Hausdorff measure
To determine the surface area of an m- dimensional surface in n-dimensional space ( m < n ), there is in mathematics various dimensions that are defined for all subsets of and "decent" on the ( non-degenerate ) m-dimensional surfaces whose heuristic acreage yield. (For the "decent" areas include in particular the submanifolds of. )
The best known of these measures is the m-dimensional Hausdorff measure, named after Felix Hausdorff; to illustrate the definition, however, the m-dimensional spherical measure should first be explained.
Definition of the spherical measure
A subset of the one considers the variables
For, where the infimum is taken over all coverings of by a countable set of m- dimensional spheres with diameters in the ... ( Diametern ). Here, the volume of the M-dimensional unit sphere ( ball of radius 1) in, equivalent to the m-dimensional area of the unit circle in the m-dimensional. The form factor ensures the correct " normalization " of the resulting surface measure. The summands are precisely the m-dimensional surface areas of the intersections of the spheres with running through the center of m-dimensional planes.
Is the m-dimensional spherical measure of then, by virtue of increasing smallness of the spheres, defined by
The refinement of the ball coverings through to 0 diameter causes an increasing convergence of m-dimensional equatorial surfaces of the balls to the exit face.
Definition of the Hausdorff dimension
For the definition of the Hausdorff dimension is achieved if instead of balls all subsets of are admitted to the covers. The diameter is defined by
For and, and you set appropriately for
But here the infimum is taken over all coverings of by a countable number of (arbitrary) subsets of ... with. Finally, we define
The outer measure. The restriction of the domain to measurable quantities provides the measure.
The terms and are even outer dimensions and have absolutely for certain quantities different values - the difference does not disappear in some " pathological " cases, in the limit to 0 - however, provide the two dimensions and ( "decent" to ) at the rectifiable m-dimensional quantities of the same value. In general, the inequality
Connection with the area formula
For the explicit calculation of the Hausdorff dimension of a parameterized surface with an area and an injective differentiable function, the area formula applies:
This is the generalized Jacobian of, and denotes the m-dimensional Lebesgue measure ( volume measure ) in.
Generalizations
(1 ) Analog is used for " non-integer dimensions " m the above definitions of and, here with the gamma function for irrational m.
The Hausdorff dimension of a subset of is then that ( uniquely determined) number m for all s
In recent decades, fractals came into the focus of popular science media. Fractals are subsets of ( " fractal " ) with fractional Hausdorff dimension; in public fractals are predominantly perceived as quantities that still distinguished not only by their fractal dimension by certain self- similarities.
(2) The definition of the M-dimensional Hausdorff measure is valid without significant changes in each space, instead of the metric; the same applies for the m-dimensional spherical degree. ( Only the absolute value function in the definition of the diameter is replaced by the underlying metric, more precisely. Is off)