Leonhard Euler | An Introduction

Leonhard Euler | An Introduction Born in Basel, Switzerland on 15th April 1707, Leonhard Euler was arguably the brightest mathematician of all time. The Swiss mathematician and physicist is considered a pioneer in many fields of mathematics. He introduced a lot of the mathematical terminology and notation used today and he is considered the father of mathematical analysis where, for instance, he introduced the notation of a mathematical function, f(x). His contributions to the field of mathematics are in analytic and differential geometry, calculus, the calculus of variation, differential equations, series and the theory of numbers. In physics, although really all his contributions to mathematics apply to physics, he introduced both rigid body mechanics and analytical mechanics (Kline, 401-402). Born to Paul Euler and Margarete Bruckner, Leonhard was the first of six children. He grew up in Riehen but attended school in Basel. Although mathematics was not taught in his school his father had kindled his interest in the subject (Paul Euler had been friends with another great mathematician at the time, Johann Bernoulli) by giving him lessons at home. Euler entered university at the age of 13 at the University of Basel. Although his official courses of study were philosophy and law, Euler met with Johann Bernoulli who advised Euler and gave him help with his mathematical studies on Saturday afternoons (Stillwell, 188). Euler lived and worked mainly in Russia and Germany. First he joined the faculty at St Petersburg Academy of Sciences were he worked at first in the medical department then he was quickly promoted to a senior position in the department of mathematics through the influence of his friend Daniel Bernoulli. He also helped the Russian government on many projects including serving in the Russian navy as a medical Lieutenant. After the death of Catherine I in1740 and because of the tough conditions that ensued, Euler moved to Germany at the invitation of Prussian King, Frederick II to the Prussian Academy of Science were he stayed for the next 25 years of his life. Euler gave much service to the Academy which compensated him generously. He sent most of his works to be published there, served as a representative as well as advising the Academy on its many scientific activities. It is there that he reached the peak of his career writing about 225 memoirs on almost every topic in physics and m athematics (Varadarajan, 11). Euler returned to St. Petersburg in 1766 under the invitation of the then czarina Catherine the Great (Catherine II) to the St Petersburg Academy. During this period he lost almost all his eyesight through a series of illnesses becoming nearly totally blind by 1771. Nevertheless, his remarkable memory saw him writing about 400 memoirs during this time. It is said that he had a large slate board fixed to his desk where he wrote in large letters so that he could view dimly what was being written. He died on the 18th day of September 1783 due to cerebral hemorrhaging. It is also recorded that he was working even to his last breath; calculations of the height of flight of a hot air balloon were found on his board (Varadarajan 13). Eulers contribution to Mathematics and Physics was a lot. His ideas in analysis led to many advances in the field. Euler is famously known for the development of function expressions like the addition of terms, proving the power series expansion, the inverse tangent function and the number e: ∑ (xn/n!)= lim ((1/0!) + (x/1!) + (x2/2!) +†¦+ (xn/n!) ) =ex The power series equation in fact helped him solve the famous 1735 Basel problem: ∑ (1/n2) = lim ((1/12) + (1/22) + (1/32) + †¦+(1/n2)) = Ï€2/6 He introduced the exponential function, e, and used it plus logarithms in analytic proofs. He also defined the complex exponential function and a special case now known as the Eulers Identity: eiφ = cos φ + isin φ And eiÏ€ + 1= 0 (Eulers Identity) In fact, De Moivres formula for complex functions is derived from Eulers formula. Similarly, De Moivre is recognized for the development of calculus of variations, formulating the Euler-Lagrange equation. He was also the first to use solve problems of number theory using methods of analysis. Thus, he pioneered the theories of hyperbolic trigonometric functions, hyper geometric series, the analytic theory of continued fractions and the q-series. In fact, his work in this field led to the progress of the prime number theorem (Dunham 81). The most prominent notation introduced by Euler is f(x) to denote the function, f that maps the variable x. In fact he is the one who introduced the notion of a function to the field of mathematics. He introduced, amongst others, the letter ∑ to mean the sum, Ï€ for the proportion regarding the perimeter of a circle up to the span or the diameter, i for the imaginary unit, √(-1) and the e (2.142†¦) to represent the base of the natural logarithm. Euler also contributed to Applied Mathematics. Interestingly enough, he developed some Mathematics applications into music by which he hoped to incorporate musical theory in mathematics. This was however, not successful. This notwithstanding, Euler did solve real-world problems by applying analytical techniques. For instance, Euler incorporated the Method of Fluxions which was developed by Newton together with Leibnizs differential calculus to develop tools that eased the application of calculus in physical problems. He is remembered for improving and furthering the numerical approximation of integrals, even coming up with the Euler approximations. More broadly, he helped to describe many applications of the constants Ï€ and e, Euler numbers, Bernoulli numbers and Venn diagrams. The Euler-Bernoulli beam equation (one of the most fundamental equations in engineering) is just one of the contributions of the mathematician to physics. He used his analytical skills in classical mechanics and used the same methods in solving celestial problems. He determined the orbits of celestial bodies and calculated the parallax of the sun. He differed with Newton (then the authority in physics) on his corpuscular theory of light. He supported the wave theory of light proposed by Hugens. Eulers contributions to graph theory are at the heart of the field of topology. He is famously known to have solved the Seven Bridges of Konigsberg problem, the solution of which is considered the first theorem of planar graph theory. He introduced the formula V-E+F=2 It is a mathematical formula relating vertices, edges and faces of a planar graph or polyhedron. The constant in the above formula in now called the Euler characteristic. Euler is also recognized for the use of closed curves in the provision of explanation concerning reasoning which is of syllogistic nature. Afterwards the illustration or diagrams were referred to as the Euler diagrams. The Number Theory is perhaps the most difficult branch of mathematics. Euler used ideas in analysis linking them with the nature of prime numbers to provide evidence that the total of all the reciprocals of prime numbers diverges. He also discovered the link between the primes Riemann zeta function, what is now called the Euler product formula for the Riemann zeta function. Euler made great strides in the Lagrange four-square theorem while proving Fermats theorem on the sum of two squares, Fermats Identities and Newtons identities. Number theory consists of several divisions which include the following: Algebraic Number Theory, Combinational Number theory, Analytic Number Theory, Transcendental number theory, Geometric number theory and lastly we have the Computational Number Theory. For his numerous contributions to academia, Euler won numerous awards. He won the Paris Academy Prize twelve times over the course of his career. He was elected as a foreign member, in 1755, of the Royal Swedish Academy of Sciences while his image has been featured on many Russian, Swiss and German postage stamps. Above all, he was respected greatly amongst his academic peers demonstrated by a statement made by the great French mathematician, Laplace to his students to read Euler since he was the master of them all (Dunham xiii). Though not all of the proofs of Euler are satisfactory in regard to the current standards or principles used in mathematics, the ideas created by him are of great importance. They have set a path to the current mathematical advancements. To conclude, we can therefore say that Euler is a very significant person in the development and advancement of Mathematics. His work has contributed a lot to mathematics up to the current period. References Dunham, William. Euler: The Master of Us All. Dolciani Mathemathical Expositions Vol. 22. MAAA, 1999. Kline, Morris. Mathematical Thoughts from Ancient to Modern Times, Vol 2. New York: Oxford University Press, 1972. Stillwell, John. Mathematics and its History. Undergraduate Texts in Mathematics. Springer, 2002. Varadarajan, V. S. Euler Through Time: A New Look at Old Themes. AMS Bookstore, 2006.

