Dimensions and different coordinate sytems

Publicerat den oktober 3, 2012 av mattelararen

Polar coordinates

The number of figures necessary for specifying the position of a point is called the dimension of the space.

For a two dimensional space it is sufficient to use two numbers (x, y) : one specifying the position on a right ot left scale and the other number giving the position on the up- and down scale. (x,y) are often referred to as the cartesian coordinates of a point.

Alternatively one can use polar coordinates where one uses the angle between an arrow pointing at the point and the length of the arrow (r, φ).

These are related by x=r cos(φ) and y =rsin(φ)

In three-dimensional space, our world for example, consequently three numbers are necessary. (x, y, z)

An alternative is the spherical-polar coordinates involving the radius, the azimuthal angle (φ) and the the declination (θ).

<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /> \begin{cases}<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /> x &= a \, \sin\varphi \, \cos\theta \\<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /> y &= a \, \sin\varphi \, \sin\theta \\<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /> z &= a \, \cos\varphi.<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /> \end{cases}<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
These coordinates can be found from the cartesian coordinates by using the formulas
r=\sqrt{x^2+y^2+z^2}  (Pythagorean theorem in three dimensions)
\theta = \mbox{arccos}\left(\frac{z}{r}\right)
\varphi = \mbox{arctan}\left(\frac{y}{x}\right)
spherical coordinates
Publicerat i Algebra, Geometri, matematik 1c, Uncategorized | Märkt , | 1 kommentar

What is mathematics?

Publicerat den september 28, 2012 av mattelararen

Perhaps after 53 lectures it’s about time that I define what I mean with mathematics?

Mathematics can be defined as the science dealing with quantities, numbers and geometrical objects in particular.

It is characterised by its logical method which consists of drawing logical conclusions from already proven results.  This chain of reasoning can then be traced all the way back to the fundamental axioms.

Kant pointed out that mathematics is a purely theoretical science.

The axiomatic system of  mathematics is independent of the physical world. Still mathematics is of outmost importance for the empirical sciences since it provides a means for constructing mathematical models able to accurately describe and predict outcomes in the  real world.

Immanuel KAnt

Publicerat i matematik 1c, Philosophy of Science | Märkt , , , , | 1 kommentar

Chinas first aircraft-carrrier

Publicerat den september 26, 2012 av mattelararen

First chinese carrier commissioned on sep. 26th.

Publicerat i Technology, Uncategorized | Märkt , | Lämna en kommentar

factorization (faktorisering)

Publicerat den september 26, 2012 av mattelararen

Factorization means to decompose a number or polynomial into a product of other objects, called factors, which when multiplied together gives the original number or polynomial.

Ex. The number 16 = 2*2*2*2 when factorized into prime numbers. Since prime-numbers can’t be factorized it is the final stage of factorization of numbers.

Also polynomials can be factorized with e.g. the conjugate rule

x4 -1 = (x2-1)(x2+1) = (x-1)(x+1)(x2+1)

or with the theorem of factorization stating that the roots of a polynomial are also factors of that polynomial.

Ex.

Consider the polynomial p(x) = (x-2)(x-3).

As can easily be seen the roots of p(x) are x = 2 and x = 3.

Multiplying the parentheses results in

p(x) = x2 – 5x + 6.

Therefore

x2 – 5x + 6 = (x-2)(x-3)

 

Therefore the roots to this polynomial are also the factors of it!

Publicerat i Gymnasiematematik(high school math), matematik 1c | Märkt , , | 2 kommentarer

Pascal’s triangle

Publicerat den september 22, 2012 av mattelararen

Numerology is an old phenomenon in many cultures. It is represented both in art an buildings. Buildings are constructed according tothe Golden Section and magical quadrats can be found in eg Albrect Durer’s painting ”Melancholy”.

A substantial amount of mathematics is hidden in Pascal’s triangle. This triangle isbased on the principle that an element in it is the sum of the nearest elements above it.

The elements in Pascal’s triangle turns out to be the binomial-coefficients. This means that they are identical to the numbers you get when you expand binomials with different exponentials. A binomial is the sum of two numbers (x + y) .

As an example we can take (x+y)^2 = x^2 + 2xy + y^2.

The coefficients here can be found in the second row of the periodic system: 1    2    1.

The  coefficients for (x + y) ^3 is given by the third row: 1  3  3   1 and so on.

This is spectacular in its own right but it doesn’t   end there: The binomialcoefficients also have a combinatorical interpretation: They give the  number of combinations if you e.g. are to select k objects out of n possible.

Example: Suppose you wish to pick 2  apples out of a set of three. You find the answer as the second element in the third row: 3.

This is called the number of combinations possible when selecting two elements from a set of three. In a combination the order is irrelevant.   According to convention it is writen as

{n \choose k} = \frac{n!}{k!(n - k)!}
Here n! = the number of permutations of n i.e. the number of possible arrangements of n objects when the order is important, k! is the number of permutations of k objects and (n-k)! the number of permutations of the difference of n and k. 
Publicerat i Uncategorized | Märkt , | 2 kommentarer

Cauchy’s integralformula

Publicerat den september 17, 2012 av mattelararen

Theorem

Suppose U is an open subset of the complex plane C, f : U → C is a holomorphic function and the closed disk D = { z : | z − z0| ≤ r} is completely contained in U. Let \gamma be the circle forming the boundary of D. Then for every a in the interior of D:

f(a) = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\, dz

where the contour integral is taken counter-clockwise.

