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Page 1
Edurev123
7. Gauss-Divergence Theorem Stoles
Theorem and Green’s to Identity
7.1 Find the work done in moving the particle on 2 round the ellipse
?? ?? ????
+
?? ?? ????
=?? ,?? =
?? under the field of force given by
????
=(?? ?? -?? +?? )??ˆ+(?? +?? -?? )??ˆ+(?? ?? -?? ?? +?? ?? )??ˆ
(2009 : 20 Marks)
Solution:
Let ?? :
?? 2
25
+
?? 2
16
=1,?? =0 denote the ellipse.
Work done =?
?? ??
?
·?? ??
By Stoke's theorem
?
?? ??
·?? ?? =??
?? (?×??
)·??ˆ????
where ?? is the surface of the ellipse
?×??
=
|
|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? 2?? -?? +?? ?? +?? -?? 3?? -2?? +4?? |
|
=-??ˆ-2??ˆ+2??ˆ
For. Snormal is along positive z-axis
? ??ˆ =??ˆ
(?×??
)·??ˆ =2
? ??
?? ?(?×??
)·??ˆ???? =??
?? ?2???? =2× Area of ellipse
=2?????? =2?? ·5·4
=40??
7.2 Using divergent theorem evaluate ?
?? ?????
·?? ????
where ????
=?? ?? ??ˆ+?? ?? ??ˆ+?? ?? ??ˆ
and ?? is
the surface of the sphere ?? ?? +?? ?? +?? ?? =?? ?? .
(2009 : 20 Marks)
Solution:
Page 2
Edurev123
7. Gauss-Divergence Theorem Stoles
Theorem and Green’s to Identity
7.1 Find the work done in moving the particle on 2 round the ellipse
?? ?? ????
+
?? ?? ????
=?? ,?? =
?? under the field of force given by
????
=(?? ?? -?? +?? )??ˆ+(?? +?? -?? )??ˆ+(?? ?? -?? ?? +?? ?? )??ˆ
(2009 : 20 Marks)
Solution:
Let ?? :
?? 2
25
+
?? 2
16
=1,?? =0 denote the ellipse.
Work done =?
?? ??
?
·?? ??
By Stoke's theorem
?
?? ??
·?? ?? =??
?? (?×??
)·??ˆ????
where ?? is the surface of the ellipse
?×??
=
|
|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? 2?? -?? +?? ?? +?? -?? 3?? -2?? +4?? |
|
=-??ˆ-2??ˆ+2??ˆ
For. Snormal is along positive z-axis
? ??ˆ =??ˆ
(?×??
)·??ˆ =2
? ??
?? ?(?×??
)·??ˆ???? =??
?? ?2???? =2× Area of ellipse
=2?????? =2?? ·5·4
=40??
7.2 Using divergent theorem evaluate ?
?? ?????
·?? ????
where ????
=?? ?? ??ˆ+?? ?? ??ˆ+?? ?? ??ˆ
and ?? is
the surface of the sphere ?? ?? +?? ?? +?? ?? =?? ?? .
(2009 : 20 Marks)
Solution:
By gauss divergence theorem
??
?? ???
·?? ??
=??
?? ?(?·??
)????
?·??
=?
?
??? ??ˆ·??
=3?? 2
+3?? 2
+3?? 2
=3(?? 2
+?? 2
+?? 2
)
?? is surface of sphere ?? 2
+?? 2
+?? 2
=?? 2
.
??? is the volume compressing the whole sphere.
??
?? ???
·?? ??
=??
?? ?(?·??
)????
=? 1(3(?? 2
+?? 2
+?? 2
)????
Converting to spherical polar coordinates.
?? 2
+?? 2
+?? 2
=?? 2
???? =?? 2
sin ?????????????? ?? =? ?
2?? ?? =0
?? ?
2?? ?? =0
?? ?
?? ?? =0
?3?? 2
·?? 2
sin ?????????????? =? ?
2?? 0
?(? ?
2?? 0
?? ?
2
0
?(3?? 4
???? )sin ?????? )????
=? ?
