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Greens Theorem on R^2
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Page 1
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)
Read MoreFAQs on Greens Theorem on R^2
| 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.
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