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Page 1
Edurev123
4. Vector Identity and Vector Equations
4.1 Show that div (grad ?? ?? )=?? (?? +?? )?? ?? -??
(2009: 12 marks)
Solution:
?·(??? ?? ) =?? (?? +1)?? ?? -2
?? 2
=?? 2
+?? 2
+?? 2
Differentiating partially with respect to ?? on both sides
2?? ??? ??? =2?? ?
??? ??? =
?? ?? ??????????????????
??? ??? =
?? ?? ;
??? ??? =
?? ?? grad ?? ?? =?
?
??? ??ˆ(?? ?? )=?
??? ?? ??? ??ˆ
=? ?? ?? ?? -1
??? ??? ??ˆ=?? ?? ?? -1
?
?? ?? ??ˆ
=?? ?? ?? -2
??
?? h?????? , ?? =?? ??ˆ+?? ??ˆ+?? ??ˆ
div (grad ?? ?? ) =?
?
??? ??ˆ·?? ?? ?? -2
? ?? ??ˆ=?
?
??? (?? ?? ?? -2
?? )
=? (?? ?? ?? -2
+?? (?? -2)?? ?? -3
·?? ??? ??? )
=3?? ?? ?? -2
+? ?? (?? -2)?? ?? -3
?? 2
?? =3?? ?? ?? -2
+?? (?? -2)?? ?? -4
? ?? 2
=?? (?? -2)?? ?? -2
+3?? ?? ?? -2
=?? (?? +1)?? ?? -2
4.2 Find the directional derivative of
?? (?? ,?? )=?? ?? ?? ?? +????
at the point (?? ,?? ) in the direction of a unit vector which makes an angle of
?? ?? with
the ?? -axis.
(2010: 12 Marks)
Solution:
Page 2
Edurev123
4. Vector Identity and Vector Equations
4.1 Show that div (grad ?? ?? )=?? (?? +?? )?? ?? -??
(2009: 12 marks)
Solution:
?·(??? ?? ) =?? (?? +1)?? ?? -2
?? 2
=?? 2
+?? 2
+?? 2
Differentiating partially with respect to ?? on both sides
2?? ??? ??? =2?? ?
??? ??? =
?? ?? ??????????????????
??? ??? =
?? ?? ;
??? ??? =
?? ?? grad ?? ?? =?
?
??? ??ˆ(?? ?? )=?
??? ?? ??? ??ˆ
=? ?? ?? ?? -1
??? ??? ??ˆ=?? ?? ?? -1
?
?? ?? ??ˆ
=?? ?? ?? -2
??
?? h?????? , ?? =?? ??ˆ+?? ??ˆ+?? ??ˆ
div (grad ?? ?? ) =?
?
??? ??ˆ·?? ?? ?? -2
? ?? ??ˆ=?
?
??? (?? ?? ?? -2
?? )
=? (?? ?? ?? -2
+?? (?? -2)?? ?? -3
·?? ??? ??? )
=3?? ?? ?? -2
+? ?? (?? -2)?? ?? -3
?? 2
?? =3?? ?? ?? -2
+?? (?? -2)?? ?? -4
? ?? 2
=?? (?? -2)?? ?? -2
+3?? ?? ?? -2
=?? (?? +1)?? ?? -2
4.2 Find the directional derivative of
?? (?? ,?? )=?? ?? ?? ?? +????
at the point (?? ,?? ) in the direction of a unit vector which makes an angle of
?? ?? with
the ?? -axis.
(2010: 12 Marks)
Solution:
?????????? : ?? (?? ,?? ) =?? 2
?? 3
+????
??? =
??? ??? ??ˆ+
??? ??? ??ˆ=
?(?? 2
?? 3
+???? )
??? ??ˆ+
?(?? 2
?? 3
+???? )
??? ??ˆ
=(2?? ?? 3
+?? )??ˆ+(3?? 2
?? 2
+?? )??ˆ
Now, (??? )
(2,1)
=(2×2×?? 3
+1)??ˆ+(3×2
2
×??ˆ
-
2
+2)??ˆ
=5?? +14??
