grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = 0 g (0) g (0) (31) grad φ(p ) p grad φ φ (P, φ(p )) xy (x, y) = (ξ(t), η(t)) ( )

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1 grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = g () g () (3) grad φ(p ) p grad φ φ (P, φ(p )) y (, y) = (ξ(t), η(t)) ( ) ξ (t) (t) := η (t) grad f(ξ(t), η(t)) (t) g(t) := f(ξ(t), η(t)) t = t g (t ) t = t g (t ) = (ξ(t ), η(t )) dξ dt (t ) + y (ξ(t ), η(t )) dη dt (t ) = grad f(ξ(t), η(t)) (t)

2 grad grad φ d l = φ(q) φ(p ) P Q. φ f(, y, z) = ξ(t) y = η(t) z = ζ(t) t t t P (ξ(t ), η(t ), ζ(t )) Q (ξ(t ), η(t ), ζ(t )) grad φ d t f (ξ(t), η(t), ζ(t)) ξ (t) l = f y (ξ(t), η(t), ζ(t)) η (t) dt = t t t f z (ξ(t), η(t), ζ(t)) ( (ξ(t), η(t), ζ(t))dξ dt (t) ζ (t) + (ξ(t), η(t), ζ(t))dη y dt (t) g(t) = f(ξ(t), η(t), ζ(t)) dg (t) = (ξ(t), η(t), ζ(t))dξ dt dt (t) + (ξ(t), η(t), ζ(t))dη y dt (t) + (ξ(t), η(t), ζ(t))dζ z dt (t) + (ξ(t), η(t), ζ(t))dζ z dt (t) ) dt grad φ d l = t t dg dt (t)dt = g(t ) g(t ) g f f φ g(t ) g(t ) = f(ξ(t ), η(t ), ζ(t )) f(ξ(t ), η(t ), ζ(t )) = φ(q) φ(p )

3 9 3 P Q a b [a, b] f() f() f() F F grad φ = F φ F F F F F?

4 9 4 F 2 F F F? P φ P φ(p ) = F d l P P φ grad φ = F. yz φ f(, y, z) F F (, y, z) = F (, y, z) F 2 (, y, z) F 3 (, y, z) (, y, z) = F (, y, z) y (, y, z) = F 2(, y, z) z (, y, z) = F 3(, y, z) f( + h, y, z) f(, y, z) (, y, z) = lim h h (a, b, c) Q (a + h, b, c) Q h Q Q h L P Q L (a + s, b, c) s h f(, y, z) φ f(a + h, b, c) f(a, b, c) = φ(q h ) φ(q) = = = L h F d l = h +L (c) F d l F (a + s, b, c) (a + s) F 2 (a + s, b, c) (b) ds F 3 (a + s, b, c) F (a + s, b, c) F 2 (a + s, b, c) ds = F 3 (a + s, b, c) h F (a + s, b, c)ds F d l

5 9 5 f(a + h, b, c) f(a, b, c) (a, b, c) = lim h h h h = lim F (a + s, b, c)ds h h = lim F (a + s, b, c)ds F (a + s, b, c)ds h h = d h F (a + s, b, c)ds = F dh (a + h, b, c) = F (a, b, c) h= h= 2 F φ φ F 2 P Q 2 P P = F d l = F d l + F d l = F d l F d l 2 2 F d l = 2 F d l φ grad (P, φ(p )) φ grad (P, φ(p )) (, y) = (ξ(t), η(t))

6 9 6 ( (t) := ξ (t) η (t) ) grad f(ξ(t), η(t)) (t) t t t P t Q t t Q φ(q) P φ(p ) φ(q) φ(p ) = t t grad f(ξ(t), η(t)) (t)dt φ(q) φ(p ) = grad φ d l 2. yz f(, y, z) = 2 sin 2y sin 3z φ grad φ 2. yz F (, y, z) = ( ) y 2 + z 2 y z F 22. y ( ) y F (, y) = F 23. y F F (, y) = 2 + y 2 ( y ) F

7 P v φ t g(t) g(t) = φ(p + t v) Q u Q + u u Q g () d φ(p + t v) φ(p + t v) φ(p ) dt = lim t t= t P φ v v P (φ) φ P v P P P + v yz φ 3 f(, y, z) P u (a, b, c) v v g(t) w g(t) = φ(p + t v) = f(a + tu, b + tv, c + tw) g () = u (a, b, c) + v y (a, b, c) + w (a, b, c) z (P, v) P v v P (φ)

