Greens Theorem on R^2 | Topic-wise Tests & Solved Examples for Mathematics PDF Download

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 reen s  heorem on  
 
 
Statement (i)  et        e a  omain regular w r t  oth axes  
(ii)    e  oun e   y  lose   urve    
(iii) P(x y) an   (x y)  e two fun tions possessing  ontinuous partial  erivative on   
 hen ?(P x   y )
 
 ? (
  
 x
 
 P
 y
) x y
 
 
     in  the value of ?,(x y
 
) x (x
 
 y) y -
 
 taken in the  lo kwise sense along the  urve forme   y y
 
 x
 
 an  
the  hor  joining (   ) an  (   ) 
Solution  he equation of line joining the points (   ) an  (   ) is 
y   
   
   
(x  ) y x 
  ?,(x y
 
) x (x
 
 y) y -
 
 ? ,(x y
 
) x (x
 
 y) y -
 
 
 ? ,(x y
 
) x (x
 
 y) y -
 
 
  
 
  
 
 
 onsi er  
 
 
y
 
 x
 
  y x
 
 
  y 
 
 
x
 
 
 
 x 
  
 
 ? (x x
 
 
) x (x
 
 x
 
 
) 
 
 
x
 
 
 
 x
 
 
 
 ? (x x
 
 
) x 
 
 
(x
 
 
 x
 
 
) x
 
 
 
  
  
       verify 
 onsi er  
 
 
y x  y  x 
  
 
 ? (x x
 
) x (x
 
 x) x
 
 
 *
x
 
 
 
x
 
 
 
x
 
 
 
x
 
 
+
 
 
 *
 x
 
 
+
 
 
  
 
 
 
 ?,(x y
 
) x (x
 
 y) y -
 
 
  
  
 
 
 
 
     
    
 
 
  
 
     in  the value of ?(x
 
 y
 
) y
 
 taken in  ounter  lo kwise (anti lo kwise ) sense along the qua ralateral with  
verti es (   ) (   ) (   ) an  (   ) 
 o Yourself         ans
   
 
 
     erify  reen s  heorem for the line integral ?(x
 
 x y
 
 y )
 
 where   is the square x   y   x a y a 
 y  ine  ntegral 
?(x
 
 x xy y )
 
 (? ? ?
      
)(x
 
 x xy y ) 
 ? x
 
 x
 
 
 ? ay y
 
 
 ? x
 
 x
 
 
   
 
a
 
 
 
a
 
 
 
a
 
 
 
 [*
x
 
 
+
 
 
 a*
y
 
 
+
 
 
 *
x
 
 
+
 
 
] 
a
 
 
 
 y  reen s  horem 
  *(x y)   x a   y a+ 
 ?(x
 
 x xy y)
 
 ?[
 
 x
(xy) 
 
 y
(x
 
)] x y 
Page 2


  
 
     
 
 
 
 
  
 
 
 
 
 
    
 
 
  
 
   
  
   
 
  
 
 
  
 
 
 
   
 
  
 
  
   
        
 
  
 
 
 
     
  
 
 
  
 
 
 
 
 
 
  
 
 
  
   
 
  
 
 
  
 
 
 
  
 
 
 
  
  
   
 
 
 
 
 
 
 
  
 
 
 
  
 
  
 
  
 
 
  
 
 
 
 
 reen s  heorem on  
 
 
Statement (i)  et        e a  omain regular w r t  oth axes  
(ii)    e  oun e   y  lose   urve    
(iii) P(x y) an   (x y)  e two fun tions possessing  ontinuous partial  erivative on   
 hen ?(P x   y )
 
 ? (
  
 x
 
 P
 y
) x y
 
 
     in  the value of ?,(x y
 
) x (x
 
 y) y -
 
 taken in the  lo kwise sense along the  urve forme   y y
 
 x
 
 an  
the  hor  joining (   ) an  (   ) 
Solution  he equation of line joining the points (   ) an  (   ) is 
y   
   
   
(x  ) y x 
  ?,(x y
 
) x (x
 
 y) y -
 
 ? ,(x y
 
) x (x
 
 y) y -
 
 
 ? ,(x y
 
) x (x
 
 y) y -
 
 
  
 
  
 
 
 onsi er  
 
 
y
 
 x
 
  y x
 
 
  y 
 
 
x
 
 
 
 x 
  
 
 ? (x x
 
 
) x (x
 
 x
 
 
) 
 
 
x
 
 
 
 x
 
 
 
