What does modal length mean in mathematics

Length (math)

The length in mathematics is a property that can be assigned to routes, paths and curves. The length of a curve is also called Arc length or Rectification line designated.

Lengths of routes

If $ A $ and $ B $ are two points in the (two-dimensional) drawing plane ($ \ R ^ 2 $) with the respective coordinates $ A (a_1 | a_2) $ and $ B (b_1 | b_2) $ then the length is the Route $ AB $ according to the Pythagorean Theorem

$ \ overline {AB} = \ sqrt {(b_1-a_1) ^ 2 + (b_2-a_2) ^ 2}. $

In the three-dimensional visual space ($ \ R ^ 3 $) with the respective coordinates $ A (a_1 | a_2 | a_3) $ and $ B (b_1 | b_2 | b_3) $ applies

$ \ overline {AB} = \ sqrt {(b_1-a_1) ^ 2 + (b_2-a_2) ^ 2 + (b_3-a_3) ^ 2}. $

There are essentially two ways of generalizing such formulas:

  • The length of the line $ AB $ is interpreted as the length of the vector $ \ overrightarrow {AB} $ and length measures are defined for vectors. The corresponding generalized concept of length for vectors is called norm.
  • The approach of looking at the distance between the end points instead of the length of the route is even more general. General concepts of distance are called metrics.

Lengths of paths

A path is a continuous mapping $ \ gamma: [a, b] \ to X $ from an interval into a topological space $ X $. In order to be able to assign a length to Wegen, however, this space must have an additional structure. In the simplest case, $ X $ is the level $ \ R ^ 2 $ or the visual space $ \ R ^ 3 $ with the usual length term for lines; Generalizations are possible for Riemannian manifolds or any metric spaces. The length of the path $ \ gamma \, $ is then called $ L (\ gamma) \, $.

Paths in the plane and in space

A path in a plane or in space is given by two or three coordinate functions:

$ t \ mapsto (x (t), y (t)) $ or $ t \ mapsto (x (t), y (t), z (t)) $ for $ a \ leq t \ leq b $.

For paths that are continuously differentiable piece by piece, the length of the path is given by the integral over the length of the derivative vector:

$ L = \ int \ limits_a ^ b \ sqrt {\ dot x (t) ^ 2 + \ dot y (t) ^ 2} \, \ mathrm dt $ or $ \ int \ limits_a ^ b \ sqrt {\ dot x (t) ^ 2 + \ dot y (t) ^ 2 + \ dot z (t) ^ 2} \, \ mathrm dt. $

motivation

The plane path $ \ begin {matrix} f (t) = (x (t), y (t)) \ end {matrix} $ is first approximated by small straight lines $ \ Delta s $, which are divided into two components $ \ Delta x $ and $ \ Delta y $ are decomposed parallel to the coordinate axes. According to the Pythagorean theorem: $ (\ Delta s) ^ 2 = (\ Delta x) ^ 2 + (\ Delta y) ^ 2 $. The total length of the path is approximated by the sum of all straight lines:

$ L = \ sum \ Delta s = \ sum \ sqrt {(\ Delta x) ^ 2 + (\ Delta y) ^ 2} = \ sum \ sqrt {\ left (\ frac {\ Delta x} {\ Delta t } \ right) ^ 2 + \ left (\ frac {\ Delta y} {\ Delta t} \ right) ^ 2} \ Delta t $

If one assumes the convergence of the facts and gives the result without an exact limit value calculation, then the length $ L $ is the sum of all infinitesimally small straight lines, thus: $ L = \ int \ mathrm {d} s = \ int \ sqrt { \ dot {x} ^ 2 + \ dot {y} ^ 2} \, \ mathrm {d} t $.

Physically, the integrand can also be understood as the amount of the instantaneous velocity and the integration variable as the time. This is probably the best way of motivating the definition of the length of a path.

