1. This document discusses changing the order of integration when evaluating double integrals over rectangles. To evaluate a double integral, we first integrate with respect to one variable while treating the other as a constant, and then integrate with respect to the remaining variable.
2. It provides definitions and properties of double integrals, including that a double integral over a rectangle exists if the integrand function is integrable over the region.
3. Examples are given to illustrate continuous and non-continuous integrand functions over different regions.
These slides contain information about Euler method,Improved Euler and Runge-kutta's method.How these methods are helpful and applied to our questions are detailed discussed in the slides.
Newton's Backward Interpolation explained with example. History of interpolation along with it's advantages and disadvantages. Applications of interpolation in computer sciences.
These slides contain information about Euler method,Improved Euler and Runge-kutta's method.How these methods are helpful and applied to our questions are detailed discussed in the slides.
Newton's Backward Interpolation explained with example. History of interpolation along with it's advantages and disadvantages. Applications of interpolation in computer sciences.
This presentation will be very helpful to learn about system of linear equations, and solving the system.It includes common terms related with the lesson and using of Cramer's rule.
Please download the PPT first and then navigate through slide with mouse clicks.
Changing variable is something we come across very often in Integration. There are many
reasons for changing variables but the main reason for changing variables is to convert the
integrand into something simpler and also to transform the region into another region which is
easy to work with. When we convert into a new set of variables it is not always easy to find the
limits. So, before we move into changing variables with multiple integrals we first need to see
how the region may change with a change of variables. In order to change variables in an
integration we will need the Jacobian of the transformation.
Dealing with Notations and conventions in tensor analysis-Einstein's summation convention covariant and contravariant and mixed tensors-algebraic operations in tensor symmetric and skew symmetric tensors-tensor calculus-Christoffel symbols-kinematics in Riemann space-Riemann-Christoffel tensor.
This includes different line drawing algorithms,circle,ellipse generating algorithms, filled area primitives,flood fill ,boundary fill algorithms,raster scan fill approaches.
This presentation will be very helpful to learn about system of linear equations, and solving the system.It includes common terms related with the lesson and using of Cramer's rule.
Please download the PPT first and then navigate through slide with mouse clicks.
Changing variable is something we come across very often in Integration. There are many
reasons for changing variables but the main reason for changing variables is to convert the
integrand into something simpler and also to transform the region into another region which is
easy to work with. When we convert into a new set of variables it is not always easy to find the
limits. So, before we move into changing variables with multiple integrals we first need to see
how the region may change with a change of variables. In order to change variables in an
integration we will need the Jacobian of the transformation.
Dealing with Notations and conventions in tensor analysis-Einstein's summation convention covariant and contravariant and mixed tensors-algebraic operations in tensor symmetric and skew symmetric tensors-tensor calculus-Christoffel symbols-kinematics in Riemann space-Riemann-Christoffel tensor.
This includes different line drawing algorithms,circle,ellipse generating algorithms, filled area primitives,flood fill ,boundary fill algorithms,raster scan fill approaches.
Consider a l-D elastic bar problem defined on [0, 4]. The domain .pdfferoz544
Consider a l-D elastic bar problem defined on [0, 4]. The domain is devided into 4 linear 2-node
elements as follows. Write down element shape functions N (x) for each element (e = 1, 2, 3),
Give a sketch of the shape functions N_i(x) = (f = 1, 2, 3, 4, 5) on the entire domain [0, 4].
Given the nodal displacement matrix d = [4 8 6 2]^T, find the displacement solutions at x = 1.5
and x = 3.6. Given the nodal displacement matrix d = [4 8 6 2]^T, find the strain (defined as
du/dx) solutions at x = 0.4 and x = 2.3.