 ? (x x
 
 
) x 
 
 
(x
 
 
 x
 
 
) x
 
 
 
  
  
       verify 
 onsi er  
 
 
y x  y  x 
  
 
 ? (x x
 
) x (x
 
 x) x
 
 
 *
x
 
 
 
x
 
 
 
x
 
 
 
x
 
 
+
 
 
 *
 x
 
 
+
 
 
  
 
 
 
 ?,(x y
 
) x (x
 
 y) y -
 
 
  
  
 
 
 
 
     
    
 
 
  
 
     in  the value of ?(x
 
 y
 
) y
 
 taken in  ounter  lo kwise (anti lo kwise ) sense along the qua ralateral with  
verti es (   ) (   ) (   ) an  (   ) 
 o Yourself         ans
   
 
 
     erify  reen s  heorem for the line integral ?(x
 
 x y
 
 y )
 
 where   is the square x   y   x a y a 
 y  ine  ntegral 
?(x
 
 x xy y )
 
 (? ? ?
      
)(x
 
 x xy y ) 
 ? x
 
 x
 
 
 ? ay y
 
 
 ? x
 
 x
 
 
   
 
a
 
 
 
a
 
 
 
a
 
 
 
 [*
x
 
 
+
 
 
 a*
y
 
 
+
 
 
 *
x
 
 
+
 
 
] 
a
 
 
 
 y  reen s  horem 
  *(x y)   x a   y a+ 
 ?(x
 
 x xy y)
 
 ?[
 
 x
(xy) 
 
 y
(x
 
)] x y 
Free coa JAM 
 
 ? ? (y  ) x y
 
 
 
 
 ? ? y y
 
 
 
 
 *
a
 
 
x+
 
 
 
a
 
 
  
     erify  reen s  heorem for the  urve ?,(x y
 
) x (x
 
 y) y -
 
 where   is the  lose   urve forme   y y x  
an  y
 
 x
 
 in the first qua rant   
 y  ine  ntegral  
alre y  one 
 y  reen s  heorem 
  { (x y)   x   x y x
 
 
} 
 ? ? [
 
 x
(x
 
 y) 
 
 y
(x y
 
)] x y
 
 
 
   
 
   
 
 ? ?( x  y ) x y
 
 
 
   
 
   
  ? *xy 
y
 
 
+
 
 
 
 
 x
 
   
  
 
  
             verify 
     erify  reen s theorem for ?(x
 
y x xy
 
 y )
 
 where   is the  urve  oun e   y y
 
 x
 
an  y x in the first  
qua rant  
 y  ine  ntegral  o Yourself 
 
 
 
 y  reen s  heorem 
  { (x y)   x   x y x
 
 
} 
?(x
 
y x xy
 
 y ) ? ? (
 
 x
(xy
 
) 
 
 y
(x
 
y)) x
 
 
 
   
 y
 
   
 
 ? ?(y
 
 x
 
) x
 
 
 
   
 y
 
   
 
 ? *
y
 
 
 x
 
y+
 
 
 
 
 x
 
   
 ?
(
 
 
(x
 
 
)
 
 
 x
 
x
 
 
 
x
 
 
 x
 
)
 
 
 x
 
   
 
 
   
               (verify) 
     erify  reen s  heorem for ?( xy x
 
) x (x y
 
) y
 
 where   is the  lose     urve   oun e   y y x
 
 an   
x y
 
 in the first qua rant  
 y  ine  ntegral  o Yourself 
 y  reen s  heorem 
  { (x y)   x   x
 
 y vx} 
?( xy x
 
) x (x y
 
) y ? ? [
 
 x
(x y
 
) 
 
 y
( xy x
 
)] x
v 
   
 
 y
 
   
 
 ? ? (   x ) x y
v 
   
 
 
   
 
 ?,y  xy -
 
 
v 
 x
 
   
 ? [vx  x
 
 
 x
 
  x
 
] x
 
   
 
 
  
               (verify) 
     erify  reen s  heorem 
(i) ?(  x
 
)y x (  y
 
)x y
 
                    x
 
 y
 
 a
 
 
(ii) ?(  y
 
) x y y
 
                                     ysinx y  sinx 
 y  ine  ntegral  o Yourself 
Page 3


  
 
     
 
 
 
 
  
 
 
