Kirchhoff's Lawv

What are Kirchhoff’s Laws?

-Kirchhoff’s laws govern the conservation of charge and energy in

electrical circuits.

Kirchhoff’s Laws

1.

The junction rule

2.

The closed loop rule

The junction rule

• At any node (junction) in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node, or: The algebraic sum of currents in a network of conductors meeting at a point is zero”.

• The sum of currents entering the junction are thus equal to the sum of currents leaving. This implies that the current is conserved (no loss of current).

 The close loop rule

• The principles of conservation of energy imply that the directed sum of the electrical potential differences (voltage) around any closed circuit is zero.

  

Sources: http://www.iit.edu/arc/workshops/pdfs/Kirchhoff_s_Circuit_Laws.pdf

Polynomial Curve Fitting

A procedure in which the basic problem is to pass a curve through a set of points, representing experimental data, in such a way that the curve shows as well as possible the relationship between the two quantities plotted. It is always possible to pass some smooth curve through all the points plotted, but since there is assumed to be some experimental error present, such a procedure would ordinarily not be desirable.

Algebraic fit versus geometric fit for curves

For algebraic analysis of data, "fitting" usually means trying to find the curve that minimizes the vertical (y-axis) displacement of a point from the curve (e.g., ordinary least squares). However for graphical and image applications geometric fitting seeks to provide the best visual fit; which usually means trying to minimize the orthogonal distance to the curve (e.g., total least squares), or to otherwise include both axes of displacement of a point from the curve. Geometric fits are not popular because they usually require non-linear and/or iterative calculations, although they have the advantage of a more aesthetic and geometrically accurate result.

Suppose a collection of date is represented by n points in the xy-plane

 

(X1,Y1), ( X2, Y2 ) ,..., (Xn, Yn)  

 

and you are asked to find a polynomial function of degree n-1 

 

 

whose graph passes through the specified points. This procedure is called polynomial curve fitting. If all x- coordinates of the points are distinct,  then there is precisely one polynomial function of degree n-1 (or less) that fits the n points. 

 

To solve for the n coefficients of p(x) , substitute each pf the n points into the polynomial function and obtain n linear equations in n variables a0, a1, 12 ,... an-1

 

 

Sources : http://www.answers.com/topic/curve-fitting#ixzz379zcn2pV

Gauss Jordan Elimination

With Gauss Jordan Elimination , you apply the Elementary row operations to a matrix to obtain ( row - equivalent ) row-echelon form. A second method of elimination, called Gauss-Jordan elimination after Carl Gauss and Wilhelm Jordan ( 1842 - 1899), continues the reduction until a reduced row-echelon  form is obtained.
 ex:


Wilhelm Jordan

Date of birth	1 March 1842
Country of nationality	Germany
Profession	Mathematician
Date of death	17 April 1899

Wilhelm Jordan was a German geodesist who did surveys in Germany and Africa and founded the German geodesy journal.Jordan was born in Ellwangen, a small town in southern Germany. He studied at the polytechnic institute in Stuttgart and after working for two years as an engineering assistant on the preliminary stages of railway construction he returned there as an assistant in geodesy. In 1868, when he was 26 years old, he was appointed a full professor at Karlsruhe. In 1874 Jordan took part in the expedition of Friedrich Gerhard Rohlfs to Libya. From 1881 until his death he was professor of geodesy and practical geometry at the Technical University Hanover. He was a prolific writer and his best known work was his Handbuch der VermessungskundeHe is remembered among mathematicians for the Gauss–Jordan elimination algorithm, with Jordan improving the stability of the algorithm so it could be applied to minimizing the squared error in the sum of a series of surveying observations. This algebraic technique appeared in the third edition of his Textbook of Geodesy.Wilhelm Jordan is not to be confused with the mathematician Camille Jordan, nor with the German physicist Pascual Jordan.





Reference : http://www.yatedo.com/p/Wilhelm+Jordan/famous/870d538051e997c0ff6d724d551aa2ab
http://pages.pacificcoast.net/~cazelais/251/gauss-jordan.pdf

GAUSSIAN ELIMINATION

Carl Friedrich Gauss

Carl Friedrich Gauss is sometimes referred to as the "Prince of Mathematicians" and the "greatest mathematician since antiquity". He has had a remarkable influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians.
Gauss was a child prodigy. There are many anecdotes concerning his precocity as a child, and he made his first ground-breaking mathematical discoveries while still a teenager.
At just three years old, he corrected an error in his father payroll calculations, and he was looking after his father’s accounts on a regular basis by the age of 5. At the age of 7, he is reported to have amazed his teachers by summing the integers from 1 to 100 almost instantly (having quickly spotted that the sum was actually 50 pairs of numbers, with each pair summing to 101, total 5,050). By the age of 12, he was already attending gymnasium and criticizing Euclid’s geometry.
Although his family was poor and working class, Gauss' intellectual abilities attracted the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum at 15, and then to the prestigious University of Göttingen (which he attended from 1795 to 1798). It was as a teenager attending university that Gauss discovered (or independently rediscovered) several important theorems.

Graphs of the density of prime numbers

At 15, Gauss was the first to find any kind of a pattern in the occurrence of prime numbers, a problem which had exercised the minds of the best mathematicians since ancient times. Although the occurrence of prime numbers appeared to be almost competely random, Gauss approached the problem from a different angle by graphing the incidence of primes as the numbers increased. He noticed a rough pattern or trend: as the numbers increased by 10, the probability of prime numbers occurring reduced by a factor of about 2 (e.g. there is a 1 in 4 chance of getting a prime in the number from 1 to 100, a 1 in 6 chance of a prime in the numbers from 1 to 1,000, a 1 in 8 chance from 1 to 10,000, 1 in 10 from 1 to 100,000, etc). However, he was quite aware that his method merely yielded an approximation and, as he could not definitively prove his findings, and kept them secret until much later in life.


Gaussian Elimination
                     It was introduce as a procedure for solving a system of linear equations.
Matrix 
        - is a rectangular array of numbers a, symbol or expression arranged  in rows and columns

Entry
        - Is the individual items in the matrix 
                    aij
Row Subscript
         -  The index i above is called row subscript because it identifies the row in which the entry lies

Column Subscript
         - The index j is called the Column Subscript because it identifies the column in which the entry lies

Square of Order n
        - A matrix with m rows and n columns (an m x n ) is said to be the size. If m=n then the matrix is called square of order n


Main Diagonal Entries
              a11, a12, a33...
Augmented Matrix
     -the matrix derived from the coefficient and constant terms of a system of linear equation
Coefficient Matrix
     - The matrix containing only the coefficients of the system
Elementary Row Operations
1. Interchange two equations
2. Multiply an equation by a non-zero constant
3.Add a multiple of an equation to another equation
 Examples:





















references : http://www.storyofmathematics.com/19th_gauss.html
                   http://www.math.dartmouth.edu/archive/m23s06/public_html/handouts/row_reduction_examples.pdf