UPSC Exam > UPSC Notes > Mathematics Optional Notes for UPSC > Vector Identity and Vector Equations
Vector Identity and Vector Equations | Mathematics Optional Notes for UPSC PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
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
Read More
|
387 videos|203 docs
|
Related Exams
About this Document
|
Dec 18, 2024 Last updated |
Document Description: Vector Identity and Vector Equations for UPSC 2024 is part of Mathematics Optional Notes for UPSC preparation.
The notes and questions for Vector Identity and Vector Equations have been prepared according to the UPSC exam syllabus. Information about Vector Identity and Vector Equations covers topics
like and Vector Identity and Vector Equations Example, for UPSC 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Vector Identity and Vector Equations.
Introduction of Vector Identity and Vector Equations in English is available as part of our Mathematics Optional Notes for UPSC
for UPSC & Vector Identity and Vector Equations in Hindi for Mathematics Optional Notes for UPSC course.
Download more important topics related with notes, lectures and mock test series for UPSC
Exam by signing up for free. UPSC: Vector Identity and Vector Equations | Mathematics Optional Notes for UPSC
Description
Full syllabus notes, lecture & questions for Vector Identity and Vector Equations | Mathematics Optional Notes for UPSC - UPSC | Plus excerises question with solution to help you revise complete syllabus for Mathematics Optional Notes for UPSC | Best notes, free PDF download
Information about Vector Identity and Vector Equations
In this doc you can find the meaning of Vector Identity and Vector Equations defined & explained in the simplest way possible. Besides explaining types of
Vector Identity and Vector Equations theory, EduRev gives you an ample number of questions to practice Vector Identity and Vector Equations tests, examples and also practice UPSC
tests
|
Explore Courses for UPSC exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
practice quizzes
,Vector Identity and Vector Equations | Mathematics Optional Notes for UPSC
,study material
,Previous Year Questions with Solutions
,Free
,Objective type Questions
,ppt
,MCQs
,Vector Identity and Vector Equations | Mathematics Optional Notes for UPSC
,Semester Notes
,Vector Identity and Vector Equations | Mathematics Optional Notes for UPSC
,Exam
,Viva Questions
,Summary
,Extra Questions
,Important questions
,video lectures
,shortcuts and tricks
,Sample Paper
,past year papers
,mock tests for examination
;
Additional Information about Vector Identity and Vector Equations for UPSC Preparation
Vector Identity and Vector Equations Free PDF Download
The Vector Identity and Vector Equations is an invaluable resource that delves deep into the core of the UPSC exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Vector Identity and Vector Equations now and kickstart your journey towards success in the UPSC exam.
Importance of Vector Identity and Vector Equations
The importance of Vector Identity and Vector Equations cannot be overstated, especially for UPSC aspirants.
This document holds the key to success in the UPSC exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Vector Identity and Vector Equations Notes
Vector Identity and Vector Equations Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Vector Identity and Vector Equations.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Vector Identity and Vector Equations Notes on EduRev are your ultimate resource for success.
Vector Identity and Vector Equations UPSC Questions
The "Vector Identity and Vector Equations UPSC Questions" guide is a valuable resource for all aspiring students preparing for the
UPSC exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Vector Identity and Vector Equations on the App
Students of UPSC can study Vector Identity and Vector Equations alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Vector Identity and Vector Equations,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Vector Identity and Vector Equations is prepared as per the latest UPSC syllabus.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up
within 7 days!
within 7 days!
Takes less than 10 seconds to signup