Solution
Using the nodal-based smoothing operation, the strains to be used in Equation (23) is assumedto
be the
smoothed
strain for node
k
dened by
e
k
e
(
x
k
)
=
k
e
(
x
)
W
(
x
x
k
)
d
where
W
=
W
W
W
is a diagonal matrix of smoothing function
W
. For simplicity, the smoothingfunction is taken as
W
(
x
x
k
)
=
1
/
A
k
,
x
k
0
,
x
/
k
where
A
k
=
k
d
is the area of smoothing domain for node
k
.Substituting into Equation and integrating by parts, the smoothed strain canbe calculated using
e
k
=
1
A
k
k
e
(
x
)
d
=
1
A
k
k
L
n
u
(
x
)
d
=
e
k
(
u
)
where
k
is the boundary of the smoothing domain for node
k
, and
L
n
is the matrix of the outwardnormal vector on
k
. Equation states the fact that the assumed strain
e
k
is a function of theassumed displacement
u
.Substituting Equation into Equation , the smoothed strain can be expressed in thefollowing
matrix form of nodal displacements:
e
k
=
i
N
in
B
i
(
x
k
)
d
i
where
N
in
is the number of nodes in the inuence domain of node
k
(including node
k
). Whenlinear shape functions are used, it is the number of nodes that is directly connected to
node
k
inthe triangular mesh (see Figure 1). In Equation , the
B
i
(
x
k
)
is termed as the
smoothed
strainmatrix that is calculated using
B
i
(
x
k
)
=
b
ix
(
x
k
)
00
b
iy
(
x
k
)
b
iy
(
x
k
)
b
ix
(
x
k
)
Using the Gauss integration along each segment of boundary
k
, we have
b
il
=
1
A
k N
s
m
=
1
N
g
n
=
1
w
n
i
(
x
mn
)
n
l
(
x
m
)
(
l
=
x
,
y
)
(30)where
N
s
is the number of segments of the boundary
k
,
N
g
is the number of Gauss points usedin each segment,
w
n
is the corresponding weight number of Gauss integration scheme, and
n
l
isthe unit outward normal corresponding to each segment on the smoothing domain boundary. In
theLC-PIM using linear shape functions,
n
g
=
1 is used. The entries in sub-matrices of the stiffnessmatrix
K
in Equation are then expressed as
K
ij
=
N
k
=
1
K
ij
(
k
where the summation means an assembly process as we practice in the FEM, and
K
ij
(
k
)
is thestiffness matrix associated with node
k
that is computed using
K
ij
(
k
)
=
k
B
T
i
D
B
j
d
=
B
T
i
D
B
j
A
k
The entries (in sub-vectors of nodal forces) of the force vector
f
in Equation can be simplyexpressed as
f
i
=
k
N
in
f
i
(
k
)
The above integration is also performed by a summation of integrals over smoothing
domains;hence,
f
i
is an assembly of nodal force vectors at the surrounding nodes of node
k
:
f
i
(
k
)
=
t
(
k
)
U
i
ˆ
t
d
+
(
k
)
U
i
b
d
Note again th.
This study deals with the active control of the dynamic response of a string with fixed ends and mass
loaded by a point mass. It has been controlled actively by means of a feed forward control method. A point mass of a
string is considered as a vibrating receiver which be forced to vibrate by a vibrating source being positioned on the
string. By analyzing the motion of a string, the equation of motion for a string was derived by using a method of
variation of parameters. To define the optimal conditions of a controller, the cost function, which denotes the dynamic
response at the point mass of a string was evaluated numerically. The possibility of reduction of a dynamic response
was found to depend on the location of a control force, the magnitude of a point mass and a forcing frequency
Comparative Study of the Effect of Different Collocation Points on Legendre-C...IOSR Journals
We seek to explore the effects of three basic types of Collocation points namely points at zeros of Legendre polynomials, equally-spaced points with boundary points inclusive and equally-spaced points with boundary point non-inclusive. Established in literature is the fact that type of collocation point influences to a large extent the results produced via collocation method (using orthogonal polynomials as basis function). We
analyse the effect of these points on the accuracy of collocation method of solving second order BVP. For equally-spaced points we further consider the effect of including the boundary points as collocation points. Numerical results are presented to depict the effect of these points and the nature of problem that is best handled by each.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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.
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!
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
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
2. Change of Order of Integration
To evaluating a double integral we integratefirst with respect
to one variable andconsidering the other variable as
constant,and then integrate with respect to theremaining
variable. In the former case, limitsof integration are
determined in the givenregion by drawing stripes parallel to y-
axiswhile in second case by drawing strips parallel to x-axis.
2
10.
R R
R R R
R R
y)dAg(x,y)dAf(x,
thenR,y)(x,y)g(x,y)f(x,If3.
y)dAg(x,y)dAf(x,y))dAg(x,y)(f(x,2.
y)dAf(x,cy)dAcf(x,1.
:properties
existslimitthisif
10
