(XUIC ~OU£.. InONJ)IJY) tJI X=lA2_IIL2 'LJ'= (). +lJIN £U~rW1 N9--t~lA==-X+j(0. 0) VIU/YfCj II = K -...

Question

(XUIC ~OU£.. InONJ)IJY) tJI X=lA2_IIL2 'LJ'= (). +lJIN £U~rW1 N9--t~lA==-X+j(0. 0) VIU/YfCj II = K - 'J )S F~d('cC .' x= r~ f) J z: r.CJv:J 1"

Robert · Accepted Answer

Sol: (1)
 
 
 
2 2
2 2

,,
, ,
2 2,
2 2,
,
4 4
,
,
8
,
If
x u v
y u v
x x
x yx y u v
J
y yu v u v
u v
u vx y
J
u vu v
x y
J uv uv
u v
x y
J uv
u v
 
 
 
   
     
 
 
 
 
 
   
 
 
 
 

Sol: (2)  
The region of integration is a triangular ABC where A (-1, 1), B (1, 1) and                       
C (0, 0) in xy plane. The new region of integration is a triangular ' ' 'A B C in uv
plane.
          xy plane 
       
 
 
,
1,1
A x y
A

 
      
 
 
,
1,1
B x y
B


        
 
 
,
0,0
C x y
C


 
          uv plane   
 
 
' ,
' ,
' ( 1 1, 1 1)
' (0, 2)
A u v
A x y x y
A
A

  
    
 
 
 
 
' ,
' ,
' (1 1,1 1)
' (2,0)
B u v
B x y x y
B
B

  
  

 
 
 
' ,
' ,
' (0 0,0 0)
' (0,0)
A u v
A x y x y
A
A

  
  


And      
 
 
1
2
          
1
2
x u v
u x y
v x y
y u v
 
      
   
     
  

 
 
 
 
,
,
1 1
, 12 2
1 1,

(XUIC ~ OU£.. InONJ)IJY /) tJI X=lA2_IIL 2 'LJ'= (). +lJ IN £U~rW1 N9--t~ lA==-X+j (0. 0) VIU/YfCj II = K - 'J b ) S F~d(' c C .' x= r~ f) J z: r.CJv:J 1"

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