Kategorier
Algebra Calculus Gymnasiematematik(high school math) Mathematical physics Uncategorized Vectors

Vektoranalys (Vectoranalysis)

Derivering av vektorer kan ske på två sätt. Antingen som skalärprodukt eller vektorprodukt. Skalärprodukten ( En.dot – product) ger en skalär som resultat. Ett exempel är beräkning av arbete som skalärprodukten av kraften och förflyttningen i kraftens riktning: W = Fs. Den kan beräknas som Bqvcos(α).

Eller i koordinatform:

Fx∙x + Fy∙y

Lorentzkraften som anger hur stor kraft en laddning, Q, som rör sig med hastigheten v i ett magnetfält B påverkas av är ett exempel på en vektorprodukt (En. cross product).
F = qvxB.
Kraften, F, är en vektor som är vinkelrät mot v och B. Dess absolutbelopp kan beräknas som Bqvsin(α). Där &alpha: är vinkeln mellan v och B-vektorerna.

den första kallas divergenten och den sistnämnda rotation,

Enligt Helmholtz sats kan en vektor u delas upp i en irrotationell och en solenoidal del. Ett irrotationellt, eller konservativt vektorfält, har en potentialfunktion. Exempel på konservativa fält är gravitationsfältet och det elektriska fältet. Ett solenoidalt fält saknar plus- och minuspoler dvs laddningar. Exempel på sådana fält är det magnetiska fältet. För den irrationella delen är rotationen av vektorfältet ∇x u = 0 medan divergensen är noll för den solenoidala delen ∇∙u=0.

Här betecknar ∇ summan av den partiella derivatan i x-led, y-led och z-led.
∇= ∂/∂x + ∂/∂y + ∂/∂z vilket tillämpat på en skalär ger gradienten.

 

Divergens (vektoranalys) – Wikipedia

Rotationen anger vridstyrkan i det magnetiska fältet medan divergensen anger källstyrkan.

I koordinatform fås:

En av Maxwells ekvationer är för övrigt just att divergensen av det magnetiska fältet är noll vilket innebär att det inte finns några magnetiska laddningar alltså isolerade nord- och Sydpoler (En. there are no magnetic poles) :

∇∙B = 0.

Deriverar men volymen får man en yta detta använda vid Gauss sats där volymsintegralen av divergensen blir ytintegralen av vektorn. ∰∇∙u dxdydz= ∯udS

Enligt Stokes sats blir ytintegralen av rotationen av en vektor lika med linjeintegralen av vektorn. ∯∇xu dS = ∲u dl.

Kategorier
Algebra matematik 1c Uncategorized

Gauss påskformel

Glad påsk alla trogna läsare! 
Påskdagen infaller, enligt ett beslut på kyrkomötet i Nicea år  325 AD den första söndagen efter första fullmåne efter 

Vårdagjämningen. 

Carl friedrich Gauss utvecklade en formel för att beräkna när denna dag infaller ett godtyckligt år. 

Gauss påskformel

1. Dela årtalet med 19, 4, och 7 och kalla resterna a, b, c. 

2. (19a+24)/30. Kalla resten d. 

3. Utför divisionen (2b+4c+6d+5)/7 och beteckna resten med e. 

4. Då infaller påskdagen den (22+d+e) mars eller om d+e >9 den (d+e-9) april. 

påsk

Kategorier
Algebra

Partialbråksuppdelning.

Partialbråksuppdelning går ut på att man skriver om ett rationellt uttryck som summan av ratioenlla uttryck av lägre gradtal

Partialbråksuppdelning ger ansatsen

\begin{displaymath}<br /><br /> \frac{1+4x+4x^{2}}{(x+5)^{2}(x+2)(1-x)}=\frac{A}{(x+5)^{2}}<br /><br /> +\frac{B}{x+5} + \frac{C}{x+2}+\frac{D}{1-x}<br /><br /> \end{displaymath}

Handpåläggning ger $A=4\cdot4^{-1}(-1)=6,$ $C=2\cdot 2^{-1}3^{-1}=5$ och $D=2\cdot 3^{-1}=3$. Sätter vi nu $x=0$ får vi $1\cdot<br /><br /> 4^{-1}2^{-1}=6\cdot 4^{-1}+B5^{-1}+5\cdot 2^{-1}+3$ eller $1=5+3B+6+3,$ dvs $B=5$.

Detta är användbart bl.a. vid integrationer där integranden är en rationell funktion som man inte kan finna någon primitiv funktion  till.

Kategorier
Algebra matematik 5

Matrices

A matrix is a square or retangular array of numbers or function of numbers that obeys certain laws. The numbers are distinguished by two subscripts ij. The first indicating the row and the second indicating the column (vertical). in which the number appears. a13 is ther number in the first row snd third column.
It is worth mentioning that it is not a number as the determinant but an array of numbers.

Aij =Bij if and only if all the corresponding elements are equal in the two matrices.

Addition is performed by adding the respective numbers in the corresponding places in each matrix wuth each other.
A+B = B´+A so matrix addition it is commutative.

Division is performed by multipying yjr elements in row i by the elemetns in column j.
Therfore matrixmultiplication is anti-commutatative and AB not equal to BA.
A diagonal matrix is a matrix with zeroes in every position except in the diagonal. The trace of such a matrix is the sum of the diagonal elements.

A diagonal matrix with ones in all position of the diagonal is called an identity matrix. 

The identity matrix is defined by the operation AI = A.
The inverse of a matrix is defined by the relation A A-1 =I.

A tensor can be thought of as a three dimensional matrix.

Kategorier
Algebra Gymnasiematematik(high school math) matematik 5

Eulers polyederformula

Definition: A graph is called planar if can be drawn in one plane without any arcs crossing each other.

Definintion: The graph G = (V,E) is called bipartite if the nodes can be divided into two disjunct parts V = V1∨ V2. where V1dosen’t have any elemenets in common with V2.

Eulers polyhedronformula: Let G = (V, E) be a planar, connected graph and let v denote the number of nodes, e the number of arcs and r be the number of surfaces. Then

v – e + r = 2.

Ex. For the dodecaedron, the number of surfaces is 12 similar pentagons. v = 20, e = 30 and r = 12.

Kategorier
Algebra Gymnasiematematik(high school math) Uncategorized

More Graphtheory

More graph-terminology:
The distance between two nodes is the shortest distance between the two nodes.
A graph that starts and ends in the same node is called a cycle or a closed circuit.
A simple path trespasses every node only once.

Let n be a node in a graph or multigraph G. The degree or valence of v is the number of arcs having an endpoint in v.
This number can be written as deg(v).
The handshaking lemma: At a large party where everybody shakes hand but not with everybody the number of persons having shaken hand an odd number of times is even.

a graph where it is allowed to pass a node several times is called a
multi-graph.

a complete graph is a graph without loops and where every pair of nodes are connected with an arc.

Ex. Let G be a loop-free graph with n nodes, such that G has 175 arcs and its complement has 56 arcs. Determine n.
Solution: The totla number of arcs in G and its complement equals the number of arcs in the complete graph Kn.
Therefore 175 + 56 = n(n-2)/2 or ”n over 2” &imp; 231 = n(n-1)/2 &imp; n=22 ∧ n=-21.

Kategorier
Algebra matematik 1c

Peculiar functions with descriptions

The cycloid describes the motion of a point on the perimeter of a wheel.

The tractoria  (or tractrix) is a curve where the length of the tangent from the tangentialpoint to the x-axis has a constant length a. Its evolute is the chainline

y = a cosh(x/a).

a particle P attached to a string PQ pulled through Q along a given curve at right angle to thee initialposition of PQ traces out a curve called dog-curve.

The equation for this curve is x= a ln{[a+{[a+&sqrt;(a22)]/y }- &sqr;(a2-y2)

Kategorier
Algebra Gymnasiematematik(high school math) matematik 2c

Solving polynomial equations, descartes theorem

A large part of the algebra courses at upper secondary-school level are devoted to solving equations or factorization of polynomials. This is often the same thing.

Some terminology. All of these are the same:
Solving a polynomial equation.
Finding roots of a polynomial equation p(x)=0.
Finding zeros of a polynomial equation p(x)
Factorizing a polynomial function p(x).

There is a factor for every root and vice versa.
(x-r) is a factor if and only if r is a root according to the Factor theorem.

How to solve equations step-by-step:
1. If solving an equation, put it in standard form with 0 on one side and simplif.
2. Know hoewmany roots to expect.
3. Find one factor or root. (Several techniques available)
4. Divide by your factor. This leaves you with a new reduced polynomial. whose degree is 1 less. For the rest of the problem you’ll work with the reduced polynomial and not the original.
5. Now you have a quadratic or linear equation which you already know how to solve.
Write down the solution.

There is no general solution for solving equations of degree 5 and higher.
Try to factorize the polynomial as much as possible.
Equations of degree four or less can be solved by stanard methods.

A plynomial of degree n will have n roots some of which may be multiple roots. This is according to the Fundamental theorem of Algebra.

Descartes Rule of Sign:
Tells you the how many positiv or negative real zeroes the polynomial has.
1. The number of positive roots of p(x)=0 is either equal to the number of variations in sign of p*(x) or less than that by an even numer.
2. The number of negative roots of p(x) = 0 is either equal to the number of variations in sign of p(-x)=0 or less than that by an even number.

Complex Roots

If a polynomial has real coefficients, then either all roots are real or there are an even number of non-real complex roots, in conjugate pairs.

For example, if 5+2i is a zero of a polynomial with real coefficients, then 5−2i must also be a zero of that polynomial. It is equally true that if (x−5−2i) is a factor then (x−5+2i) is also a factor.

This is true because when you have a factor with an imaginary part and multiply it by its complex conjugate you get a real result:

(x−5−2i)(x−5+2i) = x²−10x+25−4i² = x²−10x+29

If (x−5−2i) was a factor but (x−5+2i) was not, then the polynomial would end up with imaginaries in its coefficients, no matter what the other factors might be. If the polynomial has only real coefficients, then any complex roots must occur in conjugate pairs.

Kategorier
Algebra Gymnasiematematik(high school math)

Modular arithmetic & the Chinese Rest Theorem

modular arithmetic (sometimes called clock arithmetic) is a system of arithmetic for integers, where numbers ”wrap around” upon reaching a certain value—the modulus.

An example of this is the clock which starts repeating itself when it has reached 12. Therefore it is said that the clock follows an arithmetic modulo 12.  

Ex: If you add 7 hours to 8 o’clock the result is not 15 since when thee clock reaches zero it starts over from zero once again so the result is 7+8=3 in this arithmetic.

For a positive integer n, two integers a and b are said to be congruent modulo n,
a=b mod(n)
if the the remainder is the same when a and b are divided by n.

if their difference a − b is an integer multiple of n. The number n is called the modulus of the congruence.

a \equiv b \pmod n,\, 
e.g. 14

14≡12 (mod 2)

An example of an application of this mathematical tool is the Chinese Remainder Theorem.

Kategorier
Algebra Gymnasiematematik(high school math) matematik 1c Uncategorized

Diophantine equations

A Diophantine equation is an equation in which only integers are allowed as coefficients. Also the solutions must be integers. This can be written as ax + by = c. This is a linear diophantine equation.
For non-linar diophantine equations there is no general solution formula available.
Ex. 8x + 7y = 148.
has the solutions x=1 and y = 20.
It can be showed that in order to give integer-solutions 148 must be divisible by the Greatest Common divisor of 8 and 7 in this case 1.