UPSC Exam > UPSC Notes > Mathematics Optional Notes for UPSC > Functions of Two and Three Variables
Functions of Two and Three Variables | 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
2. Functions of Two and Three Variables
2.1 If ?? =?? ±?? .???? and ?? =?? ±?? .???? with approximately what accuracy can you
calculate the polar coordinates ?? and ?? of the point ?? (?? ,?? ) ? Express your
estimates as percentage changes of the value that ?? and ?? have at point (?? ,?? ) .
(2009 : 20 Marks)
Solution:
Approach : ?? and ?? are functions of two variables ?? and ?? . We write the total derivative of
each to calculate error from the error in the variables.
We have ?? =3,???? =0.01,?? =4,???? =0.01
By definition
?? =v?? 2
+?? 2
?? =tan
-1
?? ??
By total derivative
???? =
??? ??? ???? +
??? ??? ????
=
?? v?? 2
+?? 2
???? +
?? v?? 2
+?? 2
????
=
3×0.01
5
+
4
5
×0.01=
7
5
×0.01
Percentage error in ??
=
??? ?? ×100=
7
5
×0.01
v3
2
+4
2
×100
=
7
25
=0.28%
Similarly, ?? =tan
-1
?? ?? =tan
-1
4
3
Page 2
Edurev123
2. Functions of Two and Three Variables
2.1 If ?? =?? ±?? .???? and ?? =?? ±?? .???? with approximately what accuracy can you
calculate the polar coordinates ?? and ?? of the point ?? (?? ,?? ) ? Express your
estimates as percentage changes of the value that ?? and ?? have at point (?? ,?? ) .
(2009 : 20 Marks)
Solution:
Approach : ?? and ?? are functions of two variables ?? and ?? . We write the total derivative of
each to calculate error from the error in the variables.
We have ?? =3,???? =0.01,?? =4,???? =0.01
By definition
?? =v?? 2
+?? 2
?? =tan
-1
?? ??
By total derivative
???? =
??? ??? ???? +
??? ??? ????
=
?? v?? 2
+?? 2
???? +
?? v?? 2
+?? 2
????
=
3×0.01
5
+
4
5
×0.01=
7
5
×0.01
Percentage error in ??
=
??? ?? ×100=
7
5
×0.01
v3
2
+4
2
×100
=
7
25
=0.28%
Similarly, ?? =tan
-1
?? ?? =tan
-1
4
3
???? =
??? ??? ???? +
??? ??? ????
=
1
1+
?? 2
?? 2
-
?? ?? 2
???? +
1
1+
?? 2
?? 2
-
1
?? ????
=
-?? ?? 2
+?? 2
???? +
?? ?? 2
+?? 2
????
? ???? =
-4×0.01
25
+
3
25
×0.01=
-1
25
×0.01
? Percentage error =
1
2.5
tan
-1
4
3
×0.01×100=0.043%
2.2 Let ?? :?? ?? ??? be defined as
?? (?? ,?? )={
????
v?? ?? +?? ?? if (?? ,?? )?(?? ,?? )
?? if (?? ,?? )=(?? ,?? )
Is ?? continuous at (?? ,?? ) ? Compute partial derivatives of ?? at any point (?? ,?? ) if
exist.
(2009 : 20 Marks)
Solution:
Approach : Using definition of continuity and partial derivative.
Function being rational is continuous at all (?? ,?? ) except possibly at (0,0) .
Continuity at (0,0) :
To check whether lim
(?? ,?? )?(0,0)
?
????
v?? 2
+?? 2
=0.
Converting to polar coordinates with ?? =?? cos ?? ,?? =?? sin ?? ,lim
(?? ,?? )?(0,0)
?~lim
?? ?0
?
lim
?? ?0
?
?? 2
sin ?? cos ?? v?? 2
cos
2
?? +?? 2
sin
2
?? =lim
?? ?0
?
?? 2
sin ?? cos ?? ?? (?? is always +???? )
=lim
?? ?0
??? sin ?? cos ?? =0 for all values of ??
??? (?? ,?? ) is cont. at (0,0) as well.
Partial derivative at (?? ,?? )?(0,0)
Page 3
Edurev123
2. Functions of Two and Three Variables
2.1 If ?? =?? ±?? .???? and ?? =?? ±?? .???? with approximately what accuracy can you
calculate the polar coordinates ?? and ?? of the point ?? (?? ,?? ) ? Express your
estimates as percentage changes of the value that ?? and ?? have at point (?? ,?? ) .
(2009 : 20 Marks)
Solution:
Approach : ?? and ?? are functions of two variables ?? and ?? . We write the total derivative of
each to calculate error from the error in the variables.
We have ?? =3,???? =0.01,?? =4,???? =0.01
By definition
?? =v?? 2
+?? 2
?? =tan
-1
?? ??
By total derivative
???? =
??? ??? ???? +
??? ??? ????
=
?? v?? 2
+?? 2
???? +
?? v?? 2
+?? 2
????
=
3×0.01
5
+
4
5
×0.01=
7
5
×0.01
Percentage error in ??
=
??? ?? ×100=
7
5
×0.01
v3
2
+4
2
×100
=
7
25
=0.28%
Similarly, ?? =tan
-1
?? ?? =tan
-1
4
3
???? =
??? ??? ???? +
??? ??? ????
=
1
1+
?? 2
?? 2
-
?? ?? 2
???? +
1
1+
?? 2
?? 2
-
1
?? ????
=
-?? ?? 2
+?? 2
???? +
?? ?? 2
+?? 2
????
? ???? =
-4×0.01
25
+
3
25
×0.01=
-1
25
×0.01
? Percentage error =
1
2.5
tan
-1
4
3
×0.01×100=0.043%
2.2 Let ?? :?? ?? ??? be defined as
?? (?? ,?? )={
????
v?? ?? +?? ?? if (?? ,?? )?(?? ,?? )
?? if (?? ,?? )=(?? ,?? )
Is ?? continuous at (?? ,?? ) ? Compute partial derivatives of ?? at any point (?? ,?? ) if
exist.
(2009 : 20 Marks)
Solution:
Approach : Using definition of continuity and partial derivative.
Function being rational is continuous at all (?? ,?? ) except possibly at (0,0) .
Continuity at (0,0) :
To check whether lim
(?? ,?? )?(0,0)
?
????
v?? 2
+?? 2
=0.
Converting to polar coordinates with ?? =?? cos ?? ,?? =?? sin ?? ,lim
(?? ,?? )?(0,0)
?~lim
?? ?0
?
lim
?? ?0
?
?? 2
sin ?? cos ?? v?? 2
cos
2
?? +?? 2
sin
2
?? =lim
?? ?0
?
?? 2
sin ?? cos ?? ?? (?? is always +???? )
=lim
?? ?0
??? sin ?? cos ?? =0 for all values of ??
??? (?? ,?? ) is cont. at (0,0) as well.
Partial derivative at (?? ,?? )?(0,0)
??? ??? =
?? v?? 2
+?? 2
-???? ·
2?? 2
v?? 2
+?? 2
(?? 2
+?? 2
)
=
?? (?? 2
+?? 2
)-?? ?? 2
(?? 2
+?? 2
)
3/2
=
?? 3
(?? 2
+?? 2
)
3/2
??? ??? =
?? v?? 2
+?? 2
-???? ·
2?? 2v?? 2
+?? 2
(?? 2
+?? 2
)
=
?? 3
(?? 2
+?? 2
)
3/2
(Simply ?? ??? as it is symmetrical function in ??
and ?? )
Partial derivative at (0,0)
??? ??? =lim
h?0
?
?? (h,0)-?? (0,0)
h
=lim
h?0
?
0-0
h
=0
??? ??? =lim
?? ?0
?
?? (0,?? )-?? (0,0)
?? =lim
?? ?0
?
0-0
?? =0
2.3 A space probe in the shape of the ellipsoid ?? ?? ?? +?? ?? +?? ?? ?? =???? enters the
earth's atmosphere and its surface begins to heat. After one hour the temperature
at the point (?? ,?? ,?? ) on the probe surface is given by
?? (?? ,?? ,?? )=?? ?? ?? +?? ???? -???? ?? +??????
Find hottest point on the probe surface.
(2009 : 20 Marks)
Solution:
Approach: Use Lagrange's multipliers.
Max.?? (?? ,?? ,?? )=8?? 2
+4???? -16?? +600 …(i)
?????????????? ???? 4?? 2
+?? 2
+4?? 2
=1 …(???? )
?????? ?? =(8?? 2
+4???? -16?? +600)+?? (4?? 2
+?? 2
+4?? 2
-16)
Taking total derivative of ??
???? =(16+8?? )?????? +(4?? +2???? )???? +(8???? -16)???? +4??
For maximum/minimum
???? =0?(16+8?? )?? =0;4?? +2???? =0;8???? +4?? -16=0 …(iii)
Let ?? ?0?
?? =-
1
2
Page 4
Edurev123
2. Functions of Two and Three Variables
2.1 If ?? =?? ±?? .???? and ?? =?? ±?? .???? with approximately what accuracy can you
calculate the polar coordinates ?? and ?? of the point ?? (?? ,?? ) ? Express your
estimates as percentage changes of the value that ?? and ?? have at point (?? ,?? ) .
(2009 : 20 Marks)
Solution:
Approach : ?? and ?? are functions of two variables ?? and ?? . We write the total derivative of
each to calculate error from the error in the variables.
We have ?? =3,???? =0.01,?? =4,???? =0.01
By definition
?? =v?? 2
+?? 2
?? =tan
-1
?? ??
By total derivative
???? =
??? ??? ???? +
??? ??? ????
=
?? v?? 2
+?? 2
???? +
?? v?? 2
+?? 2
????
=
3×0.01
5
+
4
5
×0.01=
7
5
×0.01
Percentage error in ??
=
??? ?? ×100=
7
5
×0.01
v3
2
+4
2
×100
=
7
25
=0.28%
Similarly, ?? =tan
-1
?? ?? =tan
-1
4
3
???? =
??? ??? ???? +
??? ??? ????
=
1
1+
?? 2
?? 2
-
?? ?? 2
???? +
1
1+
?? 2
?? 2
-
1
?? ????
=
-?? ?? 2
+?? 2
???? +
?? ?? 2
+?? 2
????
? ???? =
-4×0.01
25
+
3
25
×0.01=
-1
25
×0.01
? Percentage error =
1
2.5
tan
-1
4
3
×0.01×100=0.043%
2.2 Let ?? :?? ?? ??? be defined as
?? (?? ,?? )={
????
v?? ?? +?? ?? if (?? ,?? )?(?? ,?? )
?? if (?? ,?? )=(?? ,?? )
Is ?? continuous at (?? ,?? ) ? Compute partial derivatives of ?? at any point (?? ,?? ) if
exist.
(2009 : 20 Marks)
Solution:
Approach : Using definition of continuity and partial derivative.
Function being rational is continuous at all (?? ,?? ) except possibly at (0,0) .
Continuity at (0,0) :
To check whether lim
(?? ,?? )?(0,0)
?
????
v?? 2
+?? 2
=0.
Converting to polar coordinates with ?? =?? cos ?? ,?? =?? sin ?? ,lim
(?? ,?? )?(0,0)
?~lim
?? ?0
?
lim
?? ?0
?
?? 2
sin ?? cos ?? v?? 2
cos
2
?? +?? 2
sin
2
?? =lim
?? ?0
?
?? 2
sin ?? cos ?? ?? (?? is always +???? )
=lim
?? ?0
??? sin ?? cos ?? =0 for all values of ??
??? (?? ,?? ) is cont. at (0,0) as well.
Partial derivative at (?? ,?? )?(0,0)
??? ??? =
?? v?? 2
+?? 2
-???? ·
2?? 2
v?? 2
+?? 2
(?? 2
+?? 2
)
=
?? (?? 2
+?? 2
)-?? ?? 2
(?? 2
+?? 2
)
3/2
=
?? 3
(?? 2
+?? 2
)
3/2
??? ??? =
?? v?? 2
+?? 2
-???? ·
2?? 2v?? 2
+?? 2
(?? 2
+?? 2
)
=
?? 3
(?? 2
+?? 2
)
3/2
(Simply ?? ??? as it is symmetrical function in ??
and ?? )
Partial derivative at (0,0)
??? ??? =lim
h?0
?
?? (h,0)-?? (0,0)
h
=lim
h?0
?
0-0
h
=0
??? ??? =lim
?? ?0
?
?? (0,?? )-?? (0,0)
?? =lim
?? ?0
?
0-0
?? =0
2.3 A space probe in the shape of the ellipsoid ?? ?? ?? +?? ?? +?? ?? ?? =???? enters the
earth's atmosphere and its surface begins to heat. After one hour the temperature
at the point (?? ,?? ,?? ) on the probe surface is given by
?? (?? ,?? ,?? )=?? ?? ?? +?? ???? -???? ?? +??????
Find hottest point on the probe surface.
(2009 : 20 Marks)
Solution:
Approach: Use Lagrange's multipliers.
Max.?? (?? ,?? ,?? )=8?? 2
+4???? -16?? +600 …(i)
?????????????? ???? 4?? 2
+?? 2
+4?? 2
=1 …(???? )
?????? ?? =(8?? 2
+4???? -16?? +600)+?? (4?? 2
+?? 2
+4?? 2
-16)
Taking total derivative of ??
???? =(16+8?? )?????? +(4?? +2???? )???? +(8???? -16)???? +4??
For maximum/minimum
???? =0?(16+8?? )?? =0;4?? +2???? =0;8???? +4?? -16=0 …(iii)
Let ?? ?0?
?? =-
1
2
4?? -?? =0
-4z+4?? =16
}?? =
16
3
;?? =
4
3
Putting this in (iii)
?? 2
=4-
1
4
(
256
9
+
64
9
)=
-44
9
<0
So this value is not possible.
? x=0
4?? +2???? =0
8???? +4?? =16
(4?? 2
-4)?? =-16
? y=
4
1-?? 2
And z=
-??
?? 2
=
2?? ?? 2
-1
Substituting in (ii)
16
(1-?? 2
)
2
+
16?? 2
(1-?? 2
)
2
=16
? 1+?? 2
=(1-?? 2
)
2
? 1+?? 2
=?? 4
-2?? 2
+1
? ?? 2
(?? 2
-3)=0
? ?? =0;?? 2
=3
?? =0??? =0,?? =4,?? =0
Let us consider ?? and ?? as independent and ?? as dependent variable.
(Note: We can do this because we have been given a relation between ?? ,?? and ?? )
From (ii) partially differentiating w.r.t. ?? .
8?? +8?? ??? ??? =0?
??? ??? =-
?? ?? ?
??? ??? =16?? +4?? ??? ??? -16
??? ??? =16?? -
4????
?? +
16?? ?? ?
2
?? ??? 2
=16+(16-4?? )(
1
?? -
?? ?? 2
·
??? ??? )
=16+(16-4?? )(
1
?? +
?? 2
?? 3
)
Page 5
Edurev123
2. Functions of Two and Three Variables
2.1 If ?? =?? ±?? .???? and ?? =?? ±?? .???? with approximately what accuracy can you
calculate the polar coordinates ?? and ?? of the point ?? (?? ,?? ) ? Express your
estimates as percentage changes of the value that ?? and ?? have at point (?? ,?? ) .
(2009 : 20 Marks)
Solution:
Approach : ?? and ?? are functions of two variables ?? and ?? . We write the total derivative of
each to calculate error from the error in the variables.
We have ?? =3,???? =0.01,?? =4,???? =0.01
By definition
?? =v?? 2
+?? 2
?? =tan
-1
?? ??
By total derivative
???? =
??? ??? ???? +
??? ??? ????
=
?? v?? 2
+?? 2
???? +
?? v?? 2
+?? 2
????
=
3×0.01
5
+
4
5
×0.01=
7
5
×0.01
Percentage error in ??
=
??? ?? ×100=
7
5
×0.01
v3
2
+4
2
×100
=
7
25
=0.28%
Similarly, ?? =tan
-1
?? ?? =tan
-1
4
3
???? =
??? ??? ???? +
??? ??? ????
=
1
1+
?? 2
?? 2
-
?? ?? 2
???? +
1
1+
?? 2
?? 2
-
1
?? ????