The proof of this statement uses the Cauchy integral theorem and similarly only requires f to be complex differentiable. Since the reciprocal of the denominator of the integrand in Cauchy’s integral formula can be expanded as a power series in the variable (a − z0), it follows that holomorphic functions are analytic. In particular f is actually infinitely differentiable, with

f^{(n)}(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{(z-a)^{n+1}}\, dz.

This formula is sometimes referred to as Cauchy’s differentiation formula.

The circle γ can be replaced by any closed rectifiable curve in U which has winding number one about a. Moreover, as for the Cauchy integral theorem, it is sufficient to require that f be holomorphic in the open region enclosed by the path and continuous on its closure.

This proof is based on the fact that the value of the integral is invariant under deformation as long as the integrand is analytic over this closed area.

Publicerat i Calculus, Imaginary numbers | Märkt | Lämna en kommentar

Cauchy’s integral theorem

Publicerat den september 12, 2012 av mattelararen

Cauchy‘s integral theorem states that an analytic function f(z) the line integral around a closed path C is zero.

∫f(z)dz = 0  .

This means that the curve integrals over 2 curves with the same endpoints for an analytic function are the same.

Publicerat i Imaginary numbers | Lämna en kommentar

The Cauchy-Riemann equations

Publicerat den september 11, 2012 av mattelararen

In order for a complex function of a  single complex variable to be differentiable it must be differentiable both parallell to the imaginary axis δy →0 and parallell to the real axis δx →0.

This condition leads to the CAuchy –Riemann equations-

The Cauchy–Riemann equations on a pair of real-valued functions of two real variables u(x,y) and v(x,y) are the two equations:

(1a)     \dfrac{ \partial u }{ \partial x } = \dfrac{ \partial v }{ \partial y } \,

and

(1b)    \dfrac{ \partial u }{ \partial y } = -\dfrac{ \partial v }{ \partial x } \,
Proof:
Suppose that

f(z) = u(z) + i \cdot v(z)

is a function of a complex number z. Then the complex derivative of ƒ at a point z0 is defined by

\lim_{\underset{h\in\mathbb{C}}{h\to 0}} \frac{f(z_0+h)-f(z_0)}{h} = f'(z_0)

provided this limit exists.

If this limit exists, then it may be computed by taking the limit as h → 0 along the real axis or imaginary axis; in either case it should give the same result. Approaching along the real axis, one finds

\lim_{\underset{h\in\mathbb{R}}{h\to 0}} \frac{f(z_0+h)-f(z_0)}{h} = \frac{\partial f}{\partial x}(z_0).

On the other hand, approaching along the imaginary axis,

\lim_{\underset{h\in \mathbb{R}}{h\to 0}} \frac{f(z_0+ih)-f(z_0)}{ih} =\frac{1}{i}\frac{\partial f}{\partial y}(z_0).

The equality of the derivative of ƒ taken along the two axes is

i\frac{\partial f}{\partial x}(z_0)=\frac{\partial f}{\partial y}(z_0),
Holomorphy is the property of a complex function of being differentiable at every point of an open and connected subset of \mathbb{C} (this is called a domain in \mathbb{C}). 
 function that is complex-differentiable in a whole domain (holomorphic) is the same as an analytic function. This is not true for real differentiable functions.
Publicerat i Advanced, Calculus, Imaginary numbers | Lämna en kommentar

de Moivre’s formula and complex-conjugation.

Publicerat den september 10, 2012 av mattelararen

(e^ix )^n = cos(nx) + i sin(nx) is called de Moivre’s formula.

The formula is named after the 17 th. century French huguenot mathematician Abraham de Moivre.

Also the variable in of a function can be a complex number. f(z) =z^2 and z = x + iy gives x^2 + 2ixy + y^2 with real part x^2 + y*2 and imaginary part 2xy.

A necessary condition for a function of a complex variable to be differentaible is that it satisfies the Cauchy -Riemann equations.

Publicerat i Gymnasiematematik(high school math), Imaginary numbers, matematik 4, matematik 5 | Märkt , , , | Lämna en kommentar

Alternative representations of complex numbers

Publicerat den september 9, 2012 av mattelararen

As mentioned in the latest post any complex number may be represented by an arrow in the complex plane. This number is unambiguously described by two numbers: its real part x and its imaginary part y. z= x+iy. This is called the Cartesian representation. (Rene Descartes)

From the figure below it is evident that

x = r cosφ

y = r sin φ  and thus z = r( cosφ + i sinφ) . This is called polar representation of the complex number z. r is the modulus of the vector z (i.e. its length) and φ is called the argument of z.    The modulus can be  computed by multiplying the  complex number z with its conjugated complex number z

By adding the  Taylor series for cos(x) and  i sin(x) we get the series expansion for the exponential function eix. This relation is called Euler’s formula.

This leads to the exponential representation of a complex number:

z= r e(iφ) 

This

 File:Complex conjugate picture.svg
Publicerat i Imaginary numbers, matematik 4, matematik 5 | Märkt , , , | Lämna en kommentar