 f x (a, b) = f y (a, b) = 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 x n = (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 P A = B = H = P H P = Hxy P x, y y = x P = Hx y = x P = Hx (x, y) = (, ) H (a, b) A, B A [ ( P = A x + H ) ] A y AB H + A y ( x + H A y ) H AB p.84 5.3.4 = AB H = f xx f yy f xy > < > A y y = x + H A y = y = x = y = P x = y = P A 47

: = AB H > AB > B A B A < : P < f(a + x, b + y) f(a, b) < (x, y) (, ) (x, y) f(a, b) A > : f(a, b) < A < 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 = ( ) H x + A C y = (a, b) A = B =, H = AB H = H < = P = A (x + HA ) y x + H A y = P = f(a + x, b + y) f(a, b) A = B = H = x, y x, y ( ) ( ) P = Ax + Hxy + By A H x = (x y) H B y x, y n n 48

5.9.4 z = x + y (x, y) (, ) x + y > (, ) (, ) f x (x, y) = f y (x, y) = (x, y) (a, b) f xx (a, b) = A f yy (a, b) = B f xy (a, b) = H = AB H > : A < A > < : (a, b) = : 5.9.5 f(x, y) = x + y x + y f x (x, y) = x f y (x, y) = y f x = x = f y = y = (, ) f xx (x, y) = f yy (x, y) = f xy (x, y) = = = 4 > A = > x, y + = f(x, y) = x 4 + y 4 f x (x, y) = 4x 3 f y (x, y) = 4y 3 49

(, ) f xx (x, y) = x f yy (x, y) = y f xy (x, y) = (, ) A = B = H = f xxx (x, y) = 4x f yyy (x, y) = 4y (, ) f x 4(x, y) = f y 4(x, y) = 4 (, ) 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 = y = x y + = y = f y f y = x + 4x 3 = x = ± 7 ( ± 7, ) x y + = y = x + f y = x 4x(x + ) + 4x 3 = 3x + 4x + 3 = f xx (x, y) = y f yy (x, y) = 4x f xy (x, y) = x 4y + 4 ( ± 7, ) f xx ( ± 7, ) = f yy ( ± 7, ) = 8 4 7 f xy ( ± 7, ) = ± 7 = AB H = (8 4 7) (± 7) = 7 <... 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 = x y 3y + x = 5

(x, y) = (, ) x, y x( ) y( ) x y 3x 3 + xy = x y 3y 3 + xy = 3(y 3 x 3 ) = y = x x 3 3x + x = x(x 3x + ) = x(x )(x ) = y = x (x, y) = (, ), (, ), (/, /) (, ) f xx (x, y) = y 6x f yy (x, y) = x 6y f xy (x, y) = 4xy + (, ) A = B = H = (, ) A = B = 4 H = 5 = ( 4) ( 4) 5 = 9 > (/, /) A = B = 5/ H = = ( 5/) ( 5/) = 9/4 > /6 : y = x x = y f(x, y) = (x y) (y x) f(x, y) +.5 +.5 +.5 +.5.5.5 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) = = ( ) = x y = t f(x, y) = t t = y = x : f(x, y) = x y f(x, y) = (y x )(y x ) = f x (x, y) = x(4x 3y) f y (x, y) = y 3x f x = x = y = (4/3)x f y = y = f y = 8 3 x 3x = 3 x = x = (, ) f xx (x, y) = 6(4x y) f yy (x, y) = f xy (x, y) = 6x (, ) = = x = f(x, y) f(, 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 f(x, y) > f(, ) = (, ) f(x, y) (, ) x < y < x f(x, y) < y < x, y > x f(x, y) > (, ) (, ) (, ) y = ( + α)x ( < α < ) f(x, y) f(x, y) = (y x )(y x ) = αx (α )x = α(α )x 4 α(α ) < x = (, ) (, ) 5

3.5.5 +.5.5 +.5.5.5.5 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 = cos x = cos y = x = π + nπ y = π + mπ n, m f y = sin x = sin y = 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) ( 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 + = < (B ) sin + = > m + n 53

5. 5.. a b = b a 5.. 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 54

: d d f(x) f(x) dx dx dn f n dxn ( ) n d f(x) 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 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 55

5..3 z = f(x, y) (x, y) dz = f x (x, y)dx + f y (x, y)dy = z z dx + x y dy........................................ ( ) df = f f dx + dy x y dx dz dx = z x + z dy y dx dx dx = dy x y dx dy dx = 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 z xy a 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..4 z = f(x, y) x, y t x = x(t), y = y(t) z t f (t) + a 56

z = f(x, y) = f(x(t), y(t)) = f (t) dz = f x (x, y)dx + f y (x, y)dy = z z dx + x y dy dx = x (t)dt = dx dt dt dy = y (t)dt = dy dt 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 =........................................ ( ) z = f (t) dt {f (t)} = dz dt z == x dx dt + z y dy dt ( z x dx dt + z y dy ) dt dt ( ) dt 57