Quiz x y i, j, k 3 A A i A j A k x y z A x A y A z x y z A A A A A A x y z P (x, y,z) r x i y j zk P r r r r r r x y z P ( x 1, y 1, z 1 )

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1 Quiz x y i, j, k 3 A A i A j A k x y z A x A y A z x y z A A A A A A x y z P (x, y,z) x i y j zk P x y z P ( x 1, y 1, z 1 ) Q ( x, y, z ) 1 OP x1i y1 j z1k OQ x i y j z k 1

2 P Q PQ 1 PQ x x y y z z P Q PQ PQ A B A B cos A B A B A B A B B A A( B C) A B + AC A A A A A, B A, B

3 A B 0 A, B A, A, A B, B, B x y z x y z A B Ax Bx AyBy Az Bz A B A B A B B A cos A B C A A A B C C A B sin A B A B A B AB A( B + C) A B + AC A B = B A 3

4 A B A B = 0 A, B A, A, A B, B, B A B A B A B i y z z y A B A B z x x z A B A B x y y x j k x y z x y z A, B A, A, A B, B, B x y z x y z A B i j k A A A x y z B B B x y z A B C A B C A B C A( BC) A B C 4

5 A ( B C ) A( B C) ( AC) B ( A B) C ( A B) C ( AC) B ( B C) A A( B C) ( A B) C 5

6 t f ( t) t f ( t) df f lim dt t0 t f f t t f t f t s( t) s s( t) s( t1) t t t 1 t ds s lim t dt t0 t f ( t) df / dt t f ( t) Taylo t df t f ( t t) f t t dt A( t) d dt A A A A t t A t lim t0 t A( t) ( x, y, z) A, A, A x y z 6

7 da da da da dt dt dt dt x y z i j k ( x, y) t ( t) xi yj t t ( t t) ( x x) i ( y y) j u d u dt lim t0 t t t t / t x y s t t t t s s x y ( t) x( t) i y( t) j z( t) k u( t) dx t dy t dz t u( t) i j k dt dt dt 7

8 dx dy dz u u dt dt dt du u α lim dt t0 t u ut t u t du m F dt [1] d( fa) df da A f dt dt dt (, ) ( t) t e e e cosi sin j 8

9 e sini cos j e, e de d de d e, e dt dt dt dt i, j e e u d d u e e dt dt d u d d 1 d d a e e dt dt dt dt dt d d m F dt dt 1 d d dt dt m F F, F [] d( A B) da db B A dt dt dt 9

10 u( t) u du dt d ( uu ) 0 dt [3] d( A B) da db B A dt dt dt L p p p mu d L dt du N N m F dt // F dl dt 0 L m k 10

11 Quiz T ( x, y, z) u( x, y, z) T ( x, y, z, t) S W h S f e h W f S e h S n 11

12 S S Sn S n h S W f h S W f ( hn 0 ) ( hn 0 ) f kg/(m s) f f v v 1

13 f ( x, y, z) v( x, y, z) f ( x, y, z) M f f S f S M f 13

14 Quiz ( x, y) f ( x, y) x f / x f x,, f x x y f x y lim x 0 x ( x, y) f ( x, y) y f / y f y,, f x y y f x y lim y 0 y f ( x, y) df x, y dx, dy f ( x, y) df f ( x dx, y dy) f ( x, y) f f df dx dy x y i + j k x y z ( x, y, z) ( x, y, z) i + j k x y z 14

15 x y z 1 1 xi yj zk A( x, y, z) A A Az x y z x y A + h 6 hi Si i1 ( h) V hi S i 6 i1 15

16 Q Q h V t ( x, y, z, t) v( x, y, z, t) ( x, y, z) t v SM S M A( x, y, z) ( x, y, z) A ( ) A v v 3 16

17 0 3 + x y z A( x, y, z) A A A A A A y z z x x y z y x z y x A i j k i j k A x y z A A A x y z ω v ( x, y) z ω k 17

18 v( x, y) vxi vy j, vx y vy x v k v v A( x, y, z) ( x, y, z) A ( A) A+ A 0 ( ) A A 18

19 A ( ) 0 F U F U E E B A B A A B ( A ) 0 B B A B A 19

20 Quiz C A A C d Ad C P Q A d Q Q d d Axdx Aydy Az dz C A A P P P Q (P) (Q) Q P d Q P C P Q C C A d Ad C C R P R C 1 R P C A d Ad A d C C1 C 0

21 a C (,, z) d ade e e sini + cos j A S A d S S ds ds ds n ds n ds h S h W f h d S s 1

22 f S f f ds v ds M f s S, v : ds (,, z) ds nds n k ds dd k : ds (,, z) ds nds n ds ddz a : ds (,, ) ds nds n a ds a sindd a

23 / 3 E( x, y, z) ( x, y, z) K E ( x, y, z), xi yj zk x y z K const. a E( x, y, z) E d S 4 K S S V A S A ds diva dv V A S d S h h ( divh ) h S W 3

24 S S hds hdv V f f ( divf ) f S M S S f ds fdv V S S W ( ) 0 S 0 4

25 S ( ) W 0 0 ( ) 0 ( ) dv 1 V 0 V ( ) dv 0 V 0 V / 3 E( x, y, z) ( x, y, z) 1 E K xi yj zk, x y z K const. S E E d S 4 K S A C A C d A A C A A C d 5

26 A A( ot A) C S A ds C A d d C A S S z ω k ( x, y, z) v v v v ω v =, v, ω 6

27 E E E 0 C E d 0 C B A B A B 0 S A S S d 0 7

28 Quiz ( u 1, u, u 3 ) u1( x, y, z) C1, u( x, y, z) C, u3( x, y, z) C3 ( C1, C, C3 const.) ( x, y) x y C1 1 y ( x, y) tan C x z z C3 ( u 1, u, u 3 ) e 1, e e 3 e e i j 1i j 0i j e1, e e 3 f ( u, u, u ) f e f e f e f 1, f f 3 f f e ( j 1,,3) e 1, e e 3 u 1, u h1, h h 3 j j u3 8

29 e 1 h1 u1, hu e, e 3 h3u3 du 1, du, du 3 ds 1, ds, ds 3 h 1, h h 3 ds h du ds h du ds 3 h 3 du ds 1, ds, ds 3 du 1, du, du 3 h 1, h h 3 dv ds1ds ds3 h1 hh3 du1dudu3 h 1, h, h 1 dv z dddz h 1, h, h sin dv sinddd e 1, e e 3 h 1, h h 3 f ( u 1, u, u 3 ) 9

30 1 f 1 f 1 f f e e e h u h u h u f 1 f f f e e e z z f 1 f 1 f f e e e sin f ( u 1, u, u 3 ) f 1, f, f 3 1 f h h h u u u h h f h h f h h f f f f = f z z f f ( f ) (sin ) f sin sin f ( u 1, u, u 3 ) f 1, f, f 3 30

31 h f h f e h h u u f 3 3 h f h f e h h u u h f h f e 3 h h u u f z f f f z 1 f f f e e e z z z sin f f e f sin e f sin e sin f f f f ( u 1, u, u 3 ) 1 hh3 f h3h1 f h1h f f h1h h3 u1 h1 u1 u h u u3 h3 u3 1 f 1 f f f z f f sin f f sin sin 31

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