Abstract Calculus is divided into two branches, one being differential and the other being integral. This paper provides an overview of calculus and examines the two branches in more detail. It also looks at the importance of calculus in the world today.
From the Paper "It must be stated that Newton's mathematics that involved 'fluxions' was one of the first forms of the area defined as 'differential calculus'. Although Newton used and preferred to use geometrical methods to algebraic equations, calculus methods had come into importance. However, calculus was not widely accepted at the time, and there were several philosophical objections to the science, but the fact remains that these objections over the years have made no difference to the application of the science."
This paper looks at the controversy over who discovered calculus and provides an explanation of why the honor should go to Isaac Newton over the claim of Gottfried Leibniz.
Abstract In considering the great controversy as to who discovered the calculus, either Newton or Leibniz, this essay argues that the accolade should go to Newton. The decision is made on the ground as to who conducted himself most honorably in the affair. There is no doubt that both scientists come to independent discovery and formulation of the calculus. The essay is at pains to point out the greatness of Leibniz, as philosopher, scientist and mathematician. It even acknowledges that Leibniz's formulation of the calculus is superior, and that this superiority derives from his related philosophy of monadology. But Leibniz certainly acts suspiciously during the controversy. The writer maintains that in contrast, Newton at all times displays magnanimity and selflessness. The writer concludes that Newton does not need accolades for his contributions to shine, and yet they shine on their own merits.
From the Paper "Calculus to Newton was merely a tool that he required to come to his universal theory of gravitation and motion, and not something that should be flouted separately. He was even reluctant to publish the revolutionary Principia, and did so only after the prodding of Edmund Halley.
"Leibniz, on the other hand, was eager to publish and propagate his findings. While we admit to his originality to a large extent, the conduct of Leibniz is highly suspicious in the proceedings. He makes no defense of his integrity, as Newton does, but instead seem entirely intent on pushing the evidence alone, as if defending himself in a court of law, and this makes us feel that he is hiding something. Subsequent scholarship does indeed reveal that he manipulated documents before being released."
Abstract This paper helps to explain and justify the creation of calculus. Isaac Newton helped to solve some of the most perplexing problems the world has imagined, and the method he used in doing so is still used for the same purpose today. Newton's creation of calculus and ideas of using it to prove the universal laws of nature made human reason the most powerful method of thought and most definite route to seeking the truth.
From the Paper ?In mathematics, a certain surprising thing happens again and again. Someone poses a simple question, a question so simple that it seems no useful result can come from answering it. And yet it turns out that the answer opens the door to all kinds of interesting developments, and gives great power to the person who understands it.? (Saywer 3) This quote from a prestigious professor of mathematics parallels the story of Isaac Newton and his development of calculus. Isaac Newton helped to solve some of the most perplexing problems the world has imagined, and the method he used in doing so is still used for the same purpose today. There is a popular myth that Newton was sitting under a tree when an apple fell from it, and he asked himself what force could pull the apple to the Earth. Whether this story is true or not is uncertain, but the image is clear of Newton getting hit on the head with the apple of epiphany. He used calculus to prove that gravity pulled objects to Earth and held the planets together, and also to prove his world-renowned three laws of motion. By forming this revolutionary method of problem solving, Newton not only paved the way for new roads in mathematics but also changed the way that people thought and sought out answers. Newton's creation of calculus and ideas of using it to prove the universal laws of nature made human reason the most powerful method of thought and most definite route to seeking the truth.
Abstract This paper presents a detailed examination of the history of calculus. The writer takes the reader on an exploratory path through the origins of mathematics and then on to the history of calculus. The people who are credited with its invention as well as the forms that it took are all included in the discussion.
From the Paper "The history of mathematics is one in which the topic follows the actual subject. Mathematics are taught by building on foundational blocks. Each block is taught and mastered and when that is completed the next block is introduced. The origin and history of mathematics follows the same path. The history of calculus is perhaps the most interesting of the mathematical techniques. The history and origin of calculus is founded in philosophy as well as science and it is one of the most fascinating of the mathematical theories and practices."
Abstract This paper reviews and discusses the literature 'Understanding the Concepts of Limit and Continuity in Calculus Courses at the High School and/or Early College Level'. According to the paper, the ten fastest growing career fields include five that are computer-related which rely heavily on the knowledge of mathematics.
Outline:
Background and Overview
Studies in the Concept of Limit and Continuity in High School and Early College-Level Calculus Courses
Interactions of Methods and Questions for Further Study
From the Paper "Clearly, then, just going through the motions and mechanics of calculus may serve to help familiarize these young learners with the basic steps involved. There are a number of studies to date, though, that have confirmed that these students typically experience a number of problems in understanding key calculus concepts that adversely affect their ability to master them (Bezuidenhout, 2001). To help facilitate the process, just as handheld calculators have been introduced into some early mathematics classes to facilitate routine calculations to allow more time for instruction, Heid (1988) suggested early on that computers could likewise be used to help beginning calculus students learn more by automating the algorithmic functions required. Although there is clearly a trade-off involved in such an approach, this author maintains that this approach would allow more class time for development of higher mathematics concepts."
