Kategorier
Fysik 1 fysikhistoria Geometri Gymnasiefysik(high school physics) Uncategorized

500 th anniversary of Leonardo da Vinci’s death.

Idag är det 500 år sedan universalgeniet Leonardo da Vinci samlades till sina fäder.

Kategorier
Geometri matematik 2c Uncategorized

Fraktaler i blåbärssoppa

Fractaler i blåbärssoppa Icke regelbundna former kan i regel beskrivas som fraktaler dvs kurvor som består av mindre kopior av sig självt. Ex är kustlinjen och moln och träd. Dimensionen för en fraktal beräknad som kvoten mellan area och sträcka behöver inte vara ett heltal. Exempelvis är dimensionen, eller måttet, för Kochs kurva 4/3. Plottar man upp kaosartade processer blir grafen en fraktal. Mandelbrotmängderna (se bild nedan) är ett exempel. Ett system, eller en process, är kaosartad om en liten ändring av startvärdena resulterar i en stor, oförutsägbar, ändring av slutvärdena. Ett exempel på detta är fjärilseffekten som innebär att om en fjäril flaxar med vingarna i Australien ändras startvärdena för atmosfärens tillståndsvariabler så att det uppstår en orkan på andra sidan jordklotet.

sidan jordklotet. mandelbrot2

Kategorier
Geometri Uncategorized

Hyperbolic functions

The hyperbolic functions have similar names to the trigonometric functions, but they are defined
in terms of the exponential function. In this unit I define the three main hyperbolic functions,
and sketch their graphs. I also discuss some identities relating these functions, and mention
their inverse functions and reciprocal functions.

The hyperbolic functions cosh x and sinh x are defined using the exponential function ex. We
shall start with cosh x. This is defined by the formula
cosh x = (ex + e−x)/2.
We can use our knowledge of the graphs of ex and e−x to sketch the graph of cosh x. First, let
us calculate the value of cosh 0. When x = 0, ex = 1 and e−x = 1.

So
cosh 0 =(ex + e−x)/2 = (1 + 1)/= 1 .
Next, let us see what happens as x gets large. We shall rewrite cosh x as
cosh x =e x/2 e−x/2.

To see how this behaves as x gets large, recall the graphs of the two exponential functions.
y = ex/2

y= e−x
As x gets larger, ex increases quickly, but −x decreases quickly. So the second part of the sum
e , ex/2 + e−x /2 gets very small as x gets large. Therefore, as x gets larger, cosh x gets closer and
closer to ex/2. We write this as
cosh x ≈
(ex+e-x)/2
for large x.
But the graph of cosh x will always stay above the graph of ex/2. This is because, even though
(e−x)/2 (the second part of the sum) gets very small, it is always greater than zero. As x gets
larger and larger the difference between the two graphs gets smaller and smaller.

As x becomes more negative, ex increases quickly, but ex decreases
quickly, so the first part of the sum ex/2 + e−x/2 gets very small. As x gets more and more
negative, cosh x gets closer and closer to e−x/2. We write this as
cosh x ≈
(e−x)
/2
for large negative x.
Again, the graph of cosh x will always stay above the graph of e−x/2 when x is negative. This is
because, even though ex/2 (the first part of the sum) gets very small, it is always greater than
zero. But as x gets more and more negative the difference between the two graphs gets smaller
We can now sketch the graph of cosh x. Notice the graph is symmetric about the y-axis, because
cosh x = cosh(−x).

Key Point
The hyperbolic function f(x) = cosh x is defined by the formula
cosh x = (ex)/2 + e−x/2
The function satisfies the conditions cosh 0 = 1 and cosh x = cosh(−x). The graph of cosh x is always above the graphs of ex/2 and e−x/2.

Kategorier
Geometri Gymnasiematematik(high school math)

The area of the circle

‘Minute Physics’ derives the area of the circle with a string of pearls and a ruler:

Kategorier
Geometri Gymnasiematematik(high school math) matematik 2c

Conical sections

280px-Conic_section_-_clean

The force of gravity determines the trajectories of the celestial bodies. Mathematical analysis reveals that there are three types of trajectories possible for a body moving in a gravitational field determined by Newton’s law of gravity.