8 9 8 φ (P, v) P v v P (φ) v, w a ( v + w) P (φ) = v P (φ) + w P (φ) (a v) P (φ) = a( v P (φ)) φ dφ U Ω (U ) U Ω (U ) φ dφ d: Ω (U ) Ω (U ) dφ (P, v) d P φ( v) 3 d P φ( v) = v P (φ) (a, b, c) v P (φ) = u u v (a, b, c) + v y (a, b, c) + w (a, b, c) z w dφ ) ( y (, y, z) 3 ( ) u d P φ( v) = (a, b, c) y (a, b, c) z (a, b, c) v w (, y, z), y, z d, dy, dz yz d f ( ) ( ) d = = y z 3 θ (P, v) θ(p, v) P v θ P ( v) P φ dφ dφ(p, v) dφ P ( v) φ p dφ P d P φ z

9 9 9 d dy 2 dz 3 φ φ yz f dφ = d + dy + y z dz dφ df. yz φ f(, y, z) grad φ y z f(, y, z)d + g(, y, z)dy + h(, y, z)dz f(, y, z) g(, y, z) h(, y, z) d grad φ P Q dφ = φ(q) φ(p ) (34) θ F f fd + gdy + hdz g (35) θ = F d l h

10 (35) (34) grad φ d l = φ(q) φ(p ) (34)

11 9 2 y 2 z 3 2 sin 2y sin 3z 2 2 cos 2y sin 3z 3 2 sin 2y cos 3z 2 3 f(, y, z) f(, y, z) (, y, z) = ( ) 3 (36) 2 + y 2 + z 2 y (, y, z) = y ( ) 3 (37) 2 + y 2 + z 2 z z (, y, z) = ( ) 3 (38) 2 + y 2 + z 2 (36) f(, y, z) y z (36) y = b z = c ψ() = f(, b, c) ψ () = ( 2 + b 2 + c 2) 3 ψ() = ( 2 + b 2 + c 2) 3 d = 2 + b 2 + c + 2 t = 2 + b 2 + c 2 y z y z y z (36) (37) + g(y, z) 2 + y 2 + z2 + h(, z) 2 + y 2 + z2

12 9 2 (38) + k(, y) 2 + y 2 + z2 (36) (37) (38) y 2 + z2 + y 2 + y 2 + z2 + z 2 + y 2 + z2 + y z 2 + y 2 + z2 2 + y 2 + z 2 + R 22 2 f(, y) f(, y) (, y) = y (, y) = y 2 f(, y) = y + g(y) y (, y) = + g (y) = g (y) y g(y) + g (y) = g (y) = 2

13 9 3 F 2 D D F D (a, b) r = a + r cos t y = b + r sin t t 2π F ( ) ( ) 2π F d b r sin t (a + r cos t) l = dt a + r cos t (b + r sin t) = = 2π 2π ((b + r sin t)r sin t + (a + r cos t)r cos t) dt ( ar cos t + br sin t + r 2 ) dt = 2πr 2 F F f(, y) f(, y) y (, y) = 2 + y 2 (39) (, y) = y 2 + y 2 (4) 2 22 (39) y y f(, y) = 2 + y 2 d = ( ) y 2 d + = y y t 2 + ydt = Arctan t + g(y) = Arctan + g(y) (4) y Arctan ( π/2, π/2) tan y = (39) (, ) = 2 + = f(, ) = ( > ) f(, ) = ( < )

14 9 4 y y = f(, ) > < f(, y) y (4) g(y) y (4) y ( y (, y) = ( ) 2 ) y 2 + y + g (y) = 2 + y 2 + g (y) (4) g (y) = g(y) = (y > ) g(y) = (y < ) y g(y) y y = F y = > y = (4) y f(, y) y > lim f(, y) = f(, ) = y lim f(, y) = lim Arctan y + y + y + = π 2 + lim f(, y) = lim Arctan y y y + = π 2 + f(, y) >, y = y = π 2 + = π 2 + (42) < <, y = f(, y) y = π 2 + = π 2 + (43) (42) (43) = π + = π +, F

15 r = r cos t y = r sin t t 2π F ( ) ( ) 2π F d (r cos t) l = dt = (r sin t) sin t r cos t r 2π dt = 2π F f(, y) = Arctan y + π 2 y > y = Arctan y + + π 2 y < 2 2

F S S S S S S S 32 S S S 32: S S rot F ds = F d l (63) S S S 0 F rot F ds = 0 S (63) S rot F S S S S S rot F F (63)

F S S S S S S S 32 S S S 32: S S rot F ds = F d l (63) S S S 0 F rot F ds = 0 S (63) S rot F S S S S S rot F F (63) 211 12 1 19 2.9 F 32 32: rot F d = F d l (63) F rot F d = 2.9.1 (63) rot F rot F F (63) 12 2 F F F (63) 33 33: (63) rot 2.9.2 (63) I = [, 1] [, 1] 12 3 34: = 1 2 1 2 1 1 = C 1 + C C 2 2 2 = C 2 + ( C )

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