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matematik 1c Uncategorized

Euclidean Geometry

Venus Jupiter conjunction
A photograph of the Venus – Jupiter conjunction 12/3 2012

I have started to read Euclid’s ‘Elements’. This is perhaps the most influential book in mathematics and was used for more than two-thousand years in schools all over the western world. My edition is written by Christian Fredrik Lindman lecturer of mathematics at the upper secondary school in Strängnäs, Sweden and it was printed in 1867.

In the first chapter many central terms and mathematical entities are defined.

Mathematics is the science dealing with quantities.

A quantity is an entity that can be increased or decreased by the addition or subtraction of more of the same quantity.

Geometry is the science dealing with quantities that have an extension in space.

Definition is a list of characteristics typical of that object

Postulat is a theorem that cannot be proved.

An axiom is an obvious statement.

Problem is a theorem  where you must show how the problem shall be solved and then prove that it is solved.

Theorem  Is a statement  of a mathematical truth that must be proved.

Corollary is a theorem that is a direct consequence of another theorem,

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Uncategorized

More vectorcalculus: Gauss theorem and Stokes theorem

Rosa ‘Flora Danica’

Since the divergence of a vectorfield provides us with the number of field lines radiating outward from the source of the vectorfield it can be intuitively understood that the volume integral of the divergenbde of F equals the surface integral of F over the closed surface A:

∫ ∇⋅F dv =  ∫F⋅dS 

Another important theorem in vaectoranalysis is Stokes Theorem
×F ds = ∫F⋅dl

Kategorier
Gymnasiematematik(high school math) matematik 1c

Number theory-more on numbers

De första talen människan använde sig av var förmodligen positiva heltalen de sk. naturliga talen. De användes för att ange kvantiteter av olika ting: fem tomater, 10 persikor etc..

Då människan började med handel kunde man bli skuld satt och då var de negativa talen användbara för att beskriva detta.

Hur skulle man fördela jaktbyten och dylikt? Då behövdes bråktal sk rationella tal.

De vetenskapliga framstegen och de indisk-arabiska siffrorna krävde större noggrannhet varvid decimaltalen infördes. Alla tal på tallinjen kallas de reella talen.

Alla cirklar är likformiga med förhållandet pi mellan omkrets och diameter. Detta tal är inte möjligt att uttrycka vare sig som decimaltal eller bråktal. pi är ett irrationellt tal.

talmängder

 

The prime-numbers can be regarded as the building blocks of the number system.

Prime numbers are numbers that can only be divided by one and itself.

Every number cn be factorized into primenumbers.

Christian Goldbach proposed that every even number can be formed as the sum of two prime numbers. Goldbach lived in Königsberg during the 17th century.

Gauss himself devoted great attention to priimenumbers and proposed the folloeing theorem: The number of primes smaller than N equals N/ln(N).

This was proven byu Charles Hermite in the 19th century.

Prime numbers smaller than a cetain number can be generated with a method from antiquity: the sieve of Erathostenes. another possibility is to generate Mersenne primenumbers 2n – 1.

Exercise 1 in Sjunnesson Ma 1c:

Är 2 * 3* 5* 7 + 3


Ett primtal?

Lösning: bryter man ut tre fås talet 3(2*5*7 + 1). Detta tal är delbart med tre eftersom tre är en faktor i talet. 3(2*5*7+1)/3 = 2*5*7 + 1. alltså är talet inget primtal då det ju är delbart med tre.

Kategorier
matematik 1c Uncategorized

The decimalsystem and some terminology

At this point it  might be wise to take a closer look at the decimalsystem which is the way we use to represent quatities in mathematics.

The decimalsystem is a positionsystem (based on powers of ten)  which means that the value of a  number is determined by its position in the number.

e.g. 333 = 3• 100 + 3• 10 + 3•1.

Thanks to this ingenous system it is possible to express all rational numbers   with just 10 digits (indo-arabian) 0 – 9. Equalling the number of fingers.

The natural numbers i.e. the positive integers are infinitely many.

This statement an be proved by adding 1 to any given candidate to being the biggest integer.

A peculiar fact is that the number of real-numbers (rational numbers + irrationalnumbers i.e. all the numbers between the real numbers) is higher than  the number of integers. But how can anyyhing be bigger than infinity?

Georg Cantor solved this problem by dividing infinity into different categories (Cardinalities): the natural numbers are countably infinite ℵ(Cardinal number)=0  wheras the real numbers are uncountably infinite ℵ=1.

The transcendental numbers however such as π and e cannot be be explicitly written with these integers and therefore one must use special signs for them.

A mathematical function is a rule that tells you how to caculate a value from a given variable. This value must be unambiguous: only one function value must correspond to a given variable value.

A continuous function is a function that can be drawn in a coordinate system without lifting the pencil from the paper.

Kategorier
Calculus Uncategorized Vectors

Divergence and curl of vectorfields

PEar tree 'Gris Bonne'
Pyrus Communis (pear tree) ‘Gris Bonne’

According to the Helmholtz-theorem a vectorfield is completely defined by the divergence and  curl of the vectorfield.

the divergence is a measure of the strength of the source of the vectorfield whereas the degree of rotation of the field is given by the curl.

The divergence is defined as  ·F  = lim Δv→0 ∫A ds/Δv i.e. the scalarproduct(dotproduct) of the nabla operator and the vector.

The ∇-operator is defined as the vector differential operator
∇=∂/∂x + ∂/∂y + ∂/∂z.

When this operates on a scalar V one obtains the gradient  V of that scalar i.e. a vector that represents both the magnitude and the direction of the maximum space rate of increase of  of that scalar.

The curl is defined by

∇xF. = (dFz/dy – dFy/dz) i + (dFx/dz – dFz/dx)j + (dFy/dx – dFx/dy) k

The electromagnetic field is defined by the divergence and curl of the Electric field vector E and the magnetic field vector B:
∇· E= ρ

∇xE=∂B/∂t

∇· B=0; This can be interpretated as stating the fact that there are no magnetic charges.
∇xB=∂D/∂t

These are the famous Maxwellian equations which gives a full description of the electromagnetic theory.
Every electromagnetic law can be deduced from them.

Kategorier
Calculus Uncategorized

Vectorproducts

Vectors can be multiplied in two ways:

1. The scalar product gives product of a vector and the projection of  the other vector upon the first one. This is calculated according to

a b = ab cos(v)

The result is a scalar.  This statement can be proved with the following calculation:

Let C= A+B and form

C C = (A+B) (A+B) =

A2 + B2+2AB

Solving for AB =(C2-A2-B2/2

which is a scalar quantity since it is made up of absolute values.

An example is the amount of work, W, done by operating against a force F a distance x.

W=F ×cos(v)

where v is the angle between F and  displacement x.

2. As the vectorproduct

a x b = ab sin(v).

This gives the area of the parallellogram formed by vectors a and b. It can be shown (it follows directly by computing the vectorproduct of  (a1,a2,a3) and  (b1,b2,b3) and Sarrus rule) that the vectorproduct is another vector forming an orthogonal coordinatesystem with a and b.

Ex The Lorentzian force in physics is given by

F =q vxB. 

Here F equals the force on the particle with charge q moving with speed v through the magnetic field B.