?? 0
?[
3?? 5
5
? ?
?? 0
?sin ?????? ]????
=
3?? 5
5
[-cos ?? ]
0
?? ?? =
6?? ?? 5
5
7.3 Find the value of ?
?? ?(????
×????
)·?? ????
taken over the upper portion of the surface
?? ?? +?? ?? -?? ???? +???? =?? and the boundary curve lies on the plane ?? =?? when
????
=(?? ?? +?? ?? -?? ?? )??ˆ+(?? ?? +?? ?? -?? ?? )??ˆ+(?? ?? +?? ?? -?? ?? )??ˆ
(2009 : 20 Mar's)
Solution:
The given surface ?? 1
?? 2
+?? 2
-2???? +???? =0
? (?? -?? )
2
+?? 2
=?? 2
-????
Page 3
Edurev123
7. Gauss-Divergence Theorem Stoles
Theorem and Green’s to Identity
7.1 Find the work done in moving the particle on 2 round the ellipse
?? ?? ????
+
?? ?? ????
=?? ,?? =
?? under the field of force given by
????
=(?? ?? -?? +?? )??ˆ+(?? +?? -?? )??ˆ+(?? ?? -?? ?? +?? ?? )??ˆ
(2009 : 20 Marks)
Solution:
Let ?? :
?? 2
25
+
?? 2
16
=1,?? =0 denote the ellipse.
Work done =?
?? ??
?
·?? ??
By Stoke's theorem
?
?? ??
·?? ?? =??
?? (?×??
)·??ˆ????
where ?? is the surface of the ellipse
?×??
=
|
|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? 2?? -?? +?? ?? +?? -?? 3?? -2?? +4?? |
|
=-??ˆ-2??ˆ+2??ˆ
For. Snormal is along positive z-axis
? ??ˆ =??ˆ
(?×??
)·??ˆ =2
? ??
?? ?(?×??
)·??ˆ???? =??
?? ?2???? =2× Area of ellipse
=2?????? =2?? ·5·4
=40??
7.2 Using divergent theorem evaluate ?
?? ?????
·?? ????
where ????
=?? ?? ??ˆ+?? ?? ??ˆ+?? ?? ??ˆ
and ?? is
the surface of the sphere ?? ?? +?? ?? +?? ?? =?? ?? .
(2009 : 20 Marks)
Solution:
By gauss divergence theorem
??
?? ???
·?? ??
=??
?? ?(?·??
)????
?·??
=?
?
??? ??ˆ·??
=3?? 2
+3?? 2
+3?? 2
=3(?? 2
+?? 2
+?? 2
)
?? is surface of sphere ?? 2
+?? 2
+?? 2
=?? 2
.
??? is the volume compressing the whole sphere.
??
?? ???
·?? ??
=??
?? ?(?·??
)????
=? 1(3(?? 2
+?? 2
+?? 2
)????
Converting to spherical polar coordinates.
?? 2
+?? 2
+?? 2
=?? 2
???? =?? 2
sin ?????????????? ?? =? ?
2?? ?? =0
?? ?
2?? ?? =0
?? ?
?? ?? =0
?3?? 2
·?? 2
sin ?????????????? =? ?
2?? 0
?(? ?
2?? 0
?? ?
2
0
?(3?? 4
???? )sin ?????? )????
=? ?
?? 0
?[
3?? 5
5
? ?
?? 0
?sin ?????? ]????
=
3?? 5
5
[-cos ?? ]
0
?? ?? =
6?? ?? 5
5
7.3 Find the value of ?
?? ?(????
×????
)·?? ????
taken over the upper portion of the surface
?? ?? +?? ?? -?? ???? +???? =?? and the boundary curve lies on the plane ?? =?? when
????
