UPSC Exam > UPSC Notes > Mathematics Optional Notes for UPSC > Gradient, Divergence and Cur in Cartesian and Cylindrical Coordinate and Directional Derivative
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Page 1
Edurev123
3. Gradient, Divergence and Cur in
Cartesian and Cylindrical Coordinate and
Directional Derivative
3.1 Find the directional derivative of :
(i) ?? ?? ?? ?? -?? ?? ?? ?? ?? ?? ?? at (?? ,-?? ,?? ) along ?? -axis
(ii) ?? ?? ???? +?? ?? ?? ?? at (?? ,-?? ,?? ) in the direction of ?? ??ˆ-??ˆ-?? ??ˆ
(???????? :?? +?? =???? Marks)
Solution:
Approach : The directional derivative in any direction is the dot product of the gradient
with that direction.
(i)
?? (?? ,?? ,?? ) =4?? ?? 3
-3?? 2
?? 2
?? 2
??? =
??? ??? ??ˆ+
??? ??? ??ˆ+
??? ??? ??ˆ
=(4?? 3
-6?? ?? 2
?? 2
)??ˆ-6?? 2
?? ?? 2
??ˆ+(12?? ?? 2
-6?? 2
?? 2
?? )??ˆ
Directional derivative along ?? -axis
=??? ·??ˆ
=12?? ?? 2
-6?? 2
?? 2
??
Directional derivative along ?? -axis at (2,-1,2)
=(??? ·??ˆ
)|
(2,-1,2)
=12·2·2
2
-6·2
2
·(-1)
2
·2=48
(ii)
?? (?? ,?? ,?? ) =?? 2
???? +4?? ?? 2
??? =(2?????? +4?? 2
)??ˆ+(?? 2
?? )??ˆ+(?? 2
?? +8???? )??ˆ
??? |
(1,-2,1)
=??ˆ+6??ˆ
? Directional derivative along (2??ˆ-??ˆ-2??ˆ
)
=(??ˆ+6??ˆ
)·(
2??ˆ-??ˆ-2??ˆ
3
)
=(
-1-12
3
)=-13
Page 2
Edurev123
3. Gradient, Divergence and Cur in
Cartesian and Cylindrical Coordinate and
Directional Derivative
3.1 Find the directional derivative of :
(i) ?? ?? ?? ?? -?? ?? ?? ?? ?? ?? ?? at (?? ,-?? ,?? ) along ?? -axis
(ii) ?? ?? ???? +?? ?? ?? ?? at (?? ,-?? ,?? ) in the direction of ?? ??ˆ-??ˆ-?? ??ˆ
(???????? :?? +?? =???? Marks)
Solution:
Approach : The directional derivative in any direction is the dot product of the gradient
with that direction.
(i)
?? (?? ,?? ,?? ) =4?? ?? 3
-3?? 2
?? 2
?? 2
??? =
??? ??? ??ˆ+
??? ??? ??ˆ+
??? ??? ??ˆ
=(4?? 3
-6?? ?? 2
?? 2
)??ˆ-6?? 2
?? ?? 2
??ˆ+(12?? ?? 2
-6?? 2
?? 2
?? )??ˆ
Directional derivative along ?? -axis
=??? ·??ˆ
=12?? ?? 2
-6?? 2
?? 2
??
Directional derivative along ?? -axis at (2,-1,2)
=(??? ·??ˆ
)|
(2,-1,2)
=12·2·2
2
-6·2
2
·(-1)
2
·2=48
(ii)
?? (?? ,?? ,?? ) =?? 2
???? +4?? ?? 2
??? =(2?????? +4?? 2
)??ˆ+(?? 2
?? )??ˆ+(?? 2
?? +8???? )??ˆ
??? |
(1,-2,1)
=??ˆ+6??ˆ
? Directional derivative along (2??ˆ-??ˆ-2??ˆ
)
=(??ˆ+6??ˆ
)·(
2??ˆ-??ˆ-2??ˆ
3
)
=(
-1-12
3
)=-13
3.2 Examine whether the vectors ?? ?? ,?? ?? and ?? ?? are copla, 1 ar, where ?? ,?? and ??
are the scalar functions defined by:
and
?? =?? +?? +?? ?? =?? ?? +?? ?? +?? ?? ?? =???? +???? +????
(2011: 15 Marks)
Solution:
??? =(
?
??? ?? +
?
??? ?? +
?
??? ???
)(?? +?? +?? )
=?? +?? +???
??? =2?? ?? +2?? ?? +2?? ???
