Jeff Loftin

Homework 8

1. Gauss BioPart I: Johann Carl Friedrich Gauss1. He was born April 30th, 1777 in Brunswick, Germany.2. He was born to a lower, working class family.3. His mother was illiterate and had not even taken the time to record the date of his birth (being illiterate certainly would not have helped!).4. He was a child genius, able to perform amazing feats of mental capacity at young ages. It is rumored that he was able to spot an arithmetic error in his fathers payroll information at just 3 years of age, quite astounding!5. At age seven he started elementary school, and was noticed for his ability almost instantly. 6. By age 11 he was moved to the Gymnasium (an upper school of sorts) for his education. It was there that he learned both Latin and High German.7. The Duke of Brunswick-Wolfenbuttel gave Gauss a stipend, allowing him to enter the Brunswick Collegium Carolinum (BCC) just a few years later. 8. It was at the BCC that Gauss independently discovered and studied Bodes law, the prime number theorem, the binomial theorem, and the law of quadratic reciprocity. Numbers continued to fascinate him for the entirety of his life.9. In 1795 Gauss left to study at Gottingen University. It was here that he met Farkas Bolyai (in 1799) and they became friends, continuing to write to each other for many years.10. In 1799 in Brunswick Gauss received his degree. At this time, the Duke also decided to continue Gauss stipend and requested that he submit a dissertation to the University of Helmstedt. 11. It is important to note that while at University Gauss discovered many important theorems, which will be mentioned later. His work at such a young age was quite impressive and contributed to many fields outside of pure mathematics.12. Because of the stipend he was receiving, Gauss dedicated himself to study and research. He published his work Disquisitiones Arithmeticae in 1801, mostly concerned with number theory.13. In that same year, he was able to predict with very close approximation the location of a new small planet that his acquaintance Zach had been attempting to track. 14. In 1805 Gauss married Johanna Ostoff. 15. Unfortunately his benefactor, the Duke of Brunswick, was killed shortly thereafter, causing Gauss to move to Gottengin (taking over the director position).16. Within a roughly one-year period his father, wife, and second son died. 17. About a year later, Gauss remarried to Minna, his wifes best friend and had three children with her.18. Though he experienced much tragedy, he was able to focus that frustration into his work. He published works on the motion of celestial bodies, but also on series work and the hypergeometric function.19. By 1818 Gauss was asked to carry out a geodesic survey of Hanover, this pleased him and he took over the project quickly.20. Between the years 1820 and 1830 he published over 70 papers.21. In 1822 Gauss won the Copenhagen University Prize with his paper Theoria Attractionis (not published until 1825).22. Though not able to completely prove it, he was interested in non-Euclidian geometry and devoted much of his free time to this subject. His contributions helped lead to others proving the existence. 23. In 1831 Weber because the physics professor at Gottingen. He and Gauss performed well together in their six years together and did enormous amounts of work. 24. In 1837, because of a political dispute, Gauss left the university and his publication activity declined because of this.25. Staying active, and because of his tremendous mental capacity, he began working with financial matters and made himself a small fortune in investments. 26. Gauss died in his sleep on February 23, 1855.

Part II:1. Disquisitiones Arithmeticae 1801This was (and could be argued still is) the most important book on number theory ever written. In this work Gauss brought together the works of several other mathematicians including Fermat, Euler, Lagrange, and Legendre while adding much of his own knowledge. He covered the subjects of elementary number theory and algebraic number theory in this work. Essentially this book consolidated most of the information of the day on number theory into one collection. This also added numerous proofs which helped fill in gaps and compose a framework for the subject to grow from.

2. Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium 1809This work presented a method for calculating the orbit of asteroids and other celestial bodies when they leave sight. This is the method that he used to find the planetoid (asteroid) Ceres to help his friend and colleague track its orbit as it went behind the sun. Gauss used conic sections to determine the path of the celestial bodies.

3. Gauss Law for MagnetismThis is also called the absence of free magnetic poles, and can be written in multiple ways. Because of the divergence theorem, the integral and differential form can be written and work just the same. This is important to the works of physics and applied mathematics because it works with electric flux and closed surfaces, and their interactions.