(?)
(2,1)
is the direction of unit vector at angle
?? 3
with ?? -axis is
(??? )
(2,1)
·(cos
?? 3
??ˆ+sin
?? 3
??ˆ)=(5??ˆ+14??ˆ)·(
1
2
??ˆ+
v3
2
??ˆ)=
5
2
+7v3
4.3 Show that the vector field defined by the vector function
???? =?????? (???? ??ˆ+???? ??ˆ+???? ??ˆ
)
is conservative.
(2010: 12 marks)
Solution:
Given :
?? =?????? (???? ??ˆ+???? ??ˆ+???? ??ˆ
)
=?? ?? 2
?? 2
??ˆ+?? 2
?? ?? 2
??ˆ+?? 2
?? 2
?? ??ˆ
?×?? =
|
|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? ?? ?? 2
?? 2
?? 2
?? ?? 2
?? 2
?? 2
?? |
|
=??ˆ(
?
??? ?? 2
?? 2
?? -
?
??? ?? 2
?? ?? 2
)-??ˆ(
?
??? ?? 2
?? 2
?? -
?
??? ?? ?? 2
?? 2
)+??ˆ
(
?
??? ?? 2
?? ?? 2
-
?
??? ?? ?? 2
?? 2
)
=??ˆ(2?? 2
?? 2
-2?? 2
?? 2
)-??ˆ(2?? ?? 2
?? -2?? ?? 2
?? )+??ˆ
(2???? ?? 2
-2???? ?? 2
)=0
as ?×?? =0.??? is conservative.
4.4 Prove that :
?????? (?? ???? )=?? (?????? ???? )+(???????? ?? )·????
where ?? is a scalar funcion.
(2011 : 20 Marks)
Solution:
Page 3
Edurev123
4. Vector Identity and Vector Equations
4.1 Show that div (grad ?? ?? )=?? (?? +?? )?? ?? -??
(2009: 12 marks)
Solution:
?·(??? ?? ) =?? (?? +1)?? ?? -2
?? 2
=?? 2
+?? 2
+?? 2
Differentiating partially with respect to ?? on both sides
2?? ??? ??? =2?? ?
??? ??? =
?? ?? ??????????????????
??? ??? =
?? ?? ;
??? ??? =
?? ?? grad ?? ?? =?
?
??? ??ˆ(?? ?? )=?
??? ?? ??? ??ˆ
=? ?? ?? ?? -1
??? ??? ??ˆ=?? ?? ?? -1
?
?? ?? ??ˆ
=?? ?? ?? -2
??
?? h?????? , ?? =?? ??ˆ+?? ??ˆ+?? ??ˆ
div (grad ?? ?? ) =?
?
??? ??ˆ·?? ?? ?? -2
? ?? ??ˆ=?
?
??? (?? ?? ?? -2
?? )
=? (?? ?? ?? -2
+?? (?? -2)?? ?? -3
·?? ??? ??? )
=3?? ?? ?? -2
+? ?? (?? -2)?? ?? -3
?? 2
?? =3?? ?? ?? -2
+?? (?? -2)?? ?? -4
? ?? 2
=?? (?? -2)?? ?? -2
+3?? ?? ?? -2
=?? (?? +1)?? ?? -2
4.2 Find the directional derivative of
?? (?? ,?? )=?? ?? ?? ?? +????
at the point (?? ,?? ) in the direction of a unit vector which makes an angle of
?? ?? with
the ?? -axis.
(2010: 12 Marks)
Solution:
?????????? : ?? (?? ,?? ) =?? 2
?? 3
+????
??? =
??? ??? ??ˆ+
??? ??? ??ˆ=
?(?? 2
?? 3
+???? )
??? ??ˆ+
?(?? 2
?? 3
+???? )
??? ??ˆ
=(2?? ?? 3
+?? )??ˆ+(3?? 2
?? 2
+?? )??ˆ
Now, (??? )
(2,1)
=(2×2×?? 3
+1)??ˆ+(3×2
2
×??ˆ
-
2
+2)??ˆ
=5?? +14??
(?)
(2,1)
is the direction of unit vector at angle
?? 3
with ?? -axis is
(??? )
(2,1)
·(cos
?? 3
??ˆ+sin
?? 3
??ˆ)=(5??ˆ+14??ˆ)·(
1
2
??ˆ+
v3
2
??ˆ)=
5
2
+7v3
4.3 Show that the vector field defined by the vector function
???? =?????? (???? ??ˆ+???? ??ˆ+???? ??ˆ
)
is conservative.
(2010: 12 marks)
Solution:
Given :
?? =?????? (???? ??ˆ+???? ??ˆ+???? ??ˆ
)
=?? ?? 2
?? 2
??ˆ+?? 2
?? ?? 2
??ˆ+?? 2
?? 2
?? ??ˆ
?×?? =
|
|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? ?? ?? 2
?? 2
?? 2
?? ?? 2
?? 2
?? 2
?? |
|
=??ˆ(
?
??? ?? 2
?? 2
?? -
?
??? ?? 2
?? ?? 2
)-??ˆ(
?
??? ?? 2
?? 2
?? -
?
??? ?? ?? 2
?? 2
)+??ˆ
(
?
??? ?? 2
?? ?? 2
-
?
??? ?? ?? 2
?? 2
)
=??ˆ(2?? 2
?? 2
-2?? 2
?? 2
)-??ˆ(2?? ?? 2
?? -2?? ?? 2
?? )+??ˆ
(2???? ?? 2
-2???? ?? 2
)=0
as ?×?? =0.??? is conservative.
4.4 Prove that :
?????? (?? ???? )=?? (?????? ???? )+(???????? ?? )·????
where ?? is a scalar funcion.
(2011 : 20 Marks)
Solution:
LHS =div (?? ???
)
RHS=?? div ?? +( gradt )·???
Take LHS :
div (?? ) =S?? ?
??? (?? ?? )
=(? ?? ??? ??? )·?? +?? ? ?? ?
ˆ
??? (?? )
=grad ?? ·?? +?? (div ?? )
=?? (div ?? )+(grad ?? )·??
= RHS
LHS = RHS. Hence Proved.
So,
4.5 If ?? and ?? are two scalar fields and ???
is a vector field, such thet
?? ???
=???????? ?? ??
find the value of ???
· curl ???
.
(2010: 10 marks)
Solution:
?????????? ?? ??
=grad ?? ? ??
=
1
?? ·grad ?? ? ??
·curl ??
=(
1
?? grad ?? )·cur (
1
?? grad ?? )
=(
1
?? grad ?? )·[(grad
1
?? )×(grad ?? )+
1
?? (curl grad ?? )](?curl (?? ??
)=(grad ?? )×??
+?? Curl ??
=(
1
?? grad ?? )·[(grad
1
?? )×(grad ?? )+0]
=
1
?? [grad ?? grad
1
?? grad ?? ]=0
[? It a vector repeats in a scalar triple product, then its value is zero].
4.6 Calculate ?? ?? (?? ?? ) and find its expression in terms of ?? and ?? , ?? being the
distance of any point (?? ,?? ,?? ) iom the origin, ?? being a constant and ?? ?? being the
Laplace operator.
(2013 : 10 Marks)
Solution:
Page 4
Edurev123
4. Vector Identity and Vector Equations
4.1 Show that div (grad ?? ?? )=?? (?? +?? )?? ?? -??
(2009: 12 marks)
Solution:
?·(??? ?? ) =?? (?? +1)?? ?? -2
?? 2
=?? 2
+?? 2
+?? 2
Differentiating partially with respect to ?? on both sides
2?? ??? ??? =2?? ?
??? ??? =
?? ?? ??????????????????
??? ??? =
?? ?? ;
??? ??? =
?? ?? grad ?? ?? =?
?
??? ??ˆ(?? ?? )=?
??? ?? ??? ??ˆ
=? ?? ?? ?? -1
??? ??? ??ˆ=?? ?? ?? -1
?
?? ?? ??ˆ
=?? ?? ?? -2
??
?? h?????? , ?? =?? ??ˆ+?? ??ˆ+?? ??ˆ
div (grad ?? ?? ) =?
?
??? ??ˆ·?? ?? ?? -2
? ?? ??ˆ=?
?
??? (?? ?? ?? -2
?? )
=? (?? ?? ?? -2
+?? (?? -2)?? ?? -3
·?? ??? ??? )
=3?? ?? ?? -2
+? ?? (?? -2)?? ?? -3
?? 2
?? =3?? ?? ?? -2
+?? (?? -2)?? ?? -4
? ?? 2
=?? (?? -2)?? ?? -2
+3?? ?? ?? -2
=?? (?? +1)?? ?? -2
4.2 Find the directional derivative of
?? (?? ,?? )=?? ?? ?? ?? +????
at the point (?? ,?? ) in the direction of a unit vector which makes an angle of
?? ?? with
the ?? -axis.
(2010: 12 Marks)
Solution:
?????????? : ?? (?? ,?? ) =?? 2
?? 3
+????
??? =
??? ??? ??ˆ+
??? ??? ??ˆ=
?(?? 2
?? 3
+???? )
??? ??ˆ+
?(?? 2
?? 3
+???? )
??? ??ˆ
=(2?? ?? 3
+?? )??ˆ+(3?? 2
?? 2
+?? )??ˆ
Now, (??? )
(2,1)
=(2×2×?? 3
+1)??ˆ+(3×2
2
×??ˆ
-
2
+2)??ˆ
=5?? +14??
(?)
(2,1)
is the direction of unit vector at angle
?? 3
with ?? -axis is
(??? )
(2,1)
·(cos
?? 3
??ˆ+sin
?? 3
??ˆ)=(5??ˆ+14??ˆ)·(
1
2
??ˆ+
v3
2
??ˆ)=
5
2
+7v3
4.3 Show that the vector field defined by the vector function
???? =?????? (???? ??ˆ+???? ??ˆ+???? ??ˆ
)
is conservative.
(2010: 12 marks)
Solution:
Given :
?? =?????? (???? ??ˆ+???? ??ˆ+???? ??ˆ
)
=?? ?? 2
?? 2
??ˆ+?? 2
?? ?? 2
??ˆ+?? 2
?? 2
?? ??ˆ
?×?? =
|
|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? ?? ?? 2
?? 2
?? 2
?? ?? 2
?? 2
?? 2
?? |
|
=??ˆ(
?
??? ?? 2
?? 2
?? -
?
??? ?? 2
?? ?? 2
)-??ˆ(
?
??? ?? 2
?? 2
?? -
?
??? ?? ?? 2
?? 2
)+??ˆ
(
?
??? ?? 2
?? ?? 2
-
?
??? ?? ?? 2
?? 2
)
=??ˆ(2?? 2
?? 2
-2?? 2
?? 2
)-??ˆ(2?? ?? 2
?? -2?? ?? 2
?? )+??ˆ
(2???? ?? 2
-2???? ?? 2
)=0
as ?×?? =0.??? is conservative.
4.4 Prove that :
?????? (?? ???? )=?? (?????? ???? )+(???????? ?? )·????
where ?? is a scalar funcion.
(2011 : 20 Marks)
Solution:
LHS =div (?? ???
)
RHS=?? div ?? +( gradt )·???
Take LHS :
div (?? ) =S?? ?
??? (?? ?? )
=(? ?? ??? ??? )·?? +?? ? ?? ?
ˆ
??? (?? )
=grad ?? ·?? +?? (div ?? )
=?? (div ?? )+(grad ?? )·??
= RHS
LHS = RHS. Hence Proved.
So,
4.5 If ?? and ?? are two scalar fields and ???
is a vector field, such thet
?? ???
=???????? ?? ??
find the value of ???
· curl ???
.
(2010: 10 marks)
Solution:
?????????? ?? ??
=grad ?? ? ??
=
1
?? ·grad ?? ? ??
·curl ??
=(
1
?? grad ?? )·cur (
1
?? grad ?? )
=(
1
?? grad ?? )·[(grad
1
?? )×(grad ?? )+
1
?? (curl grad ?? )](?curl (?? ??
)=(grad ?? )×??
+?? Curl ??
=(
1
?? grad ?? )·[(grad
1
?? )×(grad ?? )+0]
=
1
?? [grad ?? grad
1
?? grad ?? ]=0
[? It a vector repeats in a scalar triple product, then its value is zero].
4.6 Calculate ?? ?? (?? ?? ) and find its expression in terms of ?? and ?? , ?? being the
distance of any point (?? ,?? ,?? ) iom the origin, ?? being a constant and ?? ?? being the
Laplace operator.
(2013 : 10 Marks)
Solution:
?
2
(?? ?? ) =?·?(?? ?? )
?(?? ?? ) =?
?
??? (?? ?? )??ˆ=?? ?? ?? -1
?
??? ??? ??ˆ=?? ?? ?? -1
?
??? ??? ??ˆ
?????? ?? 2
=?? 2
+?? 2
+?? 2
?2?? ??? ??? =2?? ?
??? ??? =
?? ?? ? ?(?? ?? ) =?? ?? ?? -1
?
?? ?? ??ˆ=?? ?? ?? -2
??
?? h?????? ?? =?? ??ˆ+???? +?? ??ˆ
?·?(?? ?? ) =(?
?
??? ??ˆ)·(?? ?? ?? -2
?
?? ?? )
=?
?
??? (?? ?? ?? -2
?? )
=? (?? ?? ?? -2
+?? (?? -2)?? ?? -3
??? ??? ·?? )
=(3?? ?? ?? -2
+? ?? (?? -2)?? ?? -3
?? 2
?? )
=3?? ?? ?? -2
+?? (?? -2)?? ?? -4
? ?? 2
=3?? ?? ?? -2
+?? (?? -2)?? ?? -2
=?? (?? +1)?? ?? -2
4.7 Find ?? (?? ) such that ?? ?? =
???
?? ?? and ?? (?? )=?? .
(2016 : 10 Marks)
Solution:
We know that
??? =?? '
(?? )??? =?? '
(?? )
??
??
We have, ??? =
??
?? 5
? ?? '
(?? )
??
?? =
??
?? 5
? ?? [
?? '
(?? )
?? -
1
?? 5
]=0
Since, ?? ?0,? ?? (?? )=
1
?? 4
Integrating, we get
?? (?? )=
-1
3?? 3
+??
Page 5
Edurev123
4. Vector Identity and Vector Equations
4.1 Show that div (grad ?? ?? )=?? (?? +?? )?? ?? -??
(2009: 12 marks)
Solution:
?·(??? ?? ) =?? (?? +1)?? ?? -2
?? 2
=?? 2
+?? 2
+?? 2
Differentiating partially with respect to ?? on both sides
2?? ??? ??? =2?? ?
??? ??? =
?? ?? ??????????????????
??? ??? =
?? ?? ;
??? ??? =
?? ?? grad ?? ?? =?
?
??? ??ˆ(?? ?? )=?
??? ?? ??? ??ˆ
=? ?? ?? ?? -1
??? ??? ??ˆ=?? ?? ?? -1
?
?? ?? ??ˆ
=?? ?? ?? -2
??
?? h?????? , ?? =?? ??ˆ+?? ??ˆ+?? ??ˆ
div (grad ?? ?? ) =?
?
??? ??ˆ·?? ?? ?? -2
? ?? ??ˆ=?
?
??? (?? ?? ?? -2
?? )
=? (?? ?? ?? -2
+?? (?? -2)?? ?? -3
·?? ??? ??? )
=3?? ?? ?? -2
+? ?? (?? -2)?? ?? -3
?? 2
?? =3?? ?? ?? -2
+?? (?? -2)?? ?? -4
? ?? 2
=?? (?? -2)?? ?? -2
+3?? ?? ?? -2
=?? (?? +1)?? ?? -2
4.2 Find the directional derivative of
?? (?? ,?? )=?? ?? ?? ?? +????
at the point (?? ,?? ) in the direction of a unit vector which makes an angle of
?? ?? with
the ?? -axis.
(2010: 12 Marks)
Solution:
?????????? : ?? (?? ,?? ) =?? 2
?? 3
+????
??? =
??? ??? ??ˆ+
??? ??? ??ˆ=
?(?? 2
?? 3
+???? )
??? ??ˆ+
?(?? 2
?? 3
+???? )
??? ??ˆ
=(2?? ?? 3
+?? )??ˆ+(3?? 2
?? 2
+?? )??ˆ
Now, (??? )
(2,1)
=(2×2×?? 3
+1)??ˆ+(3×2
2
×??ˆ
-
2
+2)??ˆ
=5?? +14??
(?)
(2,1)
is the direction of unit vector at angle
?? 3
with ?? -axis is
(??? )
(2,1)
·(cos
?? 3
??ˆ+sin
?? 3
??ˆ)=(5??ˆ+14??ˆ)·(
1
2
??ˆ+
v3
2
??ˆ)=
5
2
+7v3
4.3 Show that the vector field defined by the vector function
???? =?????? (???? ??ˆ+???? ??ˆ+???? ??ˆ
)
is conservative.
(2010: 12 marks)
Solution:
Given :
?? =?????? (???? ??ˆ+???? ??ˆ+???? ??ˆ
)
=?? ?? 2
?? 2
??ˆ+?? 2
?? ?? 2
??ˆ+?? 2
?? 2
?? ??ˆ
?×?? =
|
|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? ?? ?? 2
?? 2
?? 2
?? ?? 2
?? 2
?? 2
?? |
|
=??ˆ(
?
??? ?? 2
?? 2
?? -
?
??? ?? 2
?? ?? 2
)-??ˆ(
?
??? ?? 2
?? 2
?? -
?
??? ?? ?? 2
?? 2
)+??ˆ
(
?
??? ?? 2
?? ?? 2
-
?
??? ?? ?? 2
?? 2
)
=??ˆ(2?? 2
?? 2
-2?? 2
?? 2
)-??ˆ(2?? ?? 2
?? -2?? ?? 2
?? )+??ˆ
(2???? ?? 2
-2???? ?? 2
)=0
as ?×?? =0.??? is conservative.
4.4 Prove that :
?????? (?? ???? )=?? (?????? ???? )+(???????? ?? )·????
where ?? is a scalar funcion.
(2011 : 20 Marks)
Solution:
LHS =div (?? ???
)
RHS=?? div ?? +( gradt )·???
Take LHS :
div (?? ) =S?? ?
??? (?? ?? )
=(? ?? ??? ??? )·?? +?? ? ?? ?
ˆ
??? (?? )
=grad ?? ·?? +?? (div ?? )
=?? (div ?? )+(grad ?? )·??
= RHS
LHS = RHS. Hence Proved.
So,
4.5 If ?? and ?? are two scalar fields and ???
is a vector field, such thet
?? ???
=???????? ?? ??
find the value of ???
· curl ???
.
(2010: 10 marks)
Solution:
?????????? ?? ??
=grad ?? ? ??
=
1
?? ·grad ?? ? ??
·curl ??
=(
1
?? grad ?? )·cur (
1
?? grad ?? )
=(
1
?? grad ?? )·[(grad
1
?? )×(grad ?? )+
1
?? (curl grad ?? )](?curl (?? ??
)=(grad ?? )×??
+?? Curl ??
=(
1
?? grad ?? )·[(grad
1
?? )×(grad ?? )+0]
=
1
?? [grad ?? grad
1
?? grad ?? ]=0
[? It a vector repeats in a scalar triple product, then its value is zero].