Similarly , ??? =(?? +?? )?? +(?? +?? )?? +(?? +?? )???
Now, ??? ,??? ,??? would be co-planar if their scalar triple product is zero.
??? ×??? =|
?? ?? ???
2?? 2?? 2?? ?? +?? ?? +?? ?? +?? |
=?? (2???? +2?? 2
-2???? -2?? 2
)+?? (2???? +2?? 2
-2?? 2
-2???? )+
???
(2?? 2
+2???? -2?? 2
-2???? ).
? ??? (??? ×??? )= (?? +?? +???
)·[(2???? +2?? 2
-2???? -2?? 2
)??
+(2?? ?? +2?? 2
-2?? 2
-2???? )?? +(2?? 2
+2???? -2?? 2
-2???? )???
]
=2???? +2?? 2
-2?? 2
-2?? 2
+2???? +2?? 2
-2?? 2
-2???? +2?? 2
+2???? -2?? 2
-2????
=0
? The vector ??? ,??? ,? ware co-planar.
3.3 If ??? be the position vector of a point, find the value(s) of ?? for which the vector
?? ?? ??? is (i) irrotational, (ii) solenoidal.
(2011: 15 Marks)
Solution:
A vector ???
is said to be solenoidal if divergence of ???
=0.
i.e.,
?·???
=0
Also, div (?? ???
)=(grad ?? )·???
+?? div ???
Page 3
Edurev123
3. Gradient, Divergence and Cur in
Cartesian and Cylindrical Coordinate and
Directional Derivative
3.1 Find the directional derivative of :
(i) ?? ?? ?? ?? -?? ?? ?? ?? ?? ?? ?? at (?? ,-?? ,?? ) along ?? -axis
(ii) ?? ?? ???? +?? ?? ?? ?? at (?? ,-?? ,?? ) in the direction of ?? ??ˆ-??ˆ-?? ??ˆ
(???????? :?? +?? =???? Marks)
Solution:
Approach : The directional derivative in any direction is the dot product of the gradient
with that direction.
(i)
?? (?? ,?? ,?? ) =4?? ?? 3
-3?? 2
?? 2
?? 2
??? =
??? ??? ??ˆ+
??? ??? ??ˆ+
??? ??? ??ˆ
=(4?? 3
-6?? ?? 2
?? 2
)??ˆ-6?? 2
?? ?? 2
??ˆ+(12?? ?? 2
-6?? 2
?? 2
?? )??ˆ
Directional derivative along ?? -axis
=??? ·??ˆ
=12?? ?? 2
-6?? 2
?? 2
??
Directional derivative along ?? -axis at (2,-1,2)
=(??? ·??ˆ
)|
(2,-1,2)
=12·2·2
2
-6·2
2
·(-1)
2
·2=48
(ii)
?? (?? ,?? ,?? ) =?? 2
???? +4?? ?? 2
??? =(2?????? +4?? 2
)??ˆ+(?? 2
?? )??ˆ+(?? 2
?? +8???? )??ˆ
??? |
(1,-2,1)
=??ˆ+6??ˆ
? Directional derivative along (2??ˆ-??ˆ-2??ˆ
)
=(??ˆ+6??ˆ
)·(
2??ˆ-??ˆ-2??ˆ
3
)
=(
-1-12
3
)=-13
3.2 Examine whether the vectors ?? ?? ,?? ?? and ?? ?? are copla, 1 ar, where ?? ,?? and ??
are the scalar functions defined by:
and
?? =?? +?? +?? ?? =?? ?? +?? ?? +?? ?? ?? =???? +???? +????
(2011: 15 Marks)
Solution:
??? =(
?
??? ?? +
?
??? ?? +
?
??? ???
)(?? +?? +?? )
=?? +?? +???
??? =2?? ?? +2?? ?? +2?? ???
Similarly , ??? =(?? +?? )?? +(?? +?? )?? +(?? +?? )???
Now, ??? ,??? ,??? would be co-planar if their scalar triple product is zero.
??? ×??? =|
?? ?? ???
2?? 2?? 2?? ?? +?? ?? +?? ?? +?? |
=?? (2???? +2?? 2
-2???? -2?? 2
)+?? (2???? +2?? 2
-2?? 2
-2???? )+
???
(2?? 2
+2???? -2?? 2
-2???? ).
? ??? (??? ×??? )= (?? +?? +???
)·[(2???? +2?? 2
-2???? -2?? 2
)??
