. p.1/14 - PDF Free Download

. p.1/14

F(x,y) = (F 1 (x,y),f 2 (x,y)) (x,y). p.2/14

F(x,y) = (F 1 (x,y),f 2 (x,y)) (x,y) (x,y) h. p.2/14

F(x,y) = (F 1 (x,y),f 2 (x,y)) (x,y) (x,y) h h { F 2 (x+ h,y) F 2 2(x h,y) F 2 1(x,y+ h)+f 2 1(x,y h)} +( ) 2. p.2/14

F(x,y) = (F 1 (x,y),f 2 (x,y)) (x,y) (x,y) h h { F 2 (x+ h,y) F 2 2(x h,y) F 2 1(x,y+ h)+f 2 1(x,y h)} +( ) 2 =h {( F 2 (x,y)+ hf 2 2x(x,y) ) ( F 2 (x,y)+( h)f 2 2x(x,y) )} h {( F 1 (x,y)+ hf 2 1y(x,y) ) ( F 1 (x,y)+( h)f 2 1y(x,y) )} +( ). p.2/14

D. p.3/14

D h 0. p.3/14

D h 0 ( F2 F dr = C D x F ) 1 ds y C D. p.3/14

[ ] C y = x y = x 2 (0 x 1) I = (xy + x 2 )dx + x 2 dy C 1. I 2. I. p.4/14

[ ] C y = x y = x 2 (0 x 1) I = (xy + x 2 )dx + x 2 dy C 1. I 2. I [ ]. p.4/14

[ ] C y = x y = x 2 (0 x 1) I = (xy + x 2 )dx + x 2 dy C 1. I 2. I [ ] 1. C 1 : r 1 (t) = (t,t 2 ), C 2 : r 2 (t) = (t,t) (0 t 1). p.4/14

[ ] C y = x y = x 2 (0 x 1) I = (xy + x 2 )dx + x 2 dy C 1. I 2. I [ ] 1. C 1 : r 1 (t) = (t,t 2 ), C 2 : r 2 (t) = (t,t) (0 t 1) I = (xy + x 2 )dx + x 2 dy + (xy + x 2 )dx + x 2 dy C 1 C 2. p.4/14

[ ] C y = x y = x 2 (0 x 1) I = (xy + x 2 )dx + x 2 dy C 1. I 2. I [ ] 1. C 1 : r 1 (t) = (t,t 2 ), C 2 : r 2 (t) = (t,t) (0 t 1) I = (xy + x 2 )dx + x 2 dy + (xy + x 2 )dx + x 2 dy C 1 C 2 = 1 0 (t 3 + t 2 + 2t 3 )dt + 0 1 (t 2 + t 2 + t 2 )dt = 1 12. p.4/14

2.C { D I = x (x2 ) } y (xy x2 ) D dxdy. p.5/14

2.C { D I = D x (x2 ) } y (xy x2 ) = xdxdy D dxdy. p.5/14

2.C { D I = D x (x2 ) } y (xy x2 ) = xdxdy D 1 { x } = xdy dx 0 x 2 1 = x(x x 2 )dx = 1 12 0 dxdy. p.5/14

[ ] D a 2 x 2 + y 2 b 2 C I = 2xydx + (x 3 y)dy C 1. I 2. I. p.6/14

[ ] D a 2 x 2 + y 2 b 2 C I = 2xydx + (x 3 y)dy C 1. I 2. I [ ]. p.6/14

[ ] D a 2 x 2 + y 2 b 2 C I = 2xydx + (x 3 y)dy C 1. I 2. I [ ] 1. C 1 :(b cost,bsin t), C 2 :(a cos t,asin t) (0 t 2π). p.6/14

[ ] D a 2 x 2 + y 2 b 2 C I = 2xydx + (x 3 y)dy C 1. I 2. I [ ] 1. C 1 :(b cost,bsin t), C 2 :(a cos t,asin t) (0 t 2π) I = 2xydx + (x 3 y)dy + 2xydx + (x 3 y)dy C 1 C 2. p.6/14

[ ] D a 2 x 2 + y 2 b 2 C I = 2xydx + (x 3 y)dy C 1. I 2. I [ ] 1. C 1 :(b cost,bsin t), C 2 :(a cos t,asin t) (0 t 2π) I = 2xydx + (x 3 y)dy + 2xydx + (x 3 y)dy C 1 C 2 2π { = 2b 2 costsin t( b sin t) + (b 3 cos 3 t b sin t)b cost } dt 0 0 { + 2a 2 cos t sin t( a sin t) + (a 3 cos 3 t a sin t)a cos t } dt 2π. p.6/14

. p.7/14

S C S C. p.7/14

S C S C S F. p.7/14

S C S C S F F dr = rotf ds C S. p.7/14

[ ]. p.8/14

[ ] ( ) E t = 0 B j rot B = 1 ε 0 c 2j. p.8/14

[ ] ( ) E t = 0 B j rot B = 1 ε 0 c 2j C S B dr = 1 j ds ε 0 c 2 C S. p.8/14

[ ] ( ) E t = 0 B j rot B = 1 ε 0 c 2j C S B dr = 1 j ds C ε 0 c 2 S C C. p.8/14

[ ] ( ) E t = 0 B j rot B = 1 ε 0 c 2j C S B dr = 1 j ds C ε 0 c 2 S C C B dr = 1 ε 0 c2(c ) C. p.8/14