4. Normal (Gaussian) DistributionThis is a commonly occurring continuous probability distribution in probability theory. Basically it is a function that tells the probability that an observation will lie between two real numbers or specific real limits. This also is important because of its interaction with the central limit theorem, which we will not get into here. This is often also called the bell curve, but is not the only bell shaped curve know.

5. Differential GeometryGauss studied curves and surfaces in three dimensions according to Euclidian spaces. He helped to lead to the discovery of non-Euclidian geometry. Essentially this was because the way that we measure the distance on a surface determines the curvature of the object itself. His study also helped to lead to many discoveries about shortest distances and also helped prove the parallel postulate was independent of Euclids other four axioms!

Part III: Referenceshttp://www-history.mcs.st-and.ac.uk/Biographies/Gauss.htmlhttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://www.thefamouspeople.com/profiles/carl-f-gauss-442.phphttp://scienceworld.wolfram.com/biography/Gauss.htmlhttp://www.math.wichita.edu/history/men/gauss.html

2. Marie-Sophie Germain BiographySophie Germain was born April 1, 1776 in France Rue Saint-Denis, Paris, France. Born to a wealthy family, she faced tremendous resistance to her learning from her family. This was because, at the time, it was frowned upon for women to learn in just about any capacity. She would escape to her fathers library to read and delve into mathematics. It is said that when she was 13, at the time of the French revolution, she thoroughly engulfed herself in the works of mathematicians like Archimedes. Without a tutor, or any formal education whatsoever, she taught herself differential calculus a rather impressive feat. Because she was a woman, she was unable to attend the Ecole Polytechnique in Paris, but through friends and the use of the pseudonym M. Le Blanc, she managed to obtain lecture notes from many courses. It was through this association that she came to meet Lagrange, who became a mentor and support system of sorts for her for many years. After some time, her work shifted from number theory and pure mathematics to more applied mathematics. Frances emperor, Napoleon issued a contest to see who could best explain the phenomenon of harmonic motion in two dimensions. Germains entry was the only one, for political reasons, and was rejected because of errors. After two more attempts, her third submission was accepted and she won the prize in 1816. This did gain her some attention, but because she was a woman, it did not change the overall feeling toward her work and studies. Her last submission also won her the French Academys grand prize because of its application to vibrating elastic surfaces. Her theory helped explain and helped to predict the reaction and unusual patterns formed by sand and powder on elastic surfaces as they are vibrated. Because of studies like hers and others closely related, the construction of the Eiffel Tower became possible. By age 55, in 1831, Germain had been struggling with breast cancer and eventually succumbed to it, dying in her home. Sadly, this was just before she was due to receive her honorary degree from Gottengin (at Gauss behest and insistence that she be granted an honorary degree for her work and contributions to mathematics).

References:http://www.math.wichita.edu/history/women/germain.htmlhttp://en.wikipedia.org/wiki/Sophie_Germainhttps://www.sdsc.edu/ScienceWomen/germain.htmlhttp://www.britannica.com/EBchecked/topic/230626/Sophie-Germain

3. What is the fundamental theorem of algebra?The fundamental theorem of algebra states that a polynomial of degree n has n roots, though there may be complex numbers used. It states that every non-constant single variable polynomial with complex coefficients has at least one complex root. This is important because it also includes polynomials that do not appear to have complex coefficients because every real number is a complex number with a zero imaginary part. The two statements above can be proven using the use of successive polynomial division.

4. What is the definition of a hypergeometric series?A hypergeometric series is a generalization of the factorial function. They are solutions to a large class of differential equations. It states that:is hypergeometric if is a rational function. Logarithmic, exponential, Laguerre polynomials, Hermite polynomials, and trigonometric functions are hypergeometric. These series show up in many differential equations that are quite important and can be applied to physics, combinatorics, and number theory.

5. Show that We will let

define

also we will be using the identity

and we then let.Using this we will apply Eulers formula for each to see that

And because

We can see that we now have

And using what we know from above

And we note that

Therefore,

We also can see that

Using this, we can see that

Show that

Show that

Using

Thus

Thus we have: