...La Ley de Gauss • Una misma ley física enunciada desde diferentes puntos de vista • Coulomb ⇔ Gauss • Son equivalentes • Pero ambas tienen situaciones para las cuales son superiores que la otra • Aquí hay encerrada una gran verdad fundamental. Es bueno tener varias maneras de mirar una misma realidad. El Concepto General de Flujo – Algo multiplicado por Area Flujo de Fluido Volumen que cruza una superficie en unidad de tiempo. Pero el elemento del tiempo no es fundamental al concepto de flujo mientras que la superficie sí. El concepto general de flujo es algo que cruza una superficie. Matemáticamente es algo multiplicado por área. En este caso v. Flujo Eléctrico Matemáticamente, es lo mismo excepto que tomamos el vector E en vez de v. Generalizamos al caso en que E no es uniforme. Definimos muchas superficies pequeñas ∆A. Flujo Eléctrico • Igual que el flujo de líquido, es el producto de algo por area, en este caso E. • La orientación de la superficie es importante. Por lo tanto, hay que usar el producto interno (cos θ). • Si E no es constante, hay que usar un integral. • Es proporcional al número de lineas de campo que cruzan una superficie. • El concepto de flujo eléctrico es nuevo para nosotros. La manera de entenderlo es a través de la analogía con flujo de fluido. Al final viene siendo esencialmente el número de lineas que cruzan una superficie. Esto puede parecer un concepto raro y lo es pero resulta que juega un papel importante en la ley de Gauss como veremos...
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...Karl Gauss: Biography Karl Gauss lived from 1777 to 1855. He was a German mathematician, physician, and astronomer. He was born in Braunschweig, Germany, on April 30th, 1777. His family was poor and uneducated. His father was a gardener and a merchant's assistant. At a young age, Gauss taught himself how to read and count, and it is said that he spotted a mistake in his father's calculations when he was only three. Throughout the rest of his early schooling, he stood out remarkably from the rest of the students, and his teachers persuaded his father to train him for a profession rather than learn trade. His skills were noticed while he was in high school, and at age 14 he was sent to the Duke of Brunswick to demonstrate. The Duke was so impressed by this boy, that he offered him a grant that lasted from then until the Duke's death in 1806. Karl began to study at the Collegium Carolinum in 1792. He went on to the University of Gottingen, and by 1799 was awarded his doctorate from the University. However, by that time most of his significant mathematical discoveries had been made, and he took up his interest in astronomy in 1801. By about 1807, Gauss began to gain recognition from countries all over the world. He was invited to work in Leningrad, was made a member of the Royal Society in London, and was invited membership to the Russian and French Academies of Sciences. However, he remained in his hometown in Germany until his death in 1855. Acomplishments During...
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...Carl Friedrich Gauss was born on April 30,1977 in Brunswick, Germany. Gauss was a mathematician and scientist who has had a major impact in mathematics during and after his lifetime and was also known as the “prince of mathematics“. At the age of seven, Carl Friedrich Gauss started elementary school and his potential was noticed immediately ,his teachers were amazed when Gauss summed the integers from one to one hundred instantly by spotting that the sum was fifty pairs of numbers each pair summing to one hundred one .The teachers at his school were so shocked that a seven year old boy could achieve this goal and it got him recognized by the Duke of Brunswick in 1792 when he was given a stipend to allow him to pursue his education .He continued his education in 1795 when he went to the University of Gottingen , but he did not earn his diploma there. However he left his mark at the university because he made a discovery of the construction of a regular 17-gon by ruler and compasses and that was a major discovery in the time of Greek mathematics. Gauss went back to Brunswick where he received his degree. The Duke of Brunswick believed in Gauss and wanted him to submit a dissertation to the University of Helmstedt .His dissertation was a discussion of the fundamental theorem of algebra. At the age of twenty four he published Disquisitions Arithmetic in which he formulated systematic and widely influential concepts and methods of number theory dealing with the relationships and...
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...well-known mathematicians such as Adrien-Marie Legendre and Carl Friedrich Gauss. She hid under the male name “M. LeBlanc”. Her decision to identify herself as a man to prove her worth in a male-dominated field of mathematics was worthwhile. Fighting against social statuses she rose to the top becoming an equal collaborator with male mathematicians toward the end of her career. She also used her male name when a paper was assigned. Legrance discovered the paper and was astonished and recognized the abilities of Germain and began to help and encourage her. In 1809 the French Academy of Sciences offered a prize for a mathematical account of the phenomena exhibited in experiments on vibrating plates conducted by the German physicist, Ernst F.F. Chladi. In 1811, she submitted an anonymous memoir, but the prize was not awarded. The competition was reopened twice more, one in 1813 and again in 1816, and she submitted a memoir on each occasion. Her third memoir, with which she finally won the prize, treated vibrations of general curved as well as plane surfaces and was published privately in 1821. During the 1820’s she worked on generalizations of her research, but removed herself from the academic community on account of her gender and largely unaware of new developments taking place in the theory of elasticity but made little progress. Meanwhile, she had actively restored her interest and wrote to Gauss outlining her strategy for a general solution. Her result first appeared in...