Abstract This paper states that the field of aerodynamics could not exist without calculus. The author discusses the most prevalent and widely used equations. The advent of the computer has greatly improved the use of these equations in the field and allowed the field of aerodynamics to become more precise.
Table of Contents
Introduction
The Myth about Bumblebee Flight
Turbulence
The Bermouli Equation
Continuity Equation
Navier-Stokes Equations
Conclusion
From the Paper "Math is the language of science. The different disciplines of math relate to different areas of science. Science needs math in order to be understood. Algebra allows us to create sentences using numbers to describe an event. Geometry and Trigonometry help us to describe shapes, and Calculus is the tool for describing change. It can be a change in angles as in vector calculus, a change in rate, a change in speed, or almost any other change."
Abstract This paper looks at the Newtonian Universe, laws of gravity, the development of calculus, the laws of motion and the idea of the clockwork universe.
From the Paper "Sir Isaac Newton was considered by many the most important figure in the development of modern science, and his accomplishments covered a broad scope, from fundamental contributions to physics and astronomy, to the invention in parallel with Liebnitz of the mathematical field of calculus and Newtonian mechanics which came to be regarded as the ultimate explanatory science. Sir Isaac Newton changed the understanding of the universe with his three laws of motion."
Tags: gravity, clockwork universe, calculus, laws of motion
Abstract This paper explains that L'Hopital, who lived during the conception of modern calculus, was taught by Bernoulli; the result of this tuition was L'Hopital's "Analyse des Infiniments Petits", which became the French reference book in the calculus for a century. The author points out that L'Hopital's name is guaranteed to survive in the memories of thousands of mathematicians because of the L'Hopital rule, which is useful when dealing with indeterminate forms. The paper relates that L'Hopital created the template by which all calculus texts would be modeled and measured against for the next three hundred years. Formulas included.
From the Paper "A natural progression from his two first works on the topic of calculus would have been a serious examination of the integral calculus. Indeed, this was a project that L'Hopital was capable of and actually began to write before his death. However, one of his contemporaries-Leibniz-made it known to L'Hopital that he also endeavored to publish a work covering the same hole in written calculus of the time. Apparently, out of respect to the mathematician who made much of his fame possible, L'Hopital abandoned the project."
Tags: derivative, calculus, Bernoulli, indeterminate, text
From the Paper "The purpose of this research is to examine "The History of Calculus" by Carl B. Boyer. The plan of the research will be to set forth the general ideas in the book, and then to explore details contained in the book that summarize the concepts of calculus that Boyer wants to emphasize.
Boyer's The History of the Calculus is put forward as one of the few histories of how the discipline of calculus evolved, apart from an explanation of how to use it mathematically. It is by positioning the ideas of mathematics and philosophy that influenced calculus that Boyer gradually moves toward an explanation of how calculus can actually be used and applied. The background of Boyer's approach appears to be the idea that a richer understanding of how calculus came to "be" in the world of ideas can lead to an understanding of how it can be employed in ... "
Abstract This paper expounds the ?Theory of Everything,? starting with the pioneering theories of Newton's "Laws of Motion" and Einstein's ?General Theory of Relativity,? developing right through to the cutting-edge "string theory" research currently being conducted around the world today. It shows the importance of fields of study as seemingly diverse as calculus, differential geometry, electromagnetism, particle physics and quantum mechanics to the development of a "Theory of Everything".
From the Paper "However, there is a fundamental discord between Einstein's "Theory of General Relativity" and quantum mechanics. Einstein saw the universe in four dimensions (the three dimensions of space plus time). The gravitational force that binds matter to the earth stems from this space-time continuum. Since quantum mechanic's subatomic particles only exist theoretically, they cannot be located in space-time and their motion can only be hypothesized. Thus, we have two theories that work individually but not together. There are also many unanswered questions. Relativity cannot tell us how the big bang created the universe or what black holes consist of. Similarly, quantum theory is not able to make order or sense of the assortment of miniscule matter it describes."
This paper discusses the life and works of one of the most instrumental contributors to mathematical and scientific theory in the last few centuries, Sir Isaac Newton.
Abstract This paper explains that Sir Isaac Newton advanced a whole new system of mathematics, including systems of physics and calculus, which were revolutionary during his time and continued to be
observed long after his death. The author points out that one of Newton's most significant contributions is his basic laws of motion often call Newton's Laws. The paper relates Isaac Newton would never have described himself as a 'scientist' because the word was not coined until more than a century after his death; he was a reclusive Hebrew scholar and Classicist who wrote more about alchemy and theology than the natural world, and his posthumous reputation is riddled with contradictions.