  • If one of the bodies has very high speed relative to the other the moving body traces out a hyperbola. The equation for this is
    x 2/a2 – y 2/b=1
  • If the speeds of the bodies are beelow a certain threshold value they move in an elliptical curve. x 2/a2 + y 2/b2 =1
  • The limiting case between the hyperbola and the ellipse is the parabola. y=ax

Those geometrical objects can be illustrated by slicing the cone according to the figure above.

Kategorier
Geometri Gymnasiematematik(high school math) matematik 2c

Similarity

Two triangles are similar if all the corresponding angles are equal.

Then the ratios of the corresponding sides to each other are the same for both triangles.

In the fig above AB/BC = DE/EF or AB/DE = BC/EF.

A list of all the cases of similarity:

Fractals are made up of similar geomtetrical objects.

See e.g. the Serpinski triangle.

Fracrtals can be used to describe objects encounterd in real life such as coastlines, trees or clouds.

Kategorier
Geometri Uncategorized

Cylindrical coordinates

Cylindrical coordinates can be considered as a hybrid between spherical coordinates and rectangular coordinates.

The coordinates of a point is given by the angle between the projection of the point in the xy-plame Φ and two distances: the distance from the point to the xy-plane (z) and the distance to the z-axis (ρ).

Lêer:Cylindrical coordinates.svg

Kategorier
Algebra Geometri matematik 1c Uncategorized

Dimensions and different coordinate sytems

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

Chord-tangent theorem

The measure of the angle formed in the intersection between the chord of a circle and the tangent to the circle is the same as the angle at the periphery of the circle.

Click here to se the proof

Kategorier
Geometri Gymnasiematematik(high school math)

Euclidean postulates, theorems and definitions 3

Geometri teorem

den 29 juli 2012

00:02

  1. Om två vinklar är vertikalvinklar är de båda vinklarna lika stora. (If two angles are vertical angles then the two angles are congruent.)
  2. Två trianglar är kongruenta (likadana) om två sidor och mellanliggande vinkel hos de båda trianglarna är kongruenta. (Two triangles are congruent if two angles and the included angle are congruent to the corresponding parts of the second triangle. S.A.S..)
  3. Två trianglar är kongruenta om två vinklar och sidan mellan dessa är kongruenta i de båda trianglarna. (Two triangles are congruent if two angles and the included side of the  first triangle are congruent to the corresponding parts of the 2nd triangle. A.S.A.)
  4. Två trianglar är kongruenta om sidorna i den första triangeln  är kongruenta med motsvarande element i den andra triangeln. (Two triangles are congruent if the sides of the first triangle are congruent to the corresponding sides of the second triangle. (S.S.S.)
  5. Om en triangel har två kongruenta sidor, har den också två kongruenta vinklar som står mot dessa sidor. Och omvänt. (If a triangle has two congruent sides then the triangle has congruent angles opposite those sides.
  6. En liksidig triangel är alla vinklar lika stora. (An equilateral triangle is equiangular. Also converse.)
  7.  Om två likbelägna vinklar bildade av en transversal är lika stora är linjerna parallella.  Och omvänt  If a pair of corresponding angles formed by a transversal of two angles are congruent then the two lines are parallell. Also converse.
  8. Om ett par alternatvinklar bildade av en transversal  mellan två linjer är kongruenta så är dessa båda linjer parallella. Omvändningen gäller också.
  9. Två linjer är parallella om de är vinkelräta mot samma linje.
  10. Om en linje är vinkelrät mot den ena av två parallella linjer är den också vinkelrät mot den andra.
  11. Om ett par intilliggande inre  vinklar bildade av en transversal mellan två linjer är supplemetära , så är linjerna parallella. Och omvänt.
  12. Storleken på yttervinkeln i en triangel är lika med summan av de båda inre vinklarna i triangeln.
  13. Summan av storleken på de tre vinklarna i en triangel är 180 grader. Alltid.
  14. Summan av de spetsiga vinklarna i en rätvinklig triangel är 90 grader