Class notes
Calculus Modules
- Course
- Institution
this is good notes I have learn and got 90 out of 100.
[Show more]
Seller
Follow
mohd2125csit1129
Content preview
KIET GROUP OF INSTITUTIONS, DELHI – NCR, GHAZIABADB. Tech. (I SEM), 2020-21
MATHEMATICS-I (KAS-103T)
MODULE 3(Differential Calculus II)
Syllabus: Partial derivatives, Total derivative, Euler’s Theorem for homogeneous functions,
Taylor, and Maclaurin’s Theorem for function of one and two variables, Maxima and Minima of
functions of several variables, Lagrange’s Method of Multipliers, Jacobians, Approximation of
errors
Course Outcomes:
S.NO. Course Outcome BL
CO 3 Associate the concept of partial differentiation for determining 2,3
maxima, minima, expansion of series and Jacobians
CONTENT
S.NO. TOPIC PAGE NO.
3.1 Partial Derivatives and its applications 2
3.1.1 First order partial derivatives 2
3.1.2 Second and higher order partial derivatives 2
3.2 Total Derivative 6
3.2.1 Composite function and its Derivative 6
3.2.2 Differentiation of an implicit function 12
3.3 Euler’s Theorem for Homogeneous function 15
3.3.1 Homogeneous Function 15
3.3.2 Euler’s Theorem for Homogeneous Function 15
3.3.3 Deduction on Euler’s Theorem for homogeneous function 19
3.4 Taylor and Maclaurin’s Theorem for functions of one and two variables 23
3.4.1 Taylor and Maclaurin’s Theorem for functions of one variable 23
3.4.1.1 Taylor’s theorem for function of one variable 23
3.4.1.2 Maclaurin’s theorem for function of one variable 23
3.4.2.1 Taylor’s Theorem for function of two variables 26
3.4.2.2 Maclaurin’s Theorem for function of two variables 26
3.5 Maxima and minima of function of two variables 32
1
, KIET GROUP OF INSTITUTIONS, DELHI – NCR, GHAZIABAD
3.5.1 Rule to find the extreme values of a function z = f(x, y) 32
3.5.2 Condition or f(x, y, z) to be maximum or minimum 32
3.6 Lagrange’s method of undetermined multipliers 38
3.6.1 Working procedure of Lagrange’s method of undetermined multipliers 38
3.7 Jacobian 44
3.7.1 Properties of Jacobians 44
3.7.2 Functional Relationship 45
3.8 Approximation of Errors 51
3.9 E- Link for more understanding 57
3.1. Partial Derivatives and its Applications: The concept of partial differentiation is applied for function of two
or more independent variables. Partial differentiation is the process of finding various orders partial derivatives of
given function. Partial derivatives are very useful in determining Maxima and Minima, Jacobian, Series expansion
of function of several variables, Error and Approximation, In solution of wave and heat equations etc.
3.1.1. First order partial derivatives:
If𝑢 = 𝑓(𝑥, 𝑦) be any function of two independent variables 𝑥, 𝑦hen first order partial derivative of u with respect to ′𝑥′
𝜕𝑢
is obtained by differentiating u with respect to ‘x’ treating y and function of ‘y’ as constant and is denoted by
𝜕𝑥
Similarly fist order partial derivative of u with respect to ‘y’ is obtained by differentiating u with respect to ‘y’ treating
𝜕𝑢
‘x’ and functions of ‘x’ as constant and is denoted by𝜕𝑦 .
𝜕𝑢 𝜕𝑢
Example-1:If 𝑢 = 𝑥 2 + 2𝑥𝑦 − 𝑦 2 + 2𝑥 − 3𝑦 + 5 then find 𝑎𝑛𝑑 .
𝜕𝑥 𝜕𝑦
Solution: Here we have given 𝑢 = 𝑥 2 + 2𝑥𝑦 − 𝑦 2 + 2𝑥 − 3𝑦 + 5
𝜕𝑢
Therefore, to get value of 𝜕𝑥 we shall differentiate u with respect to ‘x’ treating ‘y’ as constant
𝝏𝒖
= 𝟐𝒙 + 𝟐𝒚 + 𝟐
𝝏𝒙
𝜕𝑢
Also, to find we shall differentiate u with respect to ‘y’ treating ‘x’ as constant
𝜕𝑦
𝝏𝒖
𝝏𝒚
= 𝟐𝒙 − 𝟐𝒚 − 𝟑.
3.1.2. Second and higher order partial derivatives:
If u f ( x, y ) be any function of two independent variables x, y then second order partial derivative of u with respect
to ′𝑥′ is obtained by differentiating u with respect to ‘x’ twice in succession treating ‘y’ and function of ‘y’ as constant
𝜕2 𝑢
each time and is denoted by 𝜕𝑥 2.