Table of Contents
Introduction
Life of Mathematician
Significant Contributions
Comparison to Present System
Conclusion
From the Paper "Newton's laws of motion can generally be described through example and demonstration and represent a continuation of thought and inquiry into questions of physics. For example, many scientists before Newton could think of explanations for the continuation of movement of a given object due to force and velocity, but could not demonstrate it in a scientific way in the same way that Newton could with his laws of
motion. Using force and velocity, Newton made important mathematical relations that showed up in real world examples. For example, if someone was riding on a horse, they were going the same speed as the horse, then their velocity remained constant, but if something changed their velocity, it would differ from the velocity of the horse, and they
may change velocity in direction of proportionate force. Newton also contributed the commonly repeated maxim that each and every force (action) has an equal and opposite force (reaction). This is a very significant contribution because it represents a basic precept of modern physics."
Abstract This paper explains that Daniel Bernoulli used his analytical skills across a broad range of scientific disciplines including probability, hydrodynamics, the flow of blood and blood pressure and Riccati's differential equations. The author points out that Daniel Bernoulli improved mathematical physics with his recognition of many of Newton's theories and his utilization of the more powerful calculus of Leibniz. The paper relates that Bernoulli's mathematical explanation of the behavior of gas led to Boyle's law.
Table of Contents
Introduction
Bernoulli's Contributions to Mathematics
Effect of Bernoulli's Work on Today's World
From the Paper "Aerodynamics is a subdivision of fluid mechanics that deals with the motion of air and other gaseous fluids, and with the forces acting on bodies in motion relative to such fluids. Some of the examples of aerodynamic actions are: the movement of an aircraft through the air, the wind forces applied on a structure and the working of a windmill. Daniel Bernoulli's principle is the main law dictating the motion of fluids, which links an increase in flow velocity to a decrease in pressure. For instance, for the same quantity of air at the entry to the venturi tube below to flow through the restriction in the middle, the air must accelerate."
Abstract This paper examines how trust is developed or formed, what results when trust is not formed and finally, what results when trust is violated. The paper looks at two types of trust, known as calculus-based trust and identification-based trust. The paper reviews several studies and concludes that trust, although it is intangible, is crucially important in relationships with any depth and commitment.
Outline:
Abstract
Introduction
Discussion
Review of Previous Studies
Summary and Conclusion
From the Paper "Some individuals trust very easily, and often far too easily which results in others taking advantage of their vulnerability while other individuals rarely trust others and only manage to extend trust after a firm foundation has been laid to base that trust upon. While violation of trust between casual relationships may not result in a complete breakdown of the relationship and it may be able to continue on some level the violation of trust between two individuals who are intimately close often results in a breakdown leaving a chasm far too wide to cross and at best takes much time, effort and willingness for trust to re-established and this may only result if the victim of the violation has the willingness to do so."
Abstract This paper describes how Isaac Newton revolutionized modern science with his laws and theories. The paper maintains that Newton was a maverick in his way of thinking and discusses how revolutionized science with his laws of motion and gravity as well as his invention of calculus. The paper believes that Newton represents modern science as we know it.
Outline:
Newton Revolutionized Science at Cambridge
Newton's Background and Predecessors Empowered Him
Newton's Laws of Motion Rock the Science World
Newton's Influence Extends Beyond Laws of Motion
Newton Represents Modern Science As We Know It.
From the Paper "Isaac Newton is considered one of the most ingenious minds of the twentieth century. He is most remembered for his contributions to the mathematic and scientific arenas, where his work was most influential. Newton was primarily a physicist but he was also a mathematician, an astronomer, a philosopher, and a theologian. His greatest achievement is his laws of motion, a theory that changed physics forever. While looking at Newton's achievements, it is easy to fall into the trap of reading the words on the paper. What we should never forget is how he revolutionized modern science with his laws and theories. He was a real maverick. His studious background paved the way to a greater understanding of nature and her mysteries."
Abstract This essay discusses the life of Sir Issac Newton and the points of his life that brought forth his great advances in the realm of physics and mathematics.
From the Paper "As a child Sir Isaac Newton took little interest in what was being taught to his classmates (Bixby 90). Instead, he found ways to fulfill his desire to learn. He marked where the shadows fell in his yard in order to keep time, thus producing his sundial (Rattansi 12). His interest in rushing water inspired Newton to build a windmill. He created the first horseless carriage. In addition to the pursuit of his numerous boyhood interests, Newton spent time with his landlord as the apothecary and concocted remedies for the illnesses of the locals (Christianson 16)."