[ ]. p.9/14

[ ] I B r I e r. p.9/14

[ ] I B r I [ ] e r. p.9/14

[ ] I B r I [ ] r e r. p.9/14

[ ] I B r I [ ] r B ds = 2πrB (B = B ) e r. p.9/14

[ ] I B r I [ ] r B ds = 2πrB (B = B ) B = I 2πε 0 c 2 e r. p.9/14

[ ] I B r I [ ] r B ds = 2πrB (B = B ) B = I 2πε 0 c 2 i.e. B = 1 I e r 2πε 0 c 2 r e r. p.9/14

. p.10/14

F. p.10/14

F (x,y,z) h. p.10/14

F (x,y,z) h h 2{ F(x + h,y,z) i + F(x h,y,z) ( i) 2 2 +F(x,y + h,z) j + F(x,y h,z) ( j) 2 2 +F(x,y,z + h) k + F(x,y,z h ) ( k) + ( )} 2 2. p.10/14

F (x,y,z) h h 2{ F(x + h,y,z) i + F(x h,y,z) ( i) 2 2 +F(x,y + h,z) j + F(x,y h,z) ( j) 2 2 +F(x,y,z + h) k + F(x,y,z h ) ( k) + ( )} 2 2 =h 2{( F 1 (x,y,z)+ hf 2 1x(x,y,z) ) ( F 1 (x,y,z)+( h)f 2 1x(x,y,z) ) + ( F 2 (x,y,z)+ hf 2 2y(x,y,z) ) ( F 2 (x,y,z)+( h)f 2 2y(x,y,z) ) + ( F 3 (x,y,z)+ hf 2 3z(x,y,z) ) ( F 3 (x,y,z)+( h)f 2 3z(x,y,z) )} +( ). p.10/14

D F D h. p.11/14

D F D h h 0 div FdV = F ds D S D ds D S. p.11/14

[ ] O Q ϕ(x,y,z) = 1 Q 4πε 0 r ( r = x 2 + y 2 + z 2 ). p.12/14

[ ] O Q ϕ(x,y,z) = 1 Q 4πε 0 r ( r = x 2 + y 2 + z 2 ) E(x,y,z) = grad ϕ = Q 1 4πε r 2e r (e r (x,y,z) ). p.12/14

[ ] O Q ϕ(x,y,z) = 1 Q 4πε 0 r ( r = x 2 + y 2 + z 2 ) E(x,y,z) = grad ϕ = Q 1 4πε r 2e r (e r (x,y,z) ) S 0 (Q S ) E ds = S Q (Q S ) ε 0. p.12/14

[ ] O Q ϕ(x,y,z) = 1 Q 4πε 0 r ( r = x 2 + y 2 + z 2 ) E(x,y,z) = grad ϕ = Q 1 4πε r 2e r (e r (x,y,z) ) S 0 (Q S ) E ds = S Q (Q S ) ε 0 S D. p.12/14

[ ] O Q ϕ(x,y,z) = 1 Q 4πε 0 r ( r = x 2 + y 2 + z 2 ) E(x,y,z) = grad ϕ = Q 1 4πε r 2e r (e r (x,y,z) ) S 0 (Q S ) E ds = S Q (Q S ) ε 0 S D E ds = dive dv S D. p.12/14

[ ] O Q ϕ(x,y,z) = 1 Q 4πε 0 r ( r = x 2 + y 2 + z 2 ) E(x,y,z) = grad ϕ = Q 1 4πε r 2e r (e r (x,y,z) ) S 0 (Q S ) E ds = S Q (Q S ) ε 0 S D E ds = dive dv = div(grad ϕ) dv S D D. p.12/14

[ ] O Q ϕ(x,y,z) = 1 Q 4πε 0 r ( r = x 2 + y 2 + z 2 ) E(x,y,z) = grad ϕ = Q 1 4πε r 2e r (e r (x,y,z) ) S 0 (Q S ) E ds = S Q (Q S ) ε 0 S D E ds = dive dv = div(grad ϕ) dv = 0 S D D. p.12/14

S r S S S D. p.13/14

S r S S S D 0 = dive dv D. p.13/14

S r S S S D 0 = dive dv = D S S E ds. p.13/14

S r S S S D 0 = dive dv = D S S E ds = E ds E ds S S. p.13/14

S r S S S D 0 = dive dv = E ds = E ds E ds D S S S S E ds= Q 1 S 4πε 0 S r 2e r ds. p.13/14

S r S S S D 0 = dive dv = E ds = E ds E ds D S S S S E ds= Q 1 S 4πε 0 S r 2e r ds = Q 1 4πε 0 S r 2e r nds. p.13/14

S r S S S D 0 = dive dv = E ds = E ds E ds D S S S S E ds= Q 1 S 4πε 0 S r 2e r ds = Q 1 4πε 0 S r 2e r nds = Q 4πε 0 S 1 r 2dS. p.13/14

S r S S S D 0 = dive dv = E ds = E ds E ds D S S S S E ds= Q 1 S 4πε 0 S r 2e r ds = Q 1 4πε 0 S r 2e r nds = Q 1 4πε 0 S r 2dS = Q ε 0. p.13/14

S r S S S D 0 = dive dv = E ds = E ds E ds D S S S S E ds= Q 1 S 4πε 0 S r 2e r ds = Q 1 4πε 0 S r 2e r nds = Q 1 4πε 0 S r 2dS = Q ε 0 E ds = Q S ε 0. p.13/14

p.92 1 p.101 9. p.14/14