4.6 Calculate ?? ?? (?? ?? ) and find its expression in terms of ?? and ?? , ?? being the
distance of any point (?? ,?? ,?? ) iom the origin, ?? being a constant and ?? ?? being the
Laplace operator.
(2013 : 10 Marks)
Solution:
?
2
(?? ?? ) =?·?(?? ?? )
?(?? ?? ) =?
?
??? (?? ?? )??ˆ=?? ?? ?? -1
?
??? ??? ??ˆ=?? ?? ?? -1
?
??? ??? ??ˆ
?????? ?? 2
=?? 2
+?? 2
+?? 2
?2?? ??? ??? =2?? ?
??? ??? =
?? ?? ? ?(?? ?? ) =?? ?? ?? -1
?
?? ?? ??ˆ=?? ?? ?? -2
??
?? h?????? ?? =?? ??ˆ+???? +?? ??ˆ
?·?(?? ?? ) =(?
?
??? ??ˆ)·(?? ?? ?? -2
?
?? ?? )
=?
?
??? (?? ?? ?? -2
?? )
=? (?? ?? ?? -2
+?? (?? -2)?? ?? -3
??? ??? ·?? )
=(3?? ?? ?? -2
+? ?? (?? -2)?? ?? -3
?? 2
?? )
=3?? ?? ?? -2
+?? (?? -2)?? ?? -4
? ?? 2
=3?? ?? ?? -2
+?? (?? -2)?? ?? -2
=?? (?? +1)?? ?? -2
4.7 Find ?? (?? ) such that ?? ?? =
???
?? ?? and ?? (?? )=?? .
(2016 : 10 Marks)
Solution:
We know that
??? =?? '
(?? )??? =?? '
(?? )
??
??
We have, ??? =
??
?? 5
? ?? '
(?? )
??
?? =
??
?? 5
? ?? [
?? '
(?? )
?? -
1
?? 5
]=0
Since, ?? ?0,? ?? (?? )=
1
?? 4
Integrating, we get
?? (?? )=
-1
3?? 3
+??
?? (1)=0?0=
-1
3·1
+?? ?3=
1
3
?? (?? )=
1
3
(1-
1
?? 3
)
[
?=?? ?
??? +?? ?
??? +?? ?
??? ?? =???? +???? +????
?? =v?? 2
+?? 2
+?? 2
]
4.8 Show that ?? ?? [?? ·(
???
?? )]=
?? ?? ?? , where ??? =?? ??ˆ+?? ??ˆ+?? ??ˆ
[2021: 10 marks]
Solution:
We know that ?·(???? )=?? (??? )+?? (??? ) (??)
Putting ?? =?? and ?? =
1
?? 2
in this identify,
We get,
?·(
??
?? 2
) =
1
?? 2
(??? )+?? (?·
1
?? 2
)
=
3
?? 2
+?? ×[-
2
?? 3
??? ] [???? =3 and ??? (?? )=?? '
(?? )??? ] [???? =
1
?? 2
?? ]
=
3
?? 2
+?? ×(-
2
?? 3
×
1
?? ?? )
=
3
?? 2
-
2
?? 4
(?? ×?? )=
3
?? 2
-
2
?? 4
?? 2
=
1
?? 2
?
2
[?×(
?? ?? 2
)] =?
2
(
1
?? 2
)=?×(?×
1
?? 2
)
=?×(
-2
?? 3
×??? )=?×(
-2
?? 3
×
1
?? ?? )
=?×(
-2
?? 4
?? )
=(-
2
?? 4
)(?·?? )+?? ·[?(-
2
?? 4
)], using the identity (i)
=-
2
?? 4
×3+?? ·[
8
?? 5
??? ]
=-
6
?? 4
+
8
?? 6
?? =-
6
?? 4
+
8
?? 6
?? 2
=-
6
?? 4
+
8
?? 4
=
2
?? 4
=2?? -4
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