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...1. If [pic] + [pic] =[pic], what is the value of X3 + [pic]? a) 2 b) 4 c) 0 d) 6 Answer: c) 0 2. If X= [pic] + [pic] , what is the value of X2 + [pic]? a) 11 b) 10 c) 12 d)13. Answer: 10 3. If P + [pic] = 2, what is the correct value of [pic]? a) 1 b) - [pic] c) 2 d) [pic] Answer: d) [pic]. 4. If A= { 0, 1, 2} and B = { -1, 0, 1}, which one is the connect value of AUB from the following a) {0, 1} b) {0, 1, 2} c) { -1, 0, 1} d) {-1, 0, 1, 2} Answer: d) {-1, 0, 1, 2} 5. What is the correct value of [pic]{(a+b) 2 – (a-b) 2} among the following? a) 4ab b) 2(a2 + b2) c) 2ab d) a2 + b2 Answer: c) 2ab 6. If logX324 = 4, what is the value of X? a) 3[pic] b) 2[pic] c) 3 d) 81 Answer: a) 3[pic] 7. If a≠ 0, which one from the following is the correct value for (a-1)1? a) a-1 b) a c) a-2 d) a2 Answer: b) a. 8. What are the factor of (X+5) (X- 9) -15? a) (X-6) (X+10) b) (X+6) (X-10) c) (X+6) (X+10) d) (X-6) (X-10) Answer: b) (X+6) (X-10) 9. If (a + c) is the one factor of (a + b + c)2 - b2 , what is the another one? a) (a-b + c) b) (a-2b+c) c) (a + b - c) d) (a+ 2b -c) Answer: b) (a- 2b + c) 10. What is the value of (a-1 + b-1) – 1 ? a) [pic] b) [pic] c) a + b d) ab Answer: a) [pic] ...
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...Bernhard Riemann, a famous Christian mathematician, origionally entered University of Göttingen to study theology. He planned to be a pastor. However, his love of mathematics directed him down another path. Regardless of intending to enter into ministry, his love of math reflected throughout his childhood, education, and he became an incredible mathematician. Riemann was born in Breselenz, Germany in 1826. He was one of the local Lutheran pastor’s six children. He was the second eldest, having one brother and four sisters. His mother, Charlotte, died when he was twenty. His father served as his teacher until he turned ten. He had already shown incredible mathematical abilities. However, like many great minds, he struggled with shyness and nerves. Additionally, he feared public speaking. Regardless of his shortcomings, his ability to calculate was exceptional and remained apparent throughout his life. Upon beginning his education, his mathematic brilliance became even more apparent. While living with his grandmother he attended the Lyceum in Hannover. Following her death in 1842, he transferred to Johanneum Gymnasium in Lüneburg. Although his education was largely Bible, math remained his favorite. His teacher recognized his unique ability and frequently gave him advanced reading material. Once, he read the entire 900 page book by Adrien-Marie Legendre on the theory of numbers in six days. He supposedly, even memorized it. He astounded his teachers. Riemann often transcended...
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...Small Wonders By Andrew Marshall Time Asia Small Wonders by Andrew Marshall is an intriguing article that searches for answers to this fascinating question “Are child prodigies born or made?” To begin with, the author strictly defines prodigy as a child who by age 10 displays a mastery of a field. Initially, I had no doubt about this definition; however, after reading some online information, I got to know that although there was growing consensus with the author’s definition, it would be going too far to say that such a consensus exists without controversy. There may, for example, be some wisdom in the view of Radford (1990), who has argued that there are so many problems with specifying at what age, and against what standard, a child would have to perform to be called a prodigy. Personally, I much more agree with Radford’s viewpoint than the strict definition of the author. He regards age as the standard to demiliate a prodigy, while I agree with Radford that each field has its own standards, and it seems better to try to specify what is uniquely characteristic of the child prodigy than restricting a certain age to call someone a prodigy. Prodigies, as the author says, can be found anywhere and in any race or culture. However, there are some factors that prevent some child prodigies from being discovered or developed. Besides poverty, lack of education, and absence of opportunities in the past, most of which have been mentions in the text, there...