=(?? ?? +?? ?? -?? ?? )??ˆ+(?? ?? +?? ?? -?? ?? )??ˆ+(?? ?? +?? ?? -?? ?? )??ˆ
(2009 : 20 Mar's)
Solution:
The given surface ?? 1
?? 2
+?? 2
-2???? +???? =0
? (?? -?? )
2
+?? 2
=?? 2
-????
is of an inverted paraboloid as there is maximum value of zmax=a. Assumption ?? =
+???? . Had ' ?? ' been-ve, it would not be inverted paraboloid. Consider the part of the
paraboloid enclosed by ?? 1
and ?? 2
i.e., the circle of intersection in the XY-plane.
?? 2
(?? -?? )
2
+?? 2
=?? 2
,?? =0
These surfaces enclosed a volume.
Using gauss divergence theorem
? ?
?? =?? 1
+?? 2
?(?×??
)·?? ??
=??
?? ??·(?×??
)????
=0 as ?·(?×??
)=0
? ?
?? 1
?(?×??
)·???? =-?
?? 2
?(?×??
)·????
Now for ?? 2
??ˆ = -??ˆ
(outward normal is towards
downward direction).
?×??
=
|
|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? ?? 2
+?? 2
-?? 2
?? 2
+?? 2
-?? 2
?? 2
+?? 2
-?? 2
|
|
=(2?? -2?? )??ˆ+2(?? -?? )??ˆ+2(?? -?? )??ˆ
? (?×??
)·??ˆ =2(?? -?? )
?? =-? ?
?? 2
?2(?? -?? )???? =? 2(?? -?? )????
Page 4
Edurev123
7. Gauss-Divergence Theorem Stoles
Theorem and Green’s to Identity
7.1 Find the work done in moving the particle on 2 round the ellipse
?? ?? ????
+
?? ?? ????
=?? ,?? =
?? under the field of force given by
????
=(?? ?? -?? +?? )??ˆ+(?? +?? -?? )??ˆ+(?? ?? -?? ?? +?? ?? )??ˆ
(2009 : 20 Marks)
Solution:
Let ?? :
?? 2
25
+
?? 2
16
=1,?? =0 denote the ellipse.
Work done =?
?? ??
?
·?? ??
By Stoke's theorem
?
?? ??
·?? ?? =??
?? (?×??
)·??ˆ????
where ?? is the surface of the ellipse
?×??
=
|
|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? 2?? -?? +?? ?? +?? -?? 3?? -2?? +4?? |
|
=-??ˆ-2??ˆ+2??ˆ
For. Snormal is along positive z-axis
? ??ˆ =??ˆ
(?×??
)·??ˆ =2
? ??
?? ?(?×??
)·??ˆ???? =??
?? ?2???? =2× Area of ellipse
=2?????? =2?? ·5·4
=40??
7.2 Using divergent theorem evaluate ?
?? ?????
·?? ????
where ????
=?? ?? ??ˆ+?? ?? ??ˆ+?? ?? ??ˆ
and ?? is
the surface of the sphere ?? ?? +?? ?? +?? ?? =?? ?? .
(2009 : 20 Marks)
Solution:
By gauss divergence theorem
??
?? ???
·?? ??
=??
?? ?(?·??
)????
?·??
=?
?
??? ??ˆ·??
=3?? 2
+3?? 2
+3?? 2
=3(?? 2
+?? 2
+?? 2
)
?? is surface of sphere ?? 2
+?? 2
+?? 2
=?? 2
.
??? is the volume compressing the whole sphere.
??
?? ???
·?? ??
=??
?? ?(?·??
)????
=? 1(3(?? 2
+?? 2
+?? 2
)????
Converting to spherical polar coordinates.
?? 2
+?? 2
+?? 2
=?? 2
???? =?? 2
sin ?????????????? ?? =? ?
2?? ?? =0
?? ?
2?? ?? =0
?? ?
?? ?? =0
?3?? 2
·?? 2
sin ?????????????? =? ?
2?? 0
?(? ?
2?? 0
?? ?
2
0
?(3?? 4
???? )sin ?????? )????
=? ?
?? 0
?[
3?? 5
5
? ?
?? 0
?sin ?????? ]????
=
3?? 5
5
[-cos ?? ]
0
?? ?? =
6?? ?? 5
5
7.3 Find the value of ?
?? ?(????
×????