 
 
 
    
 
 
  
 
   
  
   
 
  
 
 
  
 
 
 
   
 
  
 
  
   
        
 
  
 
 
 
     
  
 
 
  
 
 
 
 
 
 
  
 
 
  
   
 
  
 
 
  
 
 
 
  
 
 
 
  
  
   
 
 
 
 
 
 
 
  
 
 
 
  
 
  
 
  
 
 
  
 
 
 
 
 reen s  heorem on  
 
 
Statement (i)  et        e a  omain regular w r t  oth axes  
(ii)    e  oun e   y  lose   urve    
(iii) P(x y) an   (x y)  e two fun tions possessing  ontinuous partial  erivative on   
 hen ?(P x   y )
 
 ? (
  
 x
 
 P
 y
) x y
 
 
     in  the value of ?,(x y
 
) x (x
 
 y) y -
 
 taken in the  lo kwise sense along the  urve forme   y y
 
 x
 
 an  
the  hor  joining (   ) an  (   ) 
Solution  he equation of line joining the points (   ) an  (   ) is 
y   
   
   
(x  ) y x 
  ?,(x y
 
) x (x
 
 y) y -
 
 ? ,(x y
 
) x (x
 
 y) y -
 
 
 ? ,(x y
 
) x (x
 
 y) y -
 
 
  
 
  
 
 
 onsi er  
 
 
y
 
 x
 
  y x
 
 
  y 
 
 
x
 
 
 
 x 
  
 
 ? (x x
 
 
) x (x
 
 x
 
 
) 
 
 
x
 
 
 
 x
 
 
 
 ? (x x
 
 
) x 
 
 
(x
 
 
 x
 
 
) x
 
 
 
  
  
       verify 
 onsi er  
 
 
y x  y  x 
  
 
 ? (x x
 
) x (x
 
 x) x
 
 
 *
x
 
 
 
x
 
 
 
x
 
 
 
x
 
 
+
 
 
 *
 x
 
 
+
 
 
  
 
 
 
 ?,(x y
 
) x (x
 
 y) y -
 
 
  
  
 
 
 
 
     
    
 
 
  
 
     in  the value of ?(x
 
 y
 
) y
 
 taken in  ounter  lo kwise (anti lo kwise ) sense along the qua ralateral with  
verti es (   ) (   ) (   ) an  (   ) 
 o Yourself         ans
   
 
 
     erify  reen s  heorem for the line integral ?(x
 
 x y
 
 y )
 
 where   is the square x   y   x a y a 
 y  ine  ntegral 
?(x
 
 x xy y )
 
 (? ? ?
      
)(x
 
 x xy y ) 
 ? x
 
 x
 
 
 ? ay y
 
 
 ? x
 
 x
 
 
   
 
a
 
 
 
a
 
 
 
a
 
 
 
 [*
x
 
 
+
 
 
 a*
y
 
 
+
 
 
 *
x
 
 
+
 
 
] 
a
 
 
 
 y  reen s  horem 
  *(x y)   x a   y a+ 
 ?(x
 
 x xy y)
 
 ?[
 
 x
(xy) 
 
 y
(x
 
)] x y 
Free coa JAM 
 
 ? ? (y  ) x y
 
 
 
 
 ? ? y y
 
 
 
 
 *
a
 
 
x+
 
 
 
a
 
 
  
     erify  reen s  heorem for the  urve ?,(x y
 
) x (x
 
 y) y -
 
 where   is the  lose   urve forme   y y x  
an  y
 
 x
 
 in the first qua rant   
 y  ine  ntegral  
alre y  one 
 y  reen s  heorem 
  { (x y)   x   x y x
 
 
} 
 ? ? [
 
 x
(x
 
 y) 
 
 y
(x y
 
)] x y
 
 
 
   
 
   
 
 ? ?( x  y ) x y
 
 
 
   
 
   
  ? *xy 
y
 
 
+
 
 
 
 
 x
 
   
  
 
  
             verify 
     erify  reen s theorem for ?(x
 
y x xy
 
 y )
 
 where   is the  urve  oun e   y y
 
 x
 
an  y x in the first  
qua rant  
 y  ine  ntegral  o Yourself 
 
 
 
 y  reen s  heorem 
  { (x y)   x   x y x
 
 
} 
?(x
 
y x xy
 
 y ) ? ? (
 
 x
(xy
 
) 
 
 y
(x
 
y)) x
 
 
 
   
 y
 
   
 
 ? ?(y
 
 x
 
) x
 
 
 
   
 y
 
   
 
 ? *
y
 
 
 x
 
y+
 
 
 
 
 x
 
   
 ?
(
 
 
(x
 
 
)
 
 
 x
 
x
 
 
 
x
 
 
 x
 
)
 
 
 x
 
   
 
 
   
               (verify) 
     erify  reen s  heorem for ?( xy x
 
) x (x y
 
) y
 
 where   is the  lose     urve   oun e   y y x
 
 an   
x y
 
 in the first qua rant  
 y  ine  ntegral  o Yourself 
 y  reen s  heorem 
  { (x y)   x   x
 
 y vx} 
?( xy x
 
) x (x y
 
) y ? ? [
 
 x
(x y
 
) 
 
 y
( xy x
 
)] x
v 
   
 
 y
 
   
 
 ? ? (   x ) x y
v 
   
 
 
   
 
 ?,y  xy -
 
 
v 
 x
 
   
 ? [vx  x
 
 
 x
 
  x
 
] x
 
   
 
 
  
               (verify) 
     erify  reen s  heorem 
(i) ?(  x
 
)y x (  y
 
)x y
 
                    x
 
 y
 
 a
 
 
(ii) ?(  y
 
) x y y
 
                                     ysinx y  sinx 
 y  ine  ntegral  o Yourself 
Free M 
 
?(  x
 
)y x (  y
 
)x y ?(
 
 x
(x xy
 
) 
 
 y
(y x
 
y)) x y 
 ?(  y
 
   x
 
) x y 
 ?(x
 
 y
 
) x y                              put x r os  y rsin    |J| r 
 ? ?(r
 
 os
 
  r
 
sin
 
 )r r
 
   
  
  
   
 ? ? r
 
 r  
 
   
  
   
 ? *
r
 
 
+
 
 
  
  
   
 
ra
 
 
        (verify) 
(ii)  o yourself 
    Show that ?(xy
 
 y x
 
y x )
 
 
  
  
 a
 
 where   is the  ar io  r a(   os )        ,anti lo kwise - 
Solution P  x
 
y 
 P
 y
  x
 
             xy
 
 
  
 x
 y
 
 
  y  reen s  heorem 
?(xy
 
 y yx
 
 x ) ?(y
 
 x
 
) x y          put x r os  y rsin   |J| r 
 ? ? r
 
 r  
 (      )
   
  
   
 ? *
r
 
 
+
 
 (      )
  
  
   
 
a
 
 
? (   os )
 
  
  
   
 
   a
 
  
         (verify) 
 
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27 docs|150 tests

FAQs on Greens Theorem on R^2 - Topic-wise Tests & Solved Examples for Mathematics

1. What is Green's Theorem in mathematics?
Ans. Green's Theorem is a fundamental result in vector calculus that relates a double integral over a region in the xy-plane to a line integral along the boundary of the region. It states that the line integral of a vector field around a simple closed curve is equal to the double integral of the curl of the vector field over the region enclosed by the curve.
2. How is Green's Theorem applied in two-dimensional mathematics?
Ans. Green's Theorem is commonly used to calculate line integrals and double integrals in two-dimensional space. It allows us to convert a difficult line integral into a more manageable double integral, or vice versa, by using the relationship between the curl of a vector field and the line integral of the vector field.
3. What is the significance of the curl in Green's Theorem?
Ans. The curl of a vector field represents the local rotation or circulation of the field at each point. In Green's Theorem, the curl of a vector field is integrated over a region, giving us information about the net circulation of the field around the boundary of the region. This circulation is then related to the line integral of the vector field using Green's Theorem.
4. Can Green's Theorem be used in three-dimensional mathematics?
Ans. No, Green's Theorem is specifically formulated for two-dimensional vector fields in the xy-plane. In three-dimensional mathematics, a similar theorem called Stokes' Theorem is used, which relates the surface integral of a vector field to the line integral of the vector field along the boundary of the surface.
5. What are some practical applications of Green's Theorem?
Ans. Green's Theorem has various applications in physics and engineering. It can be used to calculate the work done by a force field, the flow of a fluid, the circulation of a magnetic field, or the electric flux through a closed curve. It is also used in the study of fluid dynamics, electromagnetism, and heat transfer.
27 docs|150 tests
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