1. The document provides information on multiple integrals including double integrals, triple integrals, and integrals in spherical and cylindrical coordinates. It defines each type of integral and gives their general formulas.
2. Examples are provided for calculating double and triple integrals over different regions in rectangular, cylindrical, and spherical coordinate systems. The order of integration can be changed by considering strips or slices of the region.
3. Properties of the integrals include applying Fubini's theorem to change the order of integration, and relating the triple integral over a region to the double integral over the bounds and integrating over the third variable.
In this slide fourier series of Engineering Mathematics has been described. one Example is also added for you. Hope this will help you understand fourier series.
Cauchy's integral theorem, Cauchy's integral formula, Cauchy's integral formula for derivatives, Taylor's Series, Maclaurin’s Series,Laurent's Series,Singularities and zeros, Cauchy's Residue theorem,Evaluation various types of complex integrals.
In this slide fourier series of Engineering Mathematics has been described. one Example is also added for you. Hope this will help you understand fourier series.
Cauchy's integral theorem, Cauchy's integral formula, Cauchy's integral formula for derivatives, Taylor's Series, Maclaurin’s Series,Laurent's Series,Singularities and zeros, Cauchy's Residue theorem,Evaluation various types of complex integrals.
Its states Periodic function, Fourier series for disontinous function, Fourier series, Intervals, Odd and even functions, Half range fourier series etc. Mostly used as active learning assignment in Degree 3rd sem students.
Application of definite integrals,we will explore some of the many application of definite integral by using it to calculate areas between two curves, volumes, length of curves, and several other application.
Presentation on Fourier Series
contents are:-
Euler’s Formula
Functions having point of discontinuity
Change of interval
Even and Odd functions
Half Range series
Harmonic analysis
Its states Periodic function, Fourier series for disontinous function, Fourier series, Intervals, Odd and even functions, Half range fourier series etc. Mostly used as active learning assignment in Degree 3rd sem students.
Application of definite integrals,we will explore some of the many application of definite integral by using it to calculate areas between two curves, volumes, length of curves, and several other application.
Presentation on Fourier Series
contents are:-
Euler’s Formula
Functions having point of discontinuity
Change of interval
Even and Odd functions
Half Range series
Harmonic analysis
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Application of partial derivatives with two variablesSagar Patel
Application of Partial Derivatives with Two Variables
Maxima And Minima Values.
Maximum And Minimum Values.
Tangent and Normal.
Error And Approximation.
Two novel transforms, related together and called Sine and Cosine Fresnel Transforms, as well as their optical implementation are presented. Each transform combines both backward and forward light propagation in the framework of the scalar diffraction approximation. It has been proven that the Fresnel transform is the optical version of the fractional Fourier transform. Therefore the former has the same properties as the latter. While showing properties similar to those of the Fresnel transform and therefore of the fractional Fourier transform, each of the Sine and Cosine Fresnel transforms provides a real result for a real input distribution. This enables saving half of the quantity of information in the complex plane. Because of parallelism, optics offers high speed processing of digital signals. Speech signals should be first represented by images through special light modulators for example. The Sine and Cosine Fresnel transforms may be regarded respectively as the fractional Sine and Cosine transforms which are more general than the Cosine transform used in information processing and compression.
01. Differentiation-Theory & solved example Module-3.pdfRajuSingh806014
Total No. of questions in Differentiation are-
In Chapter Examples 31
Solved Examples 32
The rate of change of one quantity with respect to some another quantity has a great importance. For example the rate of change of displacement of a particle with respect to time is called its velocity and the rate of change of velocity is
called its acceleration.
The following results can easily be established using the above definition of the derivative–
d
(i) dx (constant) = 0
The rate of change of a quantity 'y' with respect to another quantity 'x' is called the derivative or differential coefficient of y with respect to x.
Let y = f(x) be a continuous function of a variable quantity x, where x is independent and y is
(ii)
(iii)
(iv)
(v)
d
dx (ax) = a
d (xn) = nxn–1
dx
d ex =ex
dx
d (ax) = ax log a
dependent variable quantity. Let x be an arbitrary small change in the value of x and y be the
dx
d
(vi) dx
e
(logex) = 1/x
corresponding change in y then lim
y
if it exists, d 1
x0 x
is called the derivative or differential coefficient of y with respect to x and it is denoted by
(vii) dx
(logax) =
x log a
dy . y', y
dx 1
or Dy.
d
(viii) dx (sin x) = cos x
So, dy dx
dy
dx
lim
x0
lim
x0
y
x
f (x x) f (x)
x
(ix) (ix)
(x) (x)
d
dx (cos x) = – sin x
d (tan x) = sec2x
dx
The process of finding derivative of a function is called differentiation.