+(2?? ?? +2?? 2
-2?? 2
-2???? )?? +(2?? 2
+2???? -2?? 2
-2???? )???
]
=2???? +2?? 2
-2?? 2
-2?? 2
+2???? +2?? 2
-2?? 2
-2???? +2?? 2
+2???? -2?? 2
-2????
=0
? The vector ??? ,??? ,? ware co-planar.
3.3 If ??? be the position vector of a point, find the value(s) of ?? for which the vector
?? ?? ??? is (i) irrotational, (ii) solenoidal.
(2011: 15 Marks)
Solution:
A vector ???
is said to be solenoidal if divergence of ???
=0.
i.e.,
?·???
=0
Also, div (?? ???
)=(grad ?? )·???
+?? div ???
??? ?? ?? will be solenoidal if
div (?? ?? ?? ) =0
? (grad ?? ?? )·?? +?? ?? div (?? ) =0
? (?? ?? ?? -1
grad ?? )·?? +?? ?? ·3 =0
? div ?? =(??ˆ
?
??? +??ˆ
?
??? +??ˆ
?
??? )(?? ??ˆ+?? ??ˆ+?? ??ˆ
)
=1+1+1=3
and grad ?? (4)=?? '
(4) grad ?? ? (?? ?? ?? -1
·
??
?? )·?? +3?? ?? =0
? ?? ?? ?? -2
(?? ·?? )+3?? ?? =0
? ?? ?? ?? -2
·?? 2
+3?? ?? =0
? ?? ?? (?? +3)=0??? =-3
A vector ?? is said to be irrotational if
?×???
=0
Also, ?×(?? ???
) =(grad ?? )×???
+?? (?×???
)
??? ?? ?? will be irrotational if
?×(?? ?? ?? ) =0
? (grad ?? ?? )×?? +?? ?? (?×?? ) =0
? (?? ?? ?? -1
·
??
?? )·?? +?? ?? ·0=0
Hence, ?? ?? ?? is irrotatlonal for all the real values of ?? .
3.4 A vector field is given by
????
=(?? ?? +?? ?? ?? )?? +(?? ?? +?? ?? ?? )??
Verify that the field ????
is irrotational or not. Find the scalar potential.
(2015 : 12 Marks)
Solution:
A vector field ??
is said to be irrotational if curl ??
=0, i.e.,
?×??
=0
?
??
×??
=
|
|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? ?? 2
+?? ?? 2
?? 2
+?? 2
?? 0
|
|
Page 4
Edurev123
3. Gradient, Divergence and Cur in
Cartesian and Cylindrical Coordinate and
Directional Derivative
3.1 Find the directional derivative of :
(i) ?? ?? ?? ?? -?? ?? ?? ?? ?? ?? ?? at (?? ,-?? ,?? ) along ?? -axis
(ii) ?? ?? ???? +?? ?? ?? ?? at (?? ,-?? ,?? ) in the direction of ?? ??ˆ-??ˆ-?? ??ˆ
(???????? :?? +?? =???? Marks)
Solution:
Approach : The directional derivative in any direction is the dot product of the gradient
with that direction.
(i)
?? (?? ,?? ,?? ) =4?? ?? 3
-3?? 2
?? 2
?? 2
??? =
??? ??? ??ˆ+
??? ??? ??ˆ+
??? ??? ??ˆ
=(4?? 3
-6?? ?? 2
?? 2
)??ˆ-6?? 2
?? ?? 2
??ˆ+(12?? ?? 2
-6?? 2
?? 2
?? )??ˆ
Directional derivative along ?? -axis
=??? ·??ˆ
=12?? ?? 2
-6?? 2
?? 2
??
Directional derivative along ?? -axis at (2,-1,2)
=(??? ·??ˆ
)|
(2,-1,2)
=12·2·2
2
-6·2
2
·(-1)
2
·2=48
(ii)
?? (?? ,?? ,?? ) =?? 2
???? +4?? ?? 2
??? =(2?????? +4?? 2
)??ˆ+(?? 2
?? )??ˆ+(?? 2
?? +8???? )??ˆ
??? |
(1,-2,1)
=??ˆ+6??ˆ
? Directional derivative along (2??ˆ-??ˆ-2??ˆ
)
=(??ˆ+6??ˆ
)·(
2??ˆ-??ˆ-2??ˆ
3
)
=(
-1-12
3
)=-13
3.2 Examine whether the vectors ?? ?? ,?? ?? and ?? ?? are copla, 1 ar, where ?? ,?? and ??
are the scalar functions defined by:
and
?? =?? +?? +?? ?? =?? ?? +?? ?? +?? ?? ?? =???? +???? +????