Examples

  • The circular line with radius $ r $
$ t \ mapsto (r \ cdot \ cos t, \ r \ cdot \ sin t) $ for $ 0 \ leq t \ leq2 \ pi $
has the length
$ \ int \ limits_0 ^ {2 \ pi} \ sqrt {r ^ 2 \ sin ^ 2t + r ^ 2 \ cos ^ 2t} \ \ mathrm dt = \ int \ limits_0 ^ {2 \ pi} r \, \ mathrm dt = 2 \ pi r. $
  • A piece of a helix with radius $ r $ and pitch $ h $
$ t \ mapsto \ left (r \ cdot \ cos t, \ r \ cdot \ sin t, \ \ tfrac {h} {2 \ pi} \ cdot t \ right) \ quad \ mathrm {f \ ddot ur} \ ; 0 \ leq t \ leq2 \ pi $
has the length
$ \ begin {align} \ int \ limits_0 ^ {2 \ pi} \ sqrt {r ^ 2 \ sin ^ 2t + r ^ 2 \ cos ^ 2t + \ left (\ tfrac h {2 \ pi} \ right) ^ 2 } \ \ mathrm dt & = \ int \ limits_0 ^ {2 \ pi} \ sqrt {r ^ 2 + \ left (\ tfrac h {2 \ pi} \ right) ^ 2} \ \ mathrm dt \ & = \ sqrt {(2 \ pi r) ^ 2 + h ^ 2} \ end {align} $

Special cases

Length of a function graph

Let the function $ f: x \ rightarrow f (x) $ be a differentiable function on $ [a, b] \ subset \ mathbb {R} $ then the length $ L $ between the points $ \ begin {matrix} A is calculated (a | f (a)) \ end {matrix} $ and $ \ begin {matrix} B (b | f (b)) \ ​​end {matrix} $ as follows:

$ L (a, b) = \ int \ limits_ {a} ^ {b} \ sqrt {1+ (f '(x)) ^ 2} \; \ mathrm {d} x \ qquad (*) $
Parametric representation

If the curve is given in the parametric representation with $ x = x (t), y = y (t) $ and we introduce the parameter $ t $ above, the expression arises with the values ​​$ \ alpha, \ beta $ of t, which belong to x = a and y = b, represent:

$ L (\ alpha, \ beta) = \ int \ limits _ {\ alpha} ^ {\ beta} \ sqrt {\ dot {x} ^ 2 + \ dot {y} ^ 2} \; \ mathrm {d} t $

(For L. one often also writes swhich can then be seen without an integral as $ \ mathrm {d} s $)

example: The circumference of a circle can be calculated with the help of $ \ begin {matrix} (*) \ end {matrix} $. A circle with the radius $ r $ satisfies the equation $ x ^ 2 + y ^ 2 = r ^ 2 $ or $ f (x) = \ sqrt {r ^ 2-x ^ 2}. $ The derivation is: $ f '(x) = \ frac {-x} {\ sqrt {r ^ 2-x ^ 2}} $.

If one applies the formula $ \ begin {matrix} (*) \ end {matrix} $, it follows:

$ L = 2 \ int \ limits _ {- r} ^ {r} \ sqrt {1+ \ frac {x ^ 2} {r ^ 2-x ^ 2}} \, \ mathrm {d} x = 2r \ int \ limits _ {- r} ^ {r} \ frac {\ mathrm {d} x} {\ sqrt {r ^ 2-x ^ 2}} \, = 2r \ arcsin (1) - 2r \ arcsin (-1) = 2 \ pi r $

Polar coordinates

If a plane path is given in polar coordinate representation $ r (\ varphi) $, that is

$ \ varphi \ mapsto (r (\ varphi) \ cos \ varphi, r (\ varphi) \ sin \ varphi) $ for $ \ varphi_0 \ leq \ varphi \ leq \ varphi_1 $,

so one gets from the product rule

$ \ frac {\ mathrm {d} x} {\ mathrm {d} \ varphi} = r ^ \ prime (\ varphi) \ cos \ varphi-r (\ varphi) \ sin \ varphi $ and
$ \ frac {\ mathrm {d} y} {\ mathrm {d} \ varphi} = r ^ \ prime (\ varphi) \ sin \ varphi + r (\ varphi) \ cos \ varphi $, therefore
$ \ left (\ frac {\ mathrm {d} x} {\ mathrm {d} \ varphi} \ right) ^ 2 + \ left (\ frac {\ mathrm {d} y} {\ mathrm {d} \ varphi } \ right) ^ 2 = \ left (r ^ \ prime (\ varphi) \ right) ^ 2 + r ^ 2 (\ varphi) $.

The length of the path in polar coordinate representation is therefore

$ L = \ int \ limits _ {\ varphi_0} ^ {\ varphi_1} \ sqrt {\ left (r ^ \ prime (\ varphi) \ right) ^ 2 + r ^ 2 (\ varphi)} \, \ mathrm {d } \ varphi $.

Paths in Riemannian manifolds

If in general $ \ gamma \ colon [a, b] \ to M $ is a piecewise differentiable path in a Riemannian manifold, then the length of $ \ gamma $ can be defined as

$ L (\ gamma) = \ int \ limits_a ^