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

Om mattelararen

Licentiate of Philosophy in atomic Physics Master of Science in Physics
Det här inlägget postades i Algebra, Geometri, matematik 1c, Uncategorized och har märkts med etiketterna , . Bokmärk permalänken.

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