(2011: 15 Marks)
Solution:
??? =(
?
??? ?? +
?
??? ?? +
?
??? ???
)(?? +?? +?? )
=?? +?? +???
??? =2?? ?? +2?? ?? +2?? ???
Similarly , ??? =(?? +?? )?? +(?? +?? )?? +(?? +?? )???
Now, ??? ,??? ,??? would be co-planar if their scalar triple product is zero.
??? ×??? =|
?? ?? ???
2?? 2?? 2?? ?? +?? ?? +?? ?? +?? |
=?? (2???? +2?? 2
-2???? -2?? 2
)+?? (2???? +2?? 2
-2?? 2
-2???? )+
???
(2?? 2
+2???? -2?? 2
-2???? ).
? ??? (??? ×??? )= (?? +?? +???
)·[(2???? +2?? 2
-2???? -2?? 2
)??
+(2?? ?? +2?? 2
-2?? 2
-2???? )?? +(2?? 2
+2???? -2?? 2
-2???? )???
]
=2???? +2?? 2
-2?? 2
-2?? 2
+2???? +2?? 2
-2?? 2
-2???? +2?? 2
+2???? -2?? 2
-2????
=0
? The vector ??? ,??? ,? ware co-planar.
3.3 If ??? be the position vector of a point, find the value(s) of ?? for which the vector
?? ?? ??? is (i) irrotational, (ii) solenoidal.
(2011: 15 Marks)
Solution:
A vector ???
is said to be solenoidal if divergence of ???
=0.
i.e.,
?·???
=0
Also, div (?? ???
)=(grad ?? )·???
+?? div ???
??? ?? ?? will be solenoidal if
div (?? ?? ?? ) =0
? (grad ?? ?? )·?? +?? ?? div (?? ) =0
? (?? ?? ?? -1
grad ?? )·?? +?? ?? ·3 =0
? div ?? =(??ˆ
?
??? +??ˆ
?
??? +??ˆ
?
??? )(?? ??ˆ+?? ??ˆ+?? ??ˆ
)
=1+1+1=3
and grad ?? (4)=?? '
(4) grad ?? ? (?? ?? ?? -1
·
??
?? )·?? +3?? ?? =0
? ?? ?? ?? -2
(?? ·?? )+3?? ?? =0
? ?? ?? ?? -2
·?? 2
+3?? ?? =0
? ?? ?? (?? +3)=0??? =-3
A vector ?? is said to be irrotational if
?×???
=0
Also, ?×(?? ???
) =(grad ?? )×???
+?? (?×???
)
??? ?? ?? will be irrotational if
?×(?? ?? ?? ) =0
? (grad ?? ?? )×?? +?? ?? (?×?? ) =0
? (?? ?? ?? -1
·
??
?? )·?? +?? ?? ·0=0
Hence, ?? ?? ?? is irrotatlonal for all the real values of ?? .
3.4 A vector field is given by
????
=(?? ?? +?? ?? ?? )?? +(?? ?? +?? ?? ?? )??
Verify that the field ????
is irrotational or not. Find the scalar potential.
(2015 : 12 Marks)
Solution:
A vector field ??
is said to be irrotational if curl ??
=0, i.e.,
?×??
=0
?
??
×??
=
|
|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? ?? 2
+?? ?? 2
?? 2
+?? 2
?? 0
|
|
=??ˆ(0-0)-??ˆ(0-0)+??ˆ
(2???? -2???? )
=0
?
???
is irrotational.
Now, it can be written as grad of a scalar field, i.e., to find ?? so that
??? =??
i.e., ??ˆ
??? ??? +??ˆ
??? ??? =(?? 2
+?? ?? 2
)??ˆ+(?? 2
+?? 2
?? )??ˆ
?
??? ??? =?? 2
+?? ?? 2
;
??? ??? =?? 2
+?? 2
??
? ?? =
?? 3
3
+
?? 2
?? 2
2
+?? (?? ) (*)
Differentiating w.r.t. ?? and comparing with (*)
??? ??? =?? 2
?? +?? '
(?? )
? ?? '
(?? )=?? 2
?? (?? )=
?? 3
3
+?? ? ?? (?? ,?? )=
?? 3
3
+
?? 3
3
+
?? 2
?? 2
2
+??
3.5 For what values of the constants ?? ,?? and ?? the vector
????