)·?? ????
taken over the upper portion of the surface
?? ?? +?? ?? -?? ???? +???? =?? and the boundary curve lies on the plane ?? =?? when
????
=(?? ?? +?? ?? -?? ?? )??ˆ+(?? ?? +?? ?? -?? ?? )??ˆ+(?? ?? +?? ?? -?? ?? )??ˆ
(2009 : 20 Mar's)
Solution:
The given surface ?? 1
?? 2
+?? 2
-2???? +???? =0
? (?? -?? )
2
+?? 2
=?? 2
-????
is of an inverted paraboloid as there is maximum value of zmax=a. Assumption ?? =
+???? . Had ' ?? ' been-ve, it would not be inverted paraboloid. Consider the part of the
paraboloid enclosed by ?? 1
and ?? 2
i.e., the circle of intersection in the XY-plane.
?? 2
(?? -?? )
2
+?? 2
=?? 2
,?? =0
These surfaces enclosed a volume.
Using gauss divergence theorem
? ?
?? =?? 1
+?? 2
?(?×??
)·?? ??
=??
?? ??·(?×??
)????
=0 as ?·(?×??
)=0
? ?
?? 1
?(?×??
)·???? =-?
?? 2
?(?×??
)·????
Now for ?? 2
??ˆ = -??ˆ
(outward normal is towards
downward direction).
?×??
=
|
|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? ?? 2
+?? 2
-?? 2
?? 2
+?? 2
-?? 2
?? 2
+?? 2
-?? 2
|
|
=(2?? -2?? )??ˆ+2(?? -?? )??ˆ+2(?? -?? )??ˆ
? (?×??
)·??ˆ =2(?? -?? )
?? =-? ?
?? 2
?2(?? -?? )???? =? 2(?? -?? )????
Converting to polar coordinates.
?? =?? +?? cos ?? ?? =?? sin ?? ???? =?????????? ?? =2? ?
2?? ?? =0
?? ?
?? ?? =0
?(?? +?? cos ?? -?? sin ?? )?????????? =2? ?
2?? ?? =0
?[
?? ?? 2
2
+
?? 3
3
(cos ?? -sin ?? )]
0
?? ????
=2? ?
2?? ?? =0
?
?? 3
2
+
?? 3
3
(cos ?? -sin ?? )????
=2·
?? 3
2
·2?? (as integral of sin ?? and cos ?? over 0 to 2?? is zero).
=2?? ?? 3
7.4 Verify Green's Theorem for
?? -?? ?????? ?????? +?? -?? ?????? ??????
the path of integration being the boundary of the square whose vertices are
(?? ,?? ),(
?? ?? ,?? ),(
?? ?? ,
?? ?? ) and (?? ,
?? ?? ) .
(2010: 20 marks)
Solution:
By Green's Theorem,
? (?????? +?????? )=? (
??? ??? -
??? ??? )????????
Given equation is ?? -?? sin ?????? +?? -?? cos ??????
Here,
Page 5
Edurev123
7. Gauss-Divergence Theorem Stoles
Theorem and Green’s to Identity
7.1 Find the work done in moving the particle on 2 round the ellipse
?? ?? ????
+
?? ?? ????
=?? ,?? =
?? under the field of force given by
????
=(?? ?? -?? +?? )??ˆ+(?? +?? -?? )??ˆ+(?? ?? -?? ?? +?? ?? )??ˆ
(2009 : 20 Marks)
Solution:
Let ?? :
?? 2
25
+
?? 2
16
=1,?? =0 denote the ellipse.
Work done =?
?? ??
?
·?? ??
By Stoke's theorem
?
?? ??
·?? ?? =??
?? (?×??
)·??ˆ????
where ?? is the surface of the ellipse
?×??
=
|
|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? 2?? -?? +?? ?? +?? -?? 3?? -2?? +4?? |
|
=-??ˆ-2??ˆ+2??ˆ
For. Snormal is along positive z-axis
? ??ˆ =??ˆ
(?×??
)·??ˆ =2
? ??
?? ?(?×??