If we again differentiate (dy/dx) with respect to x
(xi)
d (cot x) = – cosec2x
dx
d
then the new derivative so obtained is called second derivative of y with respect to x and it is
Fd2 y
(xii) dx
d
(xiii) dx
(secx)= secx tan x
(cosec x) = – cosec x cot x
denoted by
HGdx2 Jor y" or y2 or D2y. Similarly,
d 1
we can find successive derivatives of y which
(xiv) dx
(sin–1 x) = , –1< x < 1
1 x2
may be denoted by
d –1 1
d3 y d4 y
dn y
(xv) dx (cos x) = –
,–1 < x < 1
dx3 ,
dx4 , ........, dxn , ......
d
(xvi) dx
(tan–1 x) = 1
1 x2
Note : (i)
y is a ratio of two quantities y and
x
(xvii) (xvii)
d (cot–1 x) = – 1
where as dy
dx
dy
is not a ratio, it is a single
dx
d
(xviii) (xviii)
(sec–1 x) =
1 x2
1
|x| > 1
quantity i.e.
dx dy÷ dx
dx x x2 1
(ii)
dy is
dx
d (y) in which d/dx is simply a symbol
dx
(xix)
d (cosec–1 x) = – 1
dx
of operation and not 'd' divided by dx.
d
(xx) dx
(sinh x) = cosh x
d
(xxi) dx
d
(cosh x) = sinh x
Theorem V Derivative of the function of the function. If 'y' is a function of 't' and t' is a function of 'x' then
(xxii) dx
d
(tanh x) = sech2 x
dy =
dx
dy . dt
dt dx
(xxiii) dx
d
(xxiv) dx
d
(coth x) = – cosec h2 x (sech x) = – sech x tanh x
Theorem VI Derivative of parametric equations If x = (t) , y = (t) then
dy dy / dt
=
(xxv) dx
(cosech x) = – cosec hx coth x
dx dx / dt
(xxvi) (xxvi)
(xxvii) (xxvii)
d (sin h–1 x) =
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
5. Double integrals
Definition:
The expression:
y2
x2
y y1 x x1
f ( x, y )dx.dy
is called a double integral and provided the four limits
on the integral are all constant the order in which the
integrations are performed does not matter.
If the limits on one of the integrals involve the other
variable then the order in which the integrations are
performed is crucial.
6. T h e d o u b le t e g r ao f f o ve r t h e r e ct a n g le is
in
l
R
f (x ,y )d A
R
f (x ,y )d A
R
lim
|P|
0
m
n
i 1 j 1
f (xi*j, y i*j )Δ Δi j
7.
Then, by Fubini’s Theorem
,
f ( x, y ) dA
D
F ( x , y ) dA
R
b
d
a
c
F ( x, y ) dy dx
8.
We assume that all the following integrals exist.
b
a
f ( x) dx
f x, y
c
a
b
f ( x) dx
c
f ( x) dx
g x, y dA
D
f x, y dA
D
g x, y dA
D
9.
The next property of integrals says that,
if we integrate the constant function f(x, y) = 1 over a
region D, we get the area of D:
1 dA
A D
D
If D = D1 D2, where D1 and D2 don’t overlap except perhaps on their
boundaries, then
f x, y dA
D
f x, y dA
D1
f x, y dA
D2
10. Example :
1. Evaluate
(x
3y)dA
D
WhereD
Ans :
(x
{(x, y) | -1
3y)dA
1
x
1 x2
-1 2x 2
1, 2x 2
(x
y
1
x 2}
3y)dydx
D
3
x(1 x - 2x )
((1 x 2 ) 2 - (2x 2 ) 2 )dx
-1
2
1
3
3 4
x x 3 - 2x 3
3x 2
x - 4x 4 dx
-1
2
2
1 2 1 4 3
1 5 1
3
1
3
( x - x
x x - x )
1-1 2
2
4
2
2
2
1
2
2
2
11. 2. Evaluate
xydA w hereD is the region bounded by
D
x - 1 and the parabola y 2
theline y
2x
Sol :
D {(x, y) | -3 x 5, ? y
y2 - 6
{(x, y) |
2
xydA
D
4
y 1
-2
y 2 -6
2
x
2x 6}
y 1, - 2 y 4}
xydxdy 36
6
13. Properties
1. Let R {(r, ) | a
rectangle and 0
f(x, y)dA
r b,
} be a polar
2 If f is continuous on R, then
b
a
f(rcos , rsin )rdrd
R
2. Let D {(r, ) |
, h1 ( ) r
h 2 ( )} be a polor
region. If f is continuous on D then
f(x, y)dA
D
h2 ( )
h1 ( )
f(rcos , rsin )rdrd
14. Example :
(4y2
1. Evaluate
3x)dA
R
wher R
e
Sol :
R
{(x, y) | y
{(x, y) | y
0, 1 x 2
{(r, ) | 1 r
(4y
2
0, 1 x 2
y2
2, 0
3x)dA
0
15
2
4}
1
(4(rsin ) 2
R
(15sin 2
4}
}
2
0
y2
7cos )d
3rcos )rdrd
15.
