Differential geometry

M.C Escher was more than just an artist. He created art using patterns of identical shapes, that fit together with no gaps, and did not overlap, also known as tessellations. He went through many stages throughout his art, but his most lasting legacy are the tessellations he made and the mathematical impact they had. By using mathematical reasoning he was able to create the tessellations, but he had to look beyond just mathematics to dig deep into his tessellated optical illusions. M.C Escher…

Researching two math placement tests for this project was the easiest part. A math placement test is a test that students take before entering a math course in college. The placement test that I took was for Monmouth University in Monmouth, New Jersey. Monmouth is on my list of schools that I am applying to for the fall of 2018 along with James Madison, which is the other test I chose to research for this project. I took the Monmouth test not thinking I would be able to complete the assigned…

Throughout the centuries, many people have tried to find shortcuts to make their lives simple. Usually described as lazy, mathematicians came along to see the patterns that no one else was able to see. These mathematicians helped reform their societies, and bring about equations that will help others calculate what they need faster. Upon these mathematicians, Pythagoras was most likely one of the most influential mathematicians that helped shape the way that people see shapes today. Through his…

Fundamental Theorem of Calculus The Fundamental Theorem of Calculus evaluate an antiderivative at the upper and lower limits of integration and take the difference. This theorem is separated into two parts. The first part is called the first fundamental theorem of calculus and states that one of the antiderivatives of some function may be obtained as the integral of the function with a variable bound of integration. The second part of the theorem, called the second fundamental theorem of…

The fundamental theorem of Calculus: The fundamental theorem of calculus asserts the interrelated properties of integration and differentiation. It says that a function when differentiated, can be brought back by integrating (anti-derivative) or a function when integrated, can be brought back by differentiation. First theorem: Let f be a function that is integrable on [a,x] for each x in [a,b], then let c be such that a≤c≤b and define a new function A as follows, A(x)=∫_c^x▒f(x)dt, if a≤x≤b.…

Before we talk about measurement scales and why they matter we should first explain what they are. A compound measure allows us to measure different aspects of a topic people would find complex. This topic could range from stem cell research to abortion to gun control. The good thing about compound measures is they have a general reliability to them. What I mean is, if an error is made it often does not make the whole measurement go wrong. Compound measures should have several different…

1. What are the least, and most, amount of distinct zeroes of a 7th degree polynomial, given that at least one root is a complex number? Answer: If the equation is 7th degree then it has 7 roots. Those roots can be complex or real. Complex roots always come in pairs, so if it has one, then it has 2, the other one being the conjugate of the first one. This in other words, if one complex root is a + bi, then the other complex root is a – bi. If at least one root were complex, then we would have a…

Semra Özal CONNECTIONS OF LOGARITHMIC FUNCTIONS Logarithm and exponential functions have close relationship and they are inverse function of each other in a way. Before explicitly clarifying this inverse relationship, we should analyze their definitions. Logarithm means, in mathematics, “The exponent that indicates the power to which a base number is raised to produce a given number “2 Exponential function means that “mathematical function in which an…

5. Hopf-Andronov-Poincare bifurcation In this section, we shall show that the system (2) undergoes a Hopf-Andronov-Poincare bifurcation by using as a bifurcation real parameter. Without loss of generality, suppose that is a function of and . Then system (2) becomes (29) with . System (29) can be written as (30) where , and is the bifurcation real parameter. The function is a on an open set in . Let be the set of equilibria of system…

We, the students of Professor Bart Goddard’s Differential Equations class (M 427J), are writing to you about a recent incident for which he is at risk of losing his position in the university. During the 12:00 PM class on Wednesday, February 22, 2017, Professor Goddard told a cautionary tale about how, in the Holocaust, Nazi engineers used lime to prevent blood from ruining the train tracks of the trains carrying the Jews to the camps, calling on the engineering students in the audience to…