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...Describe the work of Gauss, Bolyai and Lobachevsky on non-Euclidean geometry, including mathematical details of some of their results. What impact, if any, did the rise of non-Euclidean geometry have on subsequent developments in mathematics? Word Count: 1912 Euclidean geometry is the everyday “flat” or parabolic geometry which uses the axioms from Euclid’s book The Elements. Non-Euclidean geometry includes both hyperbolic and elliptical geometry [W5] and is a construction of shapes using a curved surface rather than an n-dimensional Euclidean space. The main difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. There has been much investigation into the first five of Euclid’s postulates; mainly into proving the formulation of the fifth one, the parallel postulate, is totally independent of the previous four. The parallel postulate states “that, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” [W1] Many mathematicians have carried out extensive work into proving the parallel postulate and into the development of non-Euclidean geometry and the first to do so were the mathematicians Saccheri and Lambert. Lambert based most of his developments on previous results and conclusions by Saccheri. Saccheri looked at the three possibilities of the sum of the...
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...An Introduction To Virtual Spatial Graph Theory 1. Introduction The mathematical theory of knots studies the many ways a single loop can be tangled up in space. Since many biological molecules, such as DNA, often form loops, knot theory has been applied to biological systems with good effect. However, many biological molecules form far more complicated shapes than simple loops; proteins, for example, often contain extensive crosslinking between cystine residues, and hence from the mathematical viewpoint are far more complicated structures–spatial graphs. The study of graphs embedded in space is known as spatial graph theory, and researchers such as Flapan have obtained good results by applying it to chemical problems. However, in biological systems, proteins are often associated with membranes, meaning that some portions of the molecule are prevented from interacting with others. In the case of a simple loop, the virtual knot theory of Kauffman provides a mathematical framework for studying such systems, as it allows some crossings of strands to be labeled “virtual,” i.e. non-interacting. We hope that a merging of these two theories, called virtual spatial graph theory, will prove equally useful in the biological sciences. Knot theory studies embeddings of circles up to isotopy. There are many ways to extend the ideas of knot theory; two natural choices are the study of spatial graphs and the theory of virtual knots. The theory of spatial graphs generalizes the objects...
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...notions of distance and curvature and, therefore, described new possibilities for the geometry of space itself. Several important theorems and concepts are named after Riemann, e.g. the Riemann-Roch theorem, a key connection among topology, complex analysis and algebraic geometry. He was so prolific and original that some of his work went unnoticed (for example, Weierstrass became famous for showing a nowhere-differentiable continuous function; later it was found that Riemann had casually mentioned one in a lecture years earlier). Like his mathematical peers (Gauss, Archimedes, Newton), Riemann was intensely interested in physics. His theory unifying electricity, magnetism and light was supplanted by Maxwell's theory; however modern physics, beginning with Einstein's relativity, relies on Riemann's notions of the geometry of space. Riemann's teacher was Carl Gauss, who helped steer the young genius towards pure mathematics. Gauss selected "On the hypotheses that Lie at the Foundations of Geometry" as Riemann's first lecture; with this famous...
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...Iterative Methods for Solving Sets of Equations 2.1 The Gauss-Seidel Method The Gauss-Seidel method may be used to solve a set of linear or nonlinear algebraic equations. We will illustrate the method by solving a heat transfer problem. For steady state, no heat generation, and constant k, the heat conduction equation is simplified to Laplace equation (2T = 0 For 2-dimensional heat transfer in Cartesian coordinate [pic] + [pic] = 0 The above equation can be put in the finite difference form. We divide the medium of interest into a number of small regions and apply the heat equation to these regions. Each sub-region is assigned a reference point called a node or a nodal point. The average temperature of a nodal point is then calculated by solving the resulting equations from the energy balance. Accurate solutions can be obtained by choosing a fine mesh with a large number of nodes. We will discuss an example from Incropera’s1 text to illustrate the method. Example 2.1-1 A long column with thermal conductivity k = 1 W/m(oK is maintained at 500oK on three surfaces while the remaining surface is exposed to a convective environment with h = 10 W/m2(oK and fluid temperature T(. The cross sectional area of the column is 1 m by 1 m. Using a grid spacing (x = (y = 0.25 m, determine the steady-state temperature distribution in the column and the heat flow to the fluid per unit length of the column. Solution The cross sectional area of the column is divided...