Changing The Order of integration
Sometimes the iterated integrals with givan limits bocomes more
compliated.As we know that w.r.t. y, or may be integrated in the
reverse order.
If it is given first to integrate w.r.t. x,then to change it consider a
vertical strip line and determine the limits.
If it is given first to integrate w.r.t. y,then to change it consider a
horizontal strip line and determine the limits.
16. 1 y
(x
3. Evaluate :
22 y
2
2
y)
(x
0 0
I R :x
I R :x
1
n
1
0, x
y, y
n
2
0, x
2
0, y
y, y
2
2
y )dxdy by changing the order of integration.
0
1
1, y
2
Take a horizontal strip line.
the limits are : x y 2 - x
0
1 2 -x
I
(
2
x y
x 1
2
)dydx
0 x
y
1
x
2
y
3
2 x
x
dx
3
0
3
1
0
x
3
1
2x
2
0
2x
3
3
4
7x
3 4
7 3
3x
(2 x)
3
dx
4 1
(2 x)
4
3
12
0
2
(2
x)
(2 x)
3
3
x
x 3
3
dx
17.
18. Triple integrals
The expression:
z2
y2
x2
z z1 y y1 x x1
f ( x, y, z )dx.dy.dz
is called a triple integral and provided the six limits on
the integral are all constant the order in which the
integrations are performed does not matter.
If the limits on the integrals involve some of the
variables then the order in which the integrations are
performed is crucial.
19. Determination of volumes by multiple integrals
The element of volume is:
V
x. y. z
Giving the volume V as:
x x2 y y2 z z2
V
x. y. z
x x1 y y1 z z1
That is:
x2
y2
z2
V
dx.dy.dz
x x1 y y1 z z1
20. properties
1. If E {(x, y, z) | (x, y) D, φ1 (x, y)
then
φ 2(x,y)
f(x, y, z)dv
E
D
2. If E {(x, y, z) | a
then
f(x, y, z)dv
E
x
φ1(x,y)
z
φ 2 (x, y)}
f(x, y, z)dz dA
b, g 1 (x)
b
g1(x) φ 2(x,y)
a
g1(x) φ1(x,y)
y
g 2 (x), φ1 (x, y)
f(x, y, z)dzdydx
z
φ 2 (x, y)}
21. Example: Find the volume of the solid bounded by the
planes z = 0, x = 1, x = 2, y = −1, y = 1 and the surface z
= x2 + y2.
2
V
x2 y 2
1
dx
x 1
dy
y
2
x 1
16
3
dz
1
z 0
3
x2 y
2
y
3
1
x 1
1
x2
dx
y
2
2x2
dx
1
x 1
y 2 dy
1
2
dx
3
22. 3. Find the volume of the tetrahedron bounded by the planes
x 2y, x 0, z 0 and x 2y z 2
Sol :
D {(x, y) | 0 x 1,
x
2-x
y
}
2
2
V
2- x
2
x
0
2
2 - x - 2ydA
D
1
3
1
(2 - x - 2y)dydx
23. 2. Find the volume of the solid bounded by the plane z
and the paraboloid z 1 - x 2 - y 2
Sol : D {(r, ) | 0
r 1, 0
(1 - x 2 - y 2 )dA
V
D
2
1
0
0
2
(1 - r 2 )rdrd
2 }
0
24. formula for triple integration in cylindrical
coordinates.
f ( x, y, z )dV
E
h2 ( )
u 2 ( r cos , r sin )
h1 ( )
u1 ( r cos , r sin )
f (r cos , r sin , z )rdzdrd
To convert from cylindrical to rectangular
coordinates, we use the equations
1 x=r cosθ y=r sinθ z=z
whereas to convert from rectangular to
cylindrical coordinates, we use
2. r2=x2+y2 tan θ=
z=z
y
x
25.
26. 2
2
D
Here we use cylindrical coordinates(r,θ,z)
∴ the limits are:
x
y
i.e. r
0 r
0
2
z
1
z 1
1
2
2π 1 1
I
r
rdzdrdθ
0 0 r
2
1
r
2
(1
r ) drd
0 0
2
0
r
3
3
x
4
1
4
0
2
1
3
1
4
2
x y dV, where D is the solid bounded by the surfaces x y z
Example : Evaluate
2
2
6
2
,z
0,z 1.
27. Formula for triple integration in spherical coordinates
f ( x, y, z )dV
E
d
b
c
a
f ( p sin cos , p sin som , p cos ) p 2 sin dpd d
where E is a spherical wedge given by
E {( p, , ) a
p b,
,c
d}
29. x
Example : Evaluate
2
2
y z
2
dV over the volume of the sphere x
D
Here we use spherical co-ordinates (r,θ,z)
∴ The limits are:
0
0
r
1
0
2
2
1
2
r r
I
2
sin drd d
0 0 0
2
0
2
cos
2
1
5
0
r
5
1
5
4
5
0
2
2
y z
2
1.