Literacy Essay -- Education Learning Papers

Literacy Why is literacy of such interest to postsecondary teachers of writing? If "literacy," as it might be defined by someone outside the field of English studies, is "the ability to read and write," then why are we interested in it when our own students presumably acquired this ability in the distant past, about the time they learned to tie bows in shoelaces? That we are interested in literacy, I take it, is obvious from the proliferation of books with the word in their titles, of conferences on topics in literacy (the Modern Language Association has now held two), and of review essays such as this one, commissioned by the editors of a journal that purports to deal in issues relevant to "composition theory" and "advanced composition" at the college level. Defining Literacy I might answer my opening question by redefining "literacy" from within English studies in such a way as to make it more complex and problematic, and thus to question the presumption that college students have already acquired it. Indeed, in these proliferating discussions of literacy, its simple definition as the ability to read and write is usually rejected at the outset. But in attempting to go beyond the simple definition, scholars begin to diverge. Consider, for example, the variety of essays collected under the title, The Right to Literacy (1990), a title which implies a unitary conception of literacy and a contest only over who possesses the desired object. In this volume, Andrea A. Lunsford, Helene Moglen, and James Slevin collect twenty-nine papers from the 1988 MLA "Right to Literacy" conference. Most of the essays, therefore, are short; and ranging through them, one gets a sense of the heterogeneous mixture of topics that accrue for Engli... ...Discourse Community." Curriculum Inquiry 12 (1982): 191-207. ---. "Literacy in Culture and Cognition." A Sourcebook for Basic Writing Teachers. Ed. Theresa Enos. New York: Random, 1987. 125-37. Brandt, Deborah. Literacy as Involvement: The Acts of Writers, Readers, and Texts. Carbondale: Southern Illinois UP, 1990. Hirsch, E.D., Jr. Cultural Literacy: What Every American Needs to Know. 2nd ed. New York: Vintage, 1988. Kintgen, Eugene R., Barry M. Kroll, and Mike Rose, eds. Perspectives on Literacy. Carbondale: Southern Illinois UP, 1988. Lunsford, Andrea A., Helene Moglen, and James Slevin, eds. The Right to Literacy. New York: MLA, 1990. Stuckey, J. Elspeth. The Violence of Literacy. Portsmouth, NH: Boynton, 1991. Taylor, Denny, and Catherine Dorsey-Gaines. Growing Up Literate: Learning from Inner-City Families. Portsmouth, NH: Heinemann, 1988.

Is Hip-Hop Culture Harming Our Youth?

