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1、 2.1 1.xy dx dy 2=,x=0,y=1. e ex x yc yx x cycyxdxdy y 2 2 , 1 1,0,ln,2 12 = =+= , 0) 1(. 2 2 =+dyxdxyx=0,y=1. x y cyxy xc yc y xydydx x y + = = + =+=+= + 1ln1 1 , 11, 00 1ln 1 , 1 1ln0, 1 1 1 2 3 yxy dx dy x y 3 2 1 + + = xxy xxyxy xy y x y c cccx dx x dy y y x ydx dy 22 2 22 2 2 2 32 2 3 2 )1 (1 )
2、1)(1 (),0(ln1ln 2 1 ln1ln 2 1 1 1 , 0 1 1 1 =+ =+=+ + = + + + = + . 0 ; 0;ln ,ln,lnln 0 11 000 0)1 ()1 (4 = =+=+ = = + = =+ xycyxxy cyxxycyyxx dy y y dx x x xyxy xdyyydxx . y y yc y y x = + += =+ + 1 1, 0 , 1 1 1 1ln y y y y y x y x y 2 3 2 1 1ln ln 2 2 1 1 1 1 + + + + + 1 ( (4 4+ x x= = 10 ln1 ln l
3、n1 ln1 , 0ln 0)ln(ln:9 3 1 :8 .coslnsinln 07 lnsgnarcsin lnsgnarcsin 1 sgn 1 1 , )1 ( , 6 ln)1ln( 2 1 1 1 1 , 1 1 , 0)()( :5 3 3 2 2 2 22 2 2 2 2 2 2 c dxdy dx dy x y cy ud u u dx xx y u dx x y dy x y ydxdyyxx cdy y y y y dx dy cxy tgxdxctgydy ctgxdytgydx cxx x y cxxu dx x xdu xdx du dx du xu dx dy
4、uxyu x y y dx dy x cxarctgu dx x du u u u dx du xu dx du xu dx dy uxyu x y xy xy dx dy dxxydyxy ee ee e e e e xy u ux yx u u xy xy yx x x += = = += + = = = += = += = = = += += = = += += +=+ = + + + + =+ += + = =+ + , ) )ln ln(ln:9 3 3 1 1 . .cosos 3 3 2 x x y y u u ydxdydyy yx xx dy y y cx x tg t xg
5、 dx e e e x = = = + + 0 0 ln ln ct ctg gct ct y yg g dydy c cx x = = + + cxyxarctg cxarctgtdxdt dx dt dx dt dx dy tyx dx dy c dxdy dx dy t t yx ee ee e xy xy yx +=+ += + += +=+ = += = = + )( , 1 1 1 1 1, .11 2 2 2 )( 12 2 )( 1 yxdx dy + = cxyxarctgyx cxarctgttdxdt t t tdx dt dx dt dx dy tyx +=+ += +
6、 +=+ )( 1 1 1 1 2 2 2 U U dX dU XU X Y YX YX dX dY YyXx yxyxyx yx yx dx dy U 21 222 2 2 , 3 1 , 3 1 3 1 , 3 1 ; 012, 012 12 12 .13 2 + = =+= =+= + = cxyxarctg t tdxdxdt dt dx dt +=+ = + )( 1 1 Y X Xx x y yx x y yx x y yx dx dy 1 2 2 12 2 12 . = = + = gt = .7)5(7 2 1 77 2 1 7)7(, 7 1 ,1,5 2 5 ,14 ) 5
7、( 2 2 cxyx cxt dxdtt t t dx dt dx dt dx dy tyx yx yx dx dy yx t +=+ += = = + = + 15 18) 14() 1( 22 +=xyyx dx dy cxyxarctgdxdu u u dx du dx du dx dy xuyx yxxyyyxx dx dy +=+= + +=+=+ +=+= 6) 3 8 3 2 3 2 ( 94 1 4 9 4 1 4141 2) 14(18181612 2 2 222 16 225 26 2 2 yxxy xy dx dy + = uy xxy xy dx dy xxyy xy