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...Output For “C” code of BISECTION METHOD This Program will find the Root of the Equation f(x) = x^3 - 4x -9 ************ By Bisection Method ************ Enter the range in which root lies................. Enter the value for the lower limit - 2 Enter the value for the upper limit - 3 Enter the limit of error allowed - 0.00001 The value of function at 2.000000 is = -9.000000 The value of function at 3.000000 is = 6.000000 Iterations Start now............................ The value of function at 2.500000 is = -3.375000 The value of function at 2.750000 is = 0.796875 The value of function at 2.625000 is = -1.412109 The value of function at 2.687500 is = -0.339111 The value of function at 2.718750 is = 0.220917 The value of function at 2.703125 is = -0.061077 The value of function at 2.710938 is = 0.079423 The value of function at 2.707031 is = 0.009049 The value of function at 2.705078 is = -0.026045 The value of function at 2.706055 is = -0.008506 The value of function at 2.706543 is = 0.000270 The value of function at 2.706299 is = -0.004118 The value of function at 2.706421 is = -0.001924 The value of function at 2.706482 is = -0.000827 The value of function at 2.706512 is = -0.000279 The Approximate root of the above function is = 2.706528 and the value of function is = -0.000004 The number of Iterations are = 16 Output For “C” code of NEWTON – RAPHSON METHOD This Program will find the Root of the Equation f(x) =...
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...Professional Values and Ethics Integrity is defined as personal ethics and values of the individual involved in the decision-making process. This can create different paths to success, depending on the choices made toward each individual goal. Gandhi explains that looking into a person's thought process presents who he or she is in the present and future. He is the perfect example on ethics and values because of the way he set his goals and how he implemented to obtain his goals. Using peaceful and non-violent resistance to battle the British policies influenced other to follow his ideas, and this drove the British out of India. Later people were able to use this idea to help persuade the United States government to change the civil rights policy. Martin Luther King Junior used the same type of ideas that worked for Gandhi to help gain support to change certain laws during the 1960's (Prabhu, 2001). Ethics and values are greatly influenced by what goals and how individuals develop over time into a success. Values are what individuals cherish and work for to meet their goals in life. Values are also defined as relative worth, merit, and of most importance to that person. Values could also be described as moral principle and beliefs of a person. Ethics is derived from the Greek word ethikos, which means customs or character of a person. Ethics involves defending, developing, and recommending, the differences between right or wrong. Ethics can be addressed by asking...
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...between Y and X/Xs will be linear. PRF: Yi 0 1 X i SRF: Yi 0 1 X i Where: Yi is the estimator of actual Yi 0 is the estimator of actual 0 or the constant(intercept) 1 is the estimator of actual 1 or slope estimate i is the estimator of actual residual term i or the stochastic disturbance term The error term i captures all relevant variables not included in the model because they are not observed in the data set Classical Linear Regression Model (CLRM) It is classical in a sense that it was first developed by Gauss in 1821 and since then has served as norm or standard against which regression models that do not satisfy the Gaussian Assumptions (listed below) may be compared with. The Gaussian, standard, or classical linear regression model (CLRM), which is the cornerstone of most econometric theory. Ordinary Least Squares (OLS) Method Attributed to Carl Friedrich Gauss The method requires that we should choose 1 and 2 as estimates of 1 and 2 , respectively such that: n Q (Yi 1 2 X i ) 2 is a minimum i 1 where Q is also the SUM OF SQUARES of the (within-sample) prediction when we predict given Xi and the estimated regression...
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...Gaussian Beams Enrique J. Galvez Department of Physics and Astronomy Colgate University Copyright 2009 ii Contents 1 Fundamental Gaussian Beams 1.1 Spherical Wavefront in the Paraxial region 1.2 Formal Solution of the Wave Equation . . 1.2.1 Beam Spot w(z) . . . . . . . . . . 1.2.2 Beam Amplitude . . . . . . . . . . 1.2.3 Wavefront . . . . . . . . . . . . . . 1.2.4 Gouy Phase . . . . . . . . . . . . . 1.3 Focusing a Gaussian Beam . . . . . . . . . 1.4 Problems . . . . . . . . . . . . . . . . . . . 1 1 3 6 8 8 9 10 12 15 15 17 20 21 25 25 26 26 27 29 30 31 31 33 35 35 36 39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 High-Order Gaussian Beams 2.1 High-Order Gaussian Beams in Rectangular Coordinates 2.2 High-Order Gaussian Beams in Cylindrical Coordinates . 2.3 Irradiance and Power . . . . . . . . . . . . . . . . . . . . 2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Wave-front interference 3.1 General Formalism . . . . . . . . . . . . . . . . 3.2 Interference of Zero-order...
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