2
, KIET GROUP OF INSTITUTIONS, DELHI – NCR, GHAZIABAD
Similarly second order partial derivative of u with respect to ‘y’ is obtained by differentiating u with respect to ‘y’
𝜕2 𝑢
twice treating ‘x’ and functions of ‘x’ as constant each time and is denoted by 𝜕𝑦2 .
𝜕𝑢
Here if we differentiate 𝜕𝑦 with respect to ‘x’ treating ‘y’ as constant then we get second order derivative of u denoted
𝜕2 𝑢 𝜕𝑢
as𝜕𝑥𝜕𝑦 . Similarly differentiating 𝜕𝑥 with respect to ‘y’ treating ‘x’ and function of ‘x’ as constant we get second order
𝜕2 𝑢 𝜕2 𝑢 𝜕2 𝑢
derivative of u denoted as 𝜕𝑦𝜕𝑥 . It is interesting to see that 𝜕𝑥𝜕𝑦
= 𝜕𝑦𝜕𝑥 . In the similar way third and other higher
order partial derivatives can be evaluated.
𝜕2 𝑢 𝜕2 𝑢
Example-2: Find all first and second order partial derivatives of𝑢(𝑥, 𝑦) = 𝑥 𝑦 . Hence show that = .
𝜕𝑥𝜕𝑦 𝜕𝑦𝜕𝑥
Solution: Here we have 𝑢(𝑥, 𝑦) = 𝑥 𝑦 … … … … . (1)
Differentiating equation (1) partially with respect to ‘x’ and ‘y’ we get
𝜕𝑢
= 𝑦𝑥 𝑦−1 … … … … … … … … … …. (2)
𝜕𝑥
𝜕𝑢
= 𝑥 𝑦 𝑙𝑜𝑔𝑥 … … … … … … … … … … … . . (3)
𝜕𝑦
Differentiating equation (2) partially with respect to ‘x’ and ‘y’ we get
𝜕2𝑢
= 𝑦(𝑦 − 1)𝑥 𝑦−2 … … … … … … … … … …. (4)
𝜕𝑥 2
𝝏𝟐 𝒖
= 𝒚𝒙𝒚−𝟏 𝒍𝒐𝒈𝒙 + 𝒙𝒚−𝟏 … … … … … … … … … …. (5)
𝝏𝒚𝝏𝒙
Now differentiating equation (3) partially with respect to ‘y’ and ‘x’ we get
𝜕2𝑢
= 𝑥 𝑦 (𝑙𝑜𝑔𝑥)2 … … … … … … … … … …. (6)
𝜕𝑦 2
𝜕2𝑢 1
= 𝑦𝑥 𝑦−1 𝑙𝑜𝑔𝑥 + 𝑥 𝑦 ( )
𝜕𝑥𝜕𝑦 𝑥
𝝏𝟐 𝒖
= 𝒚𝒙𝒚−𝟏 𝒍𝒐𝒈𝒙 + 𝒙𝒚−𝟏 … … … … … … … . (7)
𝝏𝒙𝝏𝒚
𝜕2 𝑢 𝜕2 𝑢
From equations (5) & (7), it is clear that = .
𝜕𝑥𝜕𝑦 𝜕𝑦𝜕𝑥
𝜕𝑢 𝜕𝑢 𝜕𝑢
Example-3: If 𝑢 = log(𝑡𝑎𝑛𝑥 + 𝑡𝑎𝑛𝑦 + 𝑡𝑎𝑛𝑧), then prove that 𝑠𝑖𝑛2𝑥 𝜕𝑥 + 𝑠𝑖𝑛2𝑦 𝜕𝑦
+ 𝑠𝑖𝑛2𝑧 𝜕𝑧
=2.
Solution: In this problem, we have 𝑢 = log(𝑡𝑎𝑛𝑥 + 𝑡𝑎𝑛𝑦 + 𝑡𝑎𝑛𝑧) … … … … … … … (1)
Differentiating equation (1) partially with respect to ‘x’ we get
𝜕𝑢 1
= (𝑠𝑒𝑐 2 𝑥) … … … … … … … … . . (2)
𝜕𝑥 𝑡𝑎𝑛𝑥 + 𝑡𝑎𝑛𝑦 + 𝑡𝑎𝑛𝑧
3
, KIET GROUP OF INSTITUTIONS, DELHI – NCR, GHAZIABAD
Similarly differentiating (1) partially with respect to ‘y’ and ‘z’ we get
𝜕𝑢 1
= (𝑠𝑒𝑐 2 𝑦) … … … … … … … … . . (3)
𝜕𝑦 𝑡𝑎𝑛𝑥 + 𝑡𝑎𝑛𝑦 + 𝑡𝑎𝑛𝑧
𝜕𝑢 1
= (𝑠𝑒𝑐 2 𝑧) … … … … … … … … . . (4)
𝜕𝑧 𝑡𝑎𝑛𝑥 + 𝑡𝑎𝑛𝑦 + 𝑡𝑎𝑛𝑧
Multiplying equations (2), (3) and (4) with 𝑠𝑖𝑛2𝑥, 𝑠𝑖𝑛2𝑦 𝑎𝑛𝑑 𝑠𝑖𝑛2𝑧 respectively and then adding, we get
𝜕𝑢 𝜕𝑢 𝜕𝑢 1
𝑠𝑖𝑛2𝑥 𝜕𝑥
+sin2y𝜕𝑦 + 𝑠𝑖𝑛2𝑧 𝜕𝑧
= 𝑠𝑖𝑛2𝑥 {𝑡𝑎𝑛𝑥+𝑡𝑎𝑛𝑦+𝑡𝑎𝑛𝑧 (𝑠𝑒𝑐 2 𝑥)} +
1 1
𝑠𝑖𝑛2𝑦 {𝑡𝑎𝑛𝑥+𝑡𝑎𝑛𝑦+𝑡𝑎𝑛𝑧 (𝑠𝑒𝑐 2 𝑦)}+ 𝑠𝑖𝑛2𝑧 {𝑡𝑎𝑛𝑥+𝑡𝑎𝑛𝑦+𝑡𝑎𝑛𝑧 (𝑠𝑒𝑐 2 𝑧)}
𝜕𝑢 𝜕𝑢 𝜕𝑢 2𝑡𝑎𝑛𝑥 2𝑡𝑎𝑛𝑦 2𝑡𝑎𝑛𝑧
𝑠𝑖𝑛2𝑥 +sin2y + 𝑠𝑖𝑛2𝑧 = + + +
𝜕𝑥 𝜕𝑦 𝜕𝑧 𝑡𝑎𝑛𝑥+𝑡𝑎𝑛𝑦+𝑡𝑎𝑛𝑧 𝑡𝑎𝑛𝑥+𝑡𝑎𝑛𝑦+𝑡𝑎𝑛𝑧 𝑡𝑎𝑛𝑥+𝑡𝑎𝑛𝑦+𝑡𝑎𝑛𝑧
𝝏𝒖 𝝏𝒖 𝝏𝒖
𝒔𝒊𝒏𝟐𝒙 + 𝐬𝐢𝐧𝟐𝐲 + 𝒔𝒊𝒏𝟐𝒛 = 𝟐.
𝝏𝒙 𝝏𝒚 𝝏𝒛
𝜕𝑥 𝜕𝑦 𝜕𝑟 𝜕𝜃
Example-4: If 𝑥 = 𝑟𝑐𝑜𝑠𝜃, 𝑦 = 𝑟𝑠𝑖𝑛𝜃, then find (a) ( 𝜕𝑟 ) 𝑎𝑛𝑑 (𝜕𝜃 ) , and (b) (𝜕𝑥 ) 𝑎𝑛𝑑 (𝜕𝑦 ) .
𝜃 𝑟 𝑦 𝑥
Solution: Here we have
𝑥 = 𝑟𝑐𝑜𝑠𝜃 … … … … … … … … . . (1),
and 𝑦 = 𝑟𝑠𝑖𝑛𝜃 … … … … … … … … . . (2)
Differentiating equation (1) partially with respect to ‘r’ treating ‘𝜃′ as constant, we get
𝝏𝒙
( ) = 𝒄𝒐𝒔𝜽
𝝏𝒓 𝜽
Similarly differentiating (2) partially with respect to ‘𝜃′ treating ‘r’ as constant we get
𝝏𝒚
( ) = 𝒓𝒄𝒐𝒔𝜽
𝝏𝜽 𝒓
Again, from equations (1) and (2), we get
𝑟 2 = 𝑥 2 + 𝑦 2 … … … … … … … … … … … … … … (3)
𝑦
and, 𝜃 = tan−1 (𝑥 ) … … … … … … … … … … … … (4)
Differentiating equation (3) partially with respect to ‘x’ treating ‘y’ as constant, we get
𝜕𝑟 𝝏𝒓 𝒙
2𝑟 (𝜕𝑥 ) = 2𝑥 𝒐𝒓 (𝝏𝒙) = 𝒓
𝑦 𝒚
Similarly Differentiating equation (3) partially with respect to ‘y’ treating ‘x’ as constant, we get
4
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller mohd2125csit1129. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $89.69. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
56326 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now