=
-?? ?? 2
+?? 2
???? +
?? ?? 2
+?? 2
????
? ???? =
-4×0.01
25
+
3
25
×0.01=
-1
25
×0.01
? Percentage error =
1
2.5
tan
-1
4
3
×0.01×100=0.043%
2.2 Let ?? :?? ?? ??? be defined as
?? (?? ,?? )={
????
v?? ?? +?? ?? if (?? ,?? )?(?? ,?? )
?? if (?? ,?? )=(?? ,?? )
Is ?? continuous at (?? ,?? ) ? Compute partial derivatives of ?? at any point (?? ,?? ) if
exist.
(2009 : 20 Marks)
Solution:
Approach : Using definition of continuity and partial derivative.
Function being rational is continuous at all (?? ,?? ) except possibly at (0,0) .
Continuity at (0,0) :
To check whether lim
(?? ,?? )?(0,0)
?
????
v?? 2
+?? 2
=0.
Converting to polar coordinates with ?? =?? cos ?? ,?? =?? sin ?? ,lim
(?? ,?? )?(0,0)
?~lim
?? ?0
?
lim
?? ?0
?
?? 2
sin ?? cos ?? v?? 2
cos
2
?? +?? 2
sin
2
?? =lim
?? ?0
?
?? 2
sin ?? cos ?? ?? (?? is always +???? )
=lim
?? ?0
??? sin ?? cos ?? =0 for all values of ??
??? (?? ,?? ) is cont. at (0,0) as well.
Partial derivative at (?? ,?? )?(0,0)
??? ??? =
?? v?? 2
+?? 2
-???? ·
2?? 2
v?? 2
+?? 2
(?? 2
+?? 2
)
=
?? (?? 2
+?? 2
)-?? ?? 2
(?? 2
+?? 2
)
3/2
=
?? 3
(?? 2
+?? 2
)
3/2
??? ??? =
?? v?? 2
+?? 2
-???? ·
2?? 2v?? 2
+?? 2
(?? 2
+?? 2
)
=
?? 3
(?? 2
+?? 2
)
3/2
(Simply ?? ??? as it is symmetrical function in ??
and ?? )
Partial derivative at (0,0)
??? ??? =lim
h?0
?
?? (h,0)-?? (0,0)
h
=lim
h?0
?
0-0
h
=0
??? ??? =lim
?? ?0
?
?? (0,?? )-?? (0,0)
?? =lim
?? ?0
?
0-0
?? =0
2.3 A space probe in the shape of the ellipsoid ?? ?? ?? +?? ?? +?? ?? ?? =???? enters the
earth's atmosphere and its surface begins to heat. After one hour the temperature
at the point (?? ,?? ,?? ) on the probe surface is given by
?? (?? ,?? ,?? )=?? ?? ?? +?? ???? -???? ?? +??????
Find hottest point on the probe surface.
(2009 : 20 Marks)
Solution:
Approach: Use Lagrange's multipliers.
Max.?? (?? ,?? ,?? )=8?? 2
+4???? -16?? +600 …(i)
?????????????? ???? 4?? 2
+?? 2
+4?? 2
=1 …(???? )
?????? ?? =(8?? 2
+4???? -16?? +600)+?? (4?? 2
+?? 2
+4?? 2
-16)
Taking total derivative of ??
???? =(16+8?? )?????? +(4?? +2???? )???? +(8???? -16)???? +4??
For maximum/minimum
???? =0?(16+8?? )?? =0;4?? +2???? =0;8???? +4?? -16=0 …(iii)
Let ?? ?0?
?? =-
1
2
4?? -?? =0
-4z+4?? =16
}?? =
16
3
;?? =
4
3
Putting this in (iii)
?? 2
=4-
1
4
(
256
9
+
64
9
)=
-44
9
<0
So this value is not possible.
? x=0
4?? +2???? =0
8???? +4?? =16
(4?? 2
-4)?? =-16
? y=
4
1-?? 2
And z=
-??
?? 2
=
2?? ?? 2
-1
Substituting in (ii)
16
(1-?? 2
)
2
+
16?? 2
(1-?? 2
)
2
=16
? 1+?? 2
=(1-?? 2
)
2
? 1+?? 2
=?? 4
-2?? 2
+1
? ?? 2
(?? 2
-3)=0
? ?? =0;?? 2
=3
?? =0??? =0,?? =4,?? =0
Let us consider ?? and ?? as independent and ?? as dependent variable.
(Note: We can do this because we have been given a relation between ?? ,?? and ?? )
From (ii) partially differentiating w.r.t. ?? .
8?? +8?? ??? ??? =0?
??? ??? =-
?? ?? ?
??? ??? =16?? +4?? ??? ??? -16
??? ??? =16?? -
4????
?? +
16?? ?? ?
2
?? ??? 2
=16+(16-4?? )(
1
?? -
?? ?? 2
·
??? ??? )
=16+(16-4?? )(
1
?? +
?? 2
?? 3
)
At ?? =4,
?
2
?? ??? 2
>0, so this cannot be maxima.
?? 2
= 3,?? =±v3
?? =v3,?? =v3,?? =-2,?? =0
Again
?
2
?? ??? 2
>0 so this can not be a maxima.
?? =-v3,?? =-v3,?? =-2,?? =0
?
2
?? ??? 2
=16-
24
v3
<0
3o, this is a maxima.
? Hottest point is (0,-2,-v3) .
2.4 Evaluate :?? =?
?? ??????????? +???????? +?? ?? ?? ???????? where ?? is the outer surface of the
part of the sphere ?? ?? + ?? ?? +?? ?? =?? in the first quadrant.
(2009 : 20 Marks)
Solution:
Approach: Surface integral with two changing variables can be calculated by taking the
projection of the surface in the required plane. Use of polar coordinates can simplify
integration.
Read More
|
387 videos|203 docs
|
Related Exams
About this Document
|
Oct 31, 2024 Last updated |
Document Description: Functions of Two and Three Variables for UPSC 2024 is part of Mathematics Optional Notes for UPSC preparation.
The notes and questions for Functions of Two and Three Variables have been prepared according to the UPSC exam syllabus. Information about Functions of Two and Three Variables covers topics
like and Functions of Two and Three Variables Example, for UPSC 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Functions of Two and Three Variables.
Introduction of Functions of Two and Three Variables in English is available as part of our Mathematics Optional Notes for UPSC
for UPSC & Functions of Two and Three Variables 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: Functions of Two and Three Variables | Mathematics Optional Notes for UPSC
Description
Full syllabus notes, lecture & questions for Functions of Two and Three Variables | 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 Functions of Two and Three Variables
In this doc you can find the meaning of Functions of Two and Three Variables defined & explained in the simplest way possible. Besides explaining types of
Functions of Two and Three Variables theory, EduRev gives you an ample number of questions to practice Functions of Two and Three Variables 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
Previous Year Questions with Solutions
,Functions of Two and Three Variables | Mathematics Optional Notes for UPSC
,Semester Notes
,Free
,mock tests for examination
,Functions of Two and Three Variables | Mathematics Optional Notes for UPSC
,Important questions
,Exam
,past year papers
,Extra Questions
,video lectures
,ppt
,practice quizzes
,Viva Questions
,Sample Paper
,shortcuts and tricks
,MCQs
,study material
,Objective type Questions
,Functions of Two and Three Variables | Mathematics Optional Notes for UPSC
,Summary
;
Additional Information about Functions of Two and Three Variables for UPSC Preparation
Functions of Two and Three Variables Free PDF Download
The Functions of Two and Three Variables 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 Functions of Two and Three Variables now and kickstart your journey towards success in the UPSC exam.
Importance of Functions of Two and Three Variables
The importance of Functions of Two and Three Variables 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.
Functions of Two and Three Variables Notes
Functions of Two and Three Variables Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Functions of Two and Three Variables.
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, Functions of Two and Three Variables Notes on EduRev are your ultimate resource for success.
Functions of Two and Three Variables UPSC Questions
The "Functions of Two and Three Variables 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 Functions of Two and Three Variables on the App
Students of UPSC can study Functions of Two and Three Variables alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Functions of Two and Three Variables,
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 Functions of Two and Three Variables is prepared as per the latest UPSC syllabus.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup