Evaluation of integrals of the given functions along the unit circle on the complex plane. Application of the parametrization method. Evaluation of the integral of an odd function.
Introduction to integral calculus.
This slideshow deals with concept of integration. A complete explanation is provided that how integration can be written as summation. Area under the graph can be calculated through integration.
Introduction to integral calculus.
This slideshow deals with concept of integration. A complete explanation is provided that how integration can be written as summation. Area under the graph can be calculated through integration.
Ptolemy's theorem visualisation. 3D graphics.Mikołaj Hajduk
Ptolemy's theorem states the following: a convex quadrilateral can be inscribed in a circle if and only if the product of the lengths of one pair of opposite sides added to the product of the lengths of the other pair is equal to the product of the lengths of the diagonals. Thus, in a cyclic quadrilateral ABCD we have
AB*DC + AD*BC = AC*BD
The 3D picture shows an attempt at combining applied art with pure mathematics. The vase shown in the picture is an example of ceramic vessel that could exist in real world.
Still life with vases - a 3D visualisation.Mikołaj Hajduk
The picture presenting a still life with two vases made of black polished ceramics, standing on blocks of granite and limestone, placed in an empty room illuminated by sunlight going through a big window.
How to draw arcs of huge circles - description.Mikołaj Hajduk
In case you are interested in drawing circles with a huge diameter (by "huge" I mean here really big ones, measured in kilometers or so) there is a tricky idea needed because the rope-and-the-stick method won't work for circles with a diameter bigger than several dozen meters. Here you can use a method based on the well known fact from geometry stating that all angles inscribed in a circle and subtended by the same chord (lying on the same side of the chord) are equal.
Calculation of the volume of a bottle partially filled with a fluid.Mikołaj Hajduk
How to calculate the volume of a flat-bottomed, corked bottle, partially filled with a fluid having only the ruler as a measuring tool?
The 3D graphics made by me and presented below gives an answer to this question.
Permutation theorem and its use to proving inequalities.Mikołaj Hajduk
The permutation theorem is very useful when dealing with inequalities between sums of products of the two real number sequences. In numerous cases inequalities otherwise difficult to prove can be proven almost automatically.
The sum of the triangle sides lengths reciprocals vs a cyclic sum of a specif...Mikołaj Hajduk
Proof of the inequality between the sum of the reciprocals of a triangle sides lengths and a cyclic sum of a specific form. Use of a transformed inequality between the arithmetic mean and the harmonic mean.
A few digressions on the theme of indescribable numbers. Proof of the fact that there exist real numbers that may be indescribable, i.e. impossible to express in any language based on the finite alphabet of symbols.
A mathematical explanation of a simple trick that shows how to quickly calculate the cube root of a maximally 6-digit number that was given by a spectator. The spectator chooses a natural two-digit number, keeps it in secret and only information he/she gives publicly is its cube. Conjurer's task is to quickly guess the original number.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
A Strategic Approach: GenAI in EducationPeter 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.
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.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
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.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
Francesca Gottschalk - How can education support child empowerment.pptx
Complex Integral
1. Show that
C
f(z)dz = 0, where f is the given function and C is the unit circle |z| = 1.
a) f(z) = z3
− 1 + 3i
b) f(z) = z2
+
1
z − 4
c) f(z) =
sin z
(z2 − 25) (z2 + 9)
Solution by Mikołaj Hajduk: The points a) and b) can be solved with use of parametrization z(t) = eit
where t ∈ [0, 2π] and dz = ieit
dt.
Ad. a)
C
f(z)dz =
C
(z3
− 1 + 3i)dz =
2π
0
((eit
)3
− 1 + 3i)ieit
dt =
=
2π
0
e3it
ieit
+ (−1 + 3i)ieit
dt =
2π
0
ie4it
dt +
2π
0
(−1 + 3i)ieit
dt =
= i
2π
0
e4it
dt + (−1 + 3i)i
2π
0
eit
dt = i
1
4i
e4it
2π
0
+ (−1 + 3i)i
1
i
eit
2π
0
=
=
1
4
e4it
2π
0
+ (−1 + 3i)eit
2π
0
=
1
4
(e4i∗2π
− e4i∗0
) + (−1 + 3i)(ei∗2π
− ei∗0
) =
c 2015/09/17 02:27:23, Mikołaj Hajduk 1 / 4 next
2. =
1
4
((e2iπ
)4
− e0
) + (−1 + 3i)(e2iπ
− e0
) =
1
4
(1 − 1) + (−1 + 3i)(1 − 1) = 0
Ad. b)
C
f(z)dz =
C
z2
+
1
z − 4
dz =
2π
0
(eit
)2
+
1
eit − 4
ieit
dt =
=
2π
0
ie2it
eit
dt +
2π
0
ieit
eit − 4
dt = i
2π
0
e3it
dt +
2π
0
(eit
− 4)
eit − 4
dt =
= i
1
3i
e3it
2π
0
+ ln |eit
− 4|
2π
0
=
1
3
(e3i∗2π
− e3i∗0
) + (ln |ei∗2π
− 4| − ln |ei∗0
− 4|) =
=
1
3
((e2iπ
)3
− e0
) + (ln |1 − 4| − ln |1 − 4|) =
1
3
(1 − 1) + (ln 3 − ln 3) = 0
Ad. c)
Let’s notice that the function f is odd in its domain because
sin(−z)
def
=
ei(−z)
− e−i(−z)
2i
=
e−iz
− eiz
2i
= −
eiz
− e−iz
2i
def
= − sin(z)
and hence
f(−z) =
sin(−z)
((−z)2 − 25) ((−z)2 + 9)
=
− sin z
(z2 − 25) (z2 + 9)
= −f(z)
Let CU and CB denote the circle arcs corresponding to the upper and bottom halves of the circle C. We have
then
C
f(z)dz =
CU
f(z)dz +
CB
f(z)dz
c 2015/09/17 02:27:23, Mikołaj Hajduk 2 / 4 next
3. The function h : CU −→ CB defined as h(z) = −z is a bijection between points of the arcs CU and CB that
transforms the beginning point of the arc CU into the beginning point of the arc CB and the ending point of
the arc CU into the ending point of the arc CB. The arc CB is an image of the arc CU under the function h:
CB = h(CU)
c 2015/09/17 02:27:23, Mikołaj Hajduk 3 / 4 next
4. therefore, bearing in mind that f(z) is odd and h−1
(z) = −z, we get
CB
f(z)dz =
h(CU)
f(z)dz =
CU
f(h−1
(z))dz =
CU
f(−z)dz =
CU
−f(z)dz = −
CU
f(z)dz
Hence
C
f(z)dz =
CU
f(z)dz +
CB
f(z)dz =
CU
f(z)dz −
CU
f(z)dz = 0
c 2015/09/17 02:27:23, Mikołaj Hajduk 4 / 4