Is Hip-Hop Culture Harming Our Youth? Hip-hop culture is everywhere. The culture, which encompasses rapping, deejaying, break-dancing and graffiti-writing, has become so popular that it has entered mainstream fashion and modern language. It doesn't stop there. The culture permeates everything from TV commercials to toys to video games. Currently, there is even a hip-hop exhibit at the Brooklyn Museum of Art. You name it, and hip hop is there representing. However, hip hop's most potent form is its rap music–embraced by urban Blacks and suburban Whites alike.It is raw self-expression that sometimes features profane lyrics, misogyny and violence. The music, along with rap videos that often present a disturbing mix of rap, hip-hop dance styles, fashion and language, leave many people asking: â€Å"Is hip-hop culture harming our youth? â€Å". â€Å"The hip-hop culture is just like electricity,† civil rights leader the Rev. Al Sharpton told JET. â€Å"It can be used nega tively or positively. The same electric current that lights up your house can also electrocute you.It is the misuse of hip-hop culture to attack our women and promote violence. We must encourage the proper use of hip-hop culture. We are all influenced by the hip-hop generation. † Sharpton, who recently hosted a special summit on social responsibility in the hip-hop industry, labeled gangsta rappers â€Å"well-paid slaves. † Advertisement â€Å"Don't let some record executive tell you that cursing out your mama is in style. Anytime you perpetuate a slave mentality that desecrates women and that desecrates our race in the name of a record†¦. I consider you a well-paid slave. Sharpton labeled the fashion of these rappers–loose-fitting pants and sneakers without shoelaces–prison clothes. Sharpton was joined at the summit by some of the industry's superstars, including Master P, Sean â€Å"Puffy† Combs, Chuck D and RZA. No Limit Records' CEO-rapper Master P apologetically admitted that some of his lyrics might be offensive to women. â€Å"Everything that came from me like that came out of ignorance. I hope to edit myself in the future,† he reportedly said during the event. However, he asserted, â€Å"I don't believe any form of entertainment is harming our youth.It is up to parents to raise their own children and teach them. Blaming entertainment is a scapegoat. † Dr. C. DeLores Tucker, chair of the National Political Congress of Black Women, Inc. , who has been a strong critic of hip-hop culture for more than a decade, continues to cite its reported negative influence over our youth. â€Å"The glorification of pornography, wanton disregard for civil authority, misogynistic disrespect for women and a penchant for violence are the unintended impact of hip-hop culture on today's youth,† the activist revealed. I say unintended, because hip hop †¦ was intended to celebrate the revival of the age-old rhym ed recitations of life's problems and aspirations set to music. † Unfortunately, somewhere along the way, some unscrupulous elements hijacked this influential conduit to our youth and loaded it with the evil and debasing, hate-driven messages in the lyrics we now know as gangsta rap. Hence the artistry of the rappers in the streets is used by the gangstas in the suites to spread cultural garbage among our youth. Hip-hop music mogul Russell Simmons, who dubs himself the â€Å"grandfather of hip hop,† says the culture has been helpful in allowing others to understand Blacks' inner-city plight. â€Å"It is not going to go away. I will stand by the door to keep it from being censored. It is easy to try to put a lid on what you don't like. But the real test, if you are freedom-loving people, is to protect even that which we don't like from the censorship by the government,† he explained. He points out the positive contributions from the hip-hop culture. The slam poetr y settings evolved out of the hip-hop community. Some of it does not have any curse words. Young folks have found a way to express themselves that was really inspired by rap. Kids are writing. They are thinking; brilliant thoughts are coming out in this creative poetry. † Concludes Simmons, â€Å"The hip-hop culture will evolve in ways that will be known as a significant art. The rap artists are not the poets that they are yet to become, but they are indeed poets. â€Å"

Children Always Need Love, Compassion, Support And Empathy...

Chase Sullivan DR. Earnest FSL November 9, 2016 Fatherless Children always need love, compassion, support and empathy from their parents when growing up. They are better off without a father in certain circumstances such as abuse, but both boys and girls will suffer a negative effect as they develop into adults without a father in any situation. No one can agree that growing up fatherless is far from the worst thing other than Michele Weldon from the New York Times, and her article â€Å"When Children Are Better off Fatherless† as she states: In the cases where the father is far from heroic – even abusive – his absence is also the absence of the chaos, anger, pain and disruption he would bring to his family. Americans encourage women to leave abusive partners, but mothers who do this end up in a class we shame and pity. The government itself sends the message that children are better off with a father. The reality is, many children are better off without their fathers.† Ms. Weldon is right in the fact that children are better off without abusive fathers as they regularly risk beatings, psychological damage, and sexual abuse as they are near their abusive father. A child’s stress limit will always be pushed to unsafe boundaries as they constantly feel the pressure not to upset or anger their father and the fear of face unintended consequences. An unsafe home is not a place for any child to be nurtured or grow up in. Unfortunately, leaving the abusive husband or loosing aShow MoreRelatedParenting Styles And Their Effects On Children1572 Words   |  7 PagesParenting Styles and Their Effects on Children Parents play a key role in their children’s lives, including supervision, involvement, love, support, comfort, and a wide variety of discipline and punishment. The relationship between parenting styles and child is vital and impacts the child’s physical, mental, and emotional well-being. 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