=(?? +?? +???? )??ˆ+(???? +?? ?? -?? )??ˆ+(-?? +???? +?? ?? )??ˆ
is irrational. Find the divergence in cylindrical coordinates of this vector with
these values.
(2017: 10 Marks)
Solution:
Irrational ? Curl ???
=0
Page 5
Edurev123
3. Gradient, Divergence and Cur in
Cartesian and Cylindrical Coordinate and
Directional Derivative
3.1 Find the directional derivative of :
(i) ?? ?? ?? ?? -?? ?? ?? ?? ?? ?? ?? at (?? ,-?? ,?? ) along ?? -axis
(ii) ?? ?? ???? +?? ?? ?? ?? at (?? ,-?? ,?? ) in the direction of ?? ??ˆ-??ˆ-?? ??ˆ
(???????? :?? +?? =???? Marks)
Solution:
Approach : The directional derivative in any direction is the dot product of the gradient
with that direction.
(i)
?? (?? ,?? ,?? ) =4?? ?? 3
-3?? 2
?? 2
?? 2
??? =
??? ??? ??ˆ+
??? ??? ??ˆ+
??? ??? ??ˆ
=(4?? 3
-6?? ?? 2
?? 2
)??ˆ-6?? 2
?? ?? 2
??ˆ+(12?? ?? 2
-6?? 2
?? 2
?? )??ˆ
Directional derivative along ?? -axis
=??? ·??ˆ
=12?? ?? 2
-6?? 2
?? 2
??
Directional derivative along ?? -axis at (2,-1,2)
=(??? ·??ˆ
)|
(2,-1,2)
=12·2·2
2
-6·2
2
·(-1)
2
·2=48
(ii)
?? (?? ,?? ,?? ) =?? 2
???? +4?? ?? 2
??? =(2?????? +4?? 2
)??ˆ+(?? 2
?? )??ˆ+(?? 2
?? +8???? )??ˆ
??? |
(1,-2,1)
=??ˆ+6??ˆ
? Directional derivative along (2??ˆ-??ˆ-2??ˆ
)
=(??ˆ+6??ˆ
)·(
2??ˆ-??ˆ-2??ˆ
3
)
=(
-1-12
3
)=-13
3.2 Examine whether the vectors ?? ?? ,?? ?? and ?? ?? are copla, 1 ar, where ?? ,?? and ??
are the scalar functions defined by:
and
?? =?? +?? +?? ?? =?? ?? +?? ?? +?? ?? ?? =???? +???? +????
(2011: 15 Marks)
Solution:
??? =(
?
??? ?? +
?
??? ?? +
?
??? ???
)(?? +?? +?? )
=?? +?? +???
??? =2?? ?? +2?? ?? +2?? ???
Similarly , ??? =(?? +?? )?? +(?? +?? )?? +(?? +?? )???
Now, ??? ,??? ,??? would be co-planar if their scalar triple product is zero.
??? ×??? =|
?? ?? ???
2?? 2?? 2?? ?? +?? ?? +?? ?? +?? |
=?? (2???? +2?? 2
-2???? -2?? 2
)+?? (2???? +2?? 2
-2?? 2
-2???? )+
???
(2?? 2
+2???? -2?? 2
-2???? ).
? ??? (??? ×??? )= (?? +?? +???
)·[(2???? +2?? 2
-2???? -2?? 2
)??
+(2?? ?? +2?? 2
-2?? 2
-2???? )?? +(2?? 2
+2???? -2?? 2
-2???? )???
]
=2???? +2?? 2
-2?? 2
-2?? 2
+2???? +2?? 2
-2?? 2
-2???? +2?? 2
+2???? -2?? 2
-2????
=0
? The vector ??? ,??? ,? ware co-planar.
3.3 If ??? be the position vector of a point, find the value(s) of ?? for which the vector
?? ?? ??? is (i) irrotational, (ii) solenoidal.
(2011: 15 Marks)
Solution:
A vector ???
is said to be solenoidal if divergence of ???
=0.
i.e.,
?·???
=0
Also, div (?? ???
)=(grad ?? )·???
+?? div ???
??? ?? ?? will be solenoidal if
div (?? ?? ?? ) =0
? (grad ?? ?? )·?? +?? ?? div (?? ) =0
? (?? ?? ?? -1
grad ?? )·?? +?? ?? ·3 =0
? div ?? =(??ˆ
?
??? +??ˆ
?
??? +??ˆ
?
??? )(?? ??ˆ+?? ??ˆ+?? ??ˆ
)
=1+1+1=3
and grad ?? (4)=?? '
(4) grad ?? ? (?? ?? ?? -1
·
??
?? )·?? +3?? ?? =0
? ?? ?? ?? -2
(?? ·?? )+3?? ?? =0
? ?? ?? ?? -2
·?? 2
+3?? ?? =0
? ?? ?? (?? +3)=0??? =-3
A vector ?? is said to be irrotational if
?×???
=0
Also, ?×(?? ???
) =(grad ?? )×???
+?? (?×???
)
??? ?? ?? will be irrotational if
?×(?? ?? ?? ) =0
? (grad ?? ?? )×?? +?? ?? (?×?? ) =0
? (?? ?? ?? -1
·
??
?? )·?? +?? ?? ·0=0
Hence, ?? ?? ?? is irrotatlonal for all the real values of ?? .
3.4 A vector field is given by
????
=(?? ?? +?? ?? ?? )?? +(?? ?? +?? ?? ?? )??
Verify that the field ????
is irrotational or not. Find the scalar potential.
(2015 : 12 Marks)
Solution:
A vector field ??
is said to be irrotational if curl ??
=0, i.e.,
?×??
=0
?
??
×??
=
|
|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? ?? 2
+?? ?? 2
?? 2
+?? 2
?? 0
|
|
=??ˆ(0-0)-??ˆ(0-0)+??ˆ
(2???? -2???? )
=0
?
???
is irrotational.
Now, it can be written as grad of a scalar field, i.e., to find ?? so that
??? =??
i.e., ??ˆ
??? ??? +??ˆ
??? ??? =(?? 2
+?? ?? 2
)??ˆ+(?? 2
+?? 2
?? )??ˆ
?
??? ??? =?? 2
+?? ?? 2
;
??? ??? =?? 2
+?? 2
??
? ?? =
?? 3
3
+
?? 2
?? 2
2
+?? (?? ) (*)
Differentiating w.r.t. ?? and comparing with (*)
??? ??? =?? 2
?? +?? '
(?? )
? ?? '
(?? )=?? 2
?? (?? )=
?? 3
3
+?? ? ?? (?? ,?? )=
?? 3
3
+
?? 3
3
+
?? 2
?? 2
2
+??
3.5 For what values of the constants ?? ,?? and ?? the vector
????
=(?? +?? +???? )??ˆ+(???? +?? ?? -?? )??ˆ+(-?? +???? +?? ?? )??ˆ
is irrational. Find the divergence in cylindrical coordinates of this vector with
these values.
(2017: 10 Marks)
Solution:
Irrational ? Curl ???
=0
Curl ?
??
=?×???
[
?? ?? ?? ?
??? ?
??? ?
??? ?? ?? h]
( if ?? =???? +???? +h?? )
=|
|
?? ?? ?? ?
??? ?
??? ?
??? ?? +?? +???? ???? +2?? -?? -?? +???? +2?? |
|
=??(?? -(-1)-?? (-1-?? )+?? (?? -1)
=(?? +1)?? +(?? +1)?? +(?? -1)?? =0
? ?? =-1,?? =1,?? =-1
? ???
=(?? +?? -?? )?? +(?? +2?? -?? )?? +(-?? -?? +2?? )??
We find div ???
and express it in cylindrical coordinates.
???
=(?? +?? -?? )?? +(?? +2?? -?? )?? +(-?? -?? +2?? )?? div ???
=?·???
=
??? ?? ??? +
??? ?? ??? +
??? ?? ??? =1+2+2=5 (constant)
Hence, divergence in cylindrical co-ordinates =5.
3.6 Let ???? =?? ?? ??ˆ+?? ?? ??ˆ+?? ?? ??ˆ
. Show that curl (curl ???? )=?? ?????? (?????? ???? )-?? ?? ???? .
(2018: 12 Marks)
Solution:
Curl ( Curl ?? )=?×(?×?? ) .
Now, we know that
??
×(???
×??
)=(??
·??
)???
-(??
·???
)??
Given:
?? 1
??ˆ+?? 2
??ˆ+?? 3
??ˆ
=??
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Document Description: Gradient, Divergence and Cur in Cartesian and Cylindrical Coordinate and Directional Derivative for UPSC 2024 is part of Mathematics Optional Notes for UPSC preparation.
The notes and questions for Gradient, Divergence and Cur in Cartesian and Cylindrical Coordinate and Directional Derivative have been prepared according to the UPSC exam syllabus. Information about Gradient, Divergence and Cur in Cartesian and Cylindrical Coordinate and Directional Derivative covers topics
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