)·??ˆ???? =??
?? ?2???? =2× Area of ellipse
=2?????? =2?? ·5·4
=40??
7.2 Using divergent theorem evaluate ?
?? ?????
·?? ????
where ????
=?? ?? ??ˆ+?? ?? ??ˆ+?? ?? ??ˆ
and ?? is
the surface of the sphere ?? ?? +?? ?? +?? ?? =?? ?? .
(2009 : 20 Marks)
Solution:
By gauss divergence theorem
??
?? ???
·?? ??
=??
?? ?(?·??
)????
?·??
=?
?
??? ??ˆ·??
=3?? 2
+3?? 2
+3?? 2
=3(?? 2
+?? 2
+?? 2
)
?? is surface of sphere ?? 2
+?? 2
+?? 2
=?? 2
.
??? is the volume compressing the whole sphere.
??
?? ???
·?? ??
=??
?? ?(?·??
)????
=? 1(3(?? 2
+?? 2
+?? 2
)????
Converting to spherical polar coordinates.
?? 2
+?? 2
+?? 2
=?? 2
???? =?? 2
sin ?????????????? ?? =? ?
2?? ?? =0
?? ?
2?? ?? =0
?? ?
?? ?? =0
?3?? 2
·?? 2
sin ?????????????? =? ?
2?? 0
?(? ?
2?? 0
?? ?
2
0
?(3?? 4
???? )sin ?????? )????
=? ?
?? 0
?[
3?? 5
5
? ?
?? 0
?sin ?????? ]????
=
3?? 5
5
[-cos ?? ]
0
?? ?? =
6?? ?? 5
5
7.3 Find the value of ?
?? ?(????
×????
)·?? ????
taken over the upper portion of the surface
?? ?? +?? ?? -?? ???? +???? =?? and the boundary curve lies on the plane ?? =?? when
????
=(?? ?? +?? ?? -?? ?? )??ˆ+(?? ?? +?? ?? -?? ?? )??ˆ+(?? ?? +?? ?? -?? ?? )??ˆ
(2009 : 20 Mar's)
Solution:
The given surface ?? 1
?? 2
+?? 2
-2???? +???? =0
? (?? -?? )
2
+?? 2
=?? 2
-????
is of an inverted paraboloid as there is maximum value of zmax=a. Assumption ?? =
+???? . Had ' ?? ' been-ve, it would not be inverted paraboloid. Consider the part of the
paraboloid enclosed by ?? 1
and ?? 2
i.e., the circle of intersection in the XY-plane.
?? 2
(?? -?? )
2
+?? 2
=?? 2
,?? =0
These surfaces enclosed a volume.
Using gauss divergence theorem
? ?
?? =?? 1
+?? 2
?(?×??
)·?? ??
=??
?? ??·(?×??
)????
=0 as ?·(?×??
)=0
? ?
?? 1
?(?×??
)·???? =-?
?? 2
?(?×??
)·????
Now for ?? 2
??ˆ = -??ˆ
(outward normal is towards
downward direction).
?×??
=
|
|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? ?? 2
+?? 2
-?? 2
?? 2
+?? 2
-?? 2
?? 2
+?? 2
-?? 2
|
|
=(2?? -2?? )??ˆ+2(?? -?? )??ˆ+2(?? -?? )??ˆ
? (?×??
)·??ˆ =2(?? -?? )
?? =-? ?
?? 2
?2(?? -?? )???? =? 2(?? -?? )????
Converting to polar coordinates.
?? =?? +?? cos ?? ?? =?? sin ?? ???? =?????????? ?? =2? ?
2?? ?? =0
?? ?
?? ?? =0
?(?? +?? cos ?? -?? sin ?? )?????????? =2? ?
2?? ?? =0
?[
?? ?? 2
2
+
?? 3
3
(cos ?? -sin ?? )]
0
?? ????
=2? ?
2?? ?? =0
?
?? 3
2
+
?? 3
3
(cos ?? -sin ?? )????
=2·
?? 3
2
·2?? (as integral of sin ?? and cos ?? over 0 to 2?? is zero).