8、dx dy = + = + = 3 23 2233 232 223 2 2)(3 2( 2)( 12 6 3 2 63 2 2 2 22 + = + = x u x u xxu xu dx du x xa ar rc ctg tg u dxdx dudu dxdx du dxdx dy yx xxy + + + = = + + += =+ 8 8 3 3 2 3 3 2 2 ( 4 4 1 ) ) 1 14( (1 18 2 2 225 2 2 2 2 2 yxxy x + dydy x xxyy y xy y dx dy = + + = 232 2 223 2( ( 2)( 3 2 2 6
9、63 3 2 22 + = = xuxu x xu u dxdx dudu dxdxdudu dy x x y yy y = + + + + 9 9 41 1 8 81616 2 2 cxxyxy cxyxycxxyxy cxzzdx x dz dzz z zz xyxyzzzz z zz dx dz x dx dz xz z z dx dz xz dx du z x u 153373 3353373 537 2 2 332 22 )2()3( 023)2()3 ,)2()3 112 06 2312306 ) 1.( 12 6 12 63 =+ =+ =+= + = + =+= + += 17
10、. yyyx xxyx dx dy + + = 32 3 23 32 123 132 ; ; ; ; ; ) 123( ) 132( 22 22 2 2 22 22 + + = + + = yx yx dx dy yxy yxx dx dy ) 1.( 123 132 ; ; ; ; ; ; ; ; ; ; ; , 22 + + = uv uv dv du vxuy 1111 0123 0132 += =+ =+ uYvZ uv uv + + = =+ =+ z y z y dz dy yz yz 23 32 1 023 032 )2.( 23 22 23 32 2 t t dz dt z t
11、 t dz dt zt dz dt zt dz dy z y t + = + + =+= 22222 2)2(1022xyxytt= cxyxydz z dt t t t 52222 2 2 )2( 1 22 23 022+=+= + cxyxyxyxy 522222222 )2(2+=+= 123 13 22 2 + + yx y ) 1.( 1 1 1= = = =uY Yv vZ dydy 1 1 dz dt dt zt dz dy += 2 2 02 22 2t= 22 2 ; ; ; ; ; ; ; ; ; ) ) 1 1dxdx 2 23 3 13 32 2 + + + = = u
12、v v u uv v dv 11 1 0 0 0 c yx x y dx x du u u u u x u u u u x yx yx dx dy y x xdydxyxy uxyxyf dx dy y x += =+ + = = + = + = + = + += += +=+= += = =+= + = =+ = 4 ln 1 4 2 2 41 ) 2 2 ( 1 dx du uxy(2) 0.x,c 2 0y,c 2 ,c 2 dx x 1 2u du ),(2u x 1 dx du u,xy 1 dx dy y x 0sxy0y0x:(1) u)(uf(u) x 1 1)(f(u) x
13、u 1)y(f(u)dx du f(u),1 dx du y 1 y dx du dx dy x, dx dy dx dy xyxu,xy 2 2 ).2( )1 (.1 )(18. 222 22 2 2 2 2 2 2 2 2 2 4 2 2 3 3 2 2 22 22 22 x y x y x y x y x u u u u y x 19.f(x)= x xfxdtxf 0 )(, 0, 1)(. f(x)=y, = x y dtxf 0 1 )( 1 2 y y y= cx y y cx dyy dx dx dy y + =+= 2 1 ; ; ; ; ; 1 2 1 ; ; ; ; ;
14、 ; ; ; ; ; ; ; 1 ; ; ; ; ; ; ; ; ; ; 23 3 cx y + = 2 1 = x y dtxf 0 1 )( y y dx x u u u u u u u x =+ + + = = = + += ln 1 1 ) 2 2 2 2 ( ( 1 dx du 0. 0. 0y, dx x 1 1 dx dy y 2 2 2 2 3 2 2 x u y y x x f( f x)= x x xdtxf 0 , 1)( f( f(x x) )=y y, , dydy y = = 3 du = = + + = = + + x x, ,c c 2 2 2 2 2 2 2
15、2 2 2 2 2 x x y y y x x y y x yccxccxcxdt ct x 2 1 , 02)2(; ; ; ; ; ; ; ; ; ;2 2 1 0 =+=+= + 20.x(t+s)= )()(1 )()( sxtx sxtx + x(t),x (0) t=s=0x(0)= )0(1 )0()0( x xx + = )0()0(1 )0(2 xx x x(0)0x 2=- 1 x(0)=0. x (t)=)(1)(0( )()(1 )(1)( lim )()( lim 2 2 txx txtxt txtx t txttx += + = + ) )(1)(0( )( 2 txx dt tdx +=dtx tx tdx )0( )(1 )