=2?? ?? 3
7.4 Verify Green's Theorem for
?? -?? ?????? ?????? +?? -?? ?????? ??????
the path of integration being the boundary of the square whose vertices are
(?? ,?? ),(
?? ?? ,?? ),(
?? ?? ,
?? ?? ) and (?? ,
?? ?? ) .
(2010: 20 marks)
Solution:
By Green's Theorem,
? (?????? +?????? )=? (
??? ??? -
??? ??? )????????
Given equation is ?? -?? sin ?????? +?? -?? cos ??????
Here,
?? =?? -?? sin ?? ?? =?? -?? cos ??
? ?(?? -?? sin ?????? +?? -?? cos ?????? )=?(
??? -?? cos ?? ??? -
??? -?? sin ?? ??? )???????? (By Green's Theorem)
R.H.S. : ?(
??? -?? cos ?? ??? -
??? -?? sin ?? ??? )????????
=? (-?? -?? cos ?? -?? -?? cos ?? )???? ?? '
?? =-2? ?
?? /2
?? =0
?? ?
?? /2
?? =0
??? -?? cos ?????????? =-2[
?? -?? -1
]
0
?? /2
[sin ?? ]
0
?? /2
=
+2
+1
(?? -?? /2
-1)(1-0)=2(?? -?? /2
-1)
L.H.S. : ??? -?? sin ?????? +?? -?? cos ??????
In figure, from ?? to ?? :
? (?? -?? sin ?????? +?? -?? cos ?????? ) =? 0???? +0=0
as sin ?? =0
?????? ???? =0
from ?? to ?? :
? (?? -?? sin ?????? +?? -?? cos ?????? ) =0+? ?
?? /2
?? =0
??? -?? /2
cos?????? (???? =0;?? =
?? 2
)
=?? -?? /2
[sin ?? ]
0
?? /2
=?? -?? /2
from ?? to ?? :
? (?? -?? sin ?????? +?? -?? cos ?????? ) =? ?
0
?? =
?? 2
??? -?? ????
=-[?? -?? ]
?? /2
0
=-[1-?? -?? /2
]=?? -?? /2
-1
from ?? to ?? :
? (?? -?? sin ?????? +?? -?? cos ?????? )=0+? ?? -?? cos ??????
=? ?
0
?? =
?? 2
?cos?? ???? =[sin?? ]
?? /2
0
=0-1=-1
Line integral along ?? ??? ??? ??? ??? ???? 0+?? -?? /2
+?? -?? /2
-1-1
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FAQs on Gauss-Divergence Theorem Stoles Theorem and Green’s to Identity - Mathematics Optional Notes for UPSC
| 1. What is the Gauss-Divergence Theorem? |  |
Ans. The Gauss-Divergence Theorem, also known as Gauss's Theorem, states that the outward flux of a vector field through a closed surface is equal to the divergence of the vector field over the enclosed volume.
| 2. Can you explain Stoles Theorem? | |
Ans. Stoles Theorem, also known as Stokes' Theorem, relates the surface integral of the curl of a vector field over a surface to the line integral of the vector field over the boundary of the surface.
| 3. How is Green's Identity used in mathematics? | |
Ans. Green's Identity is a fundamental theorem in vector calculus that relates a double integral over a region to a line integral around the boundary of the region. It is used to establish relationships between line integrals and double integrals in various mathematical applications.
| 4. What are the practical applications of the Gauss-Divergence Theorem? | |
Ans. The Gauss-Divergence Theorem has various practical applications in physics and engineering, such as fluid dynamics, electromagnetism, and heat transfer. It is used to analyze the flow of fluids, the distribution of electric fields, and the transfer of heat in different systems.
| 5. How can one apply Stokes' Theorem in real-life situations? | |
Ans. Stokes' Theorem is used in various real-life situations, such as in electromagnetic theory to calculate the circulation of electric and magnetic fields around closed loops. It is also applied in fluid dynamics to analyze the circulation of fluid flow around closed paths.
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Nov 21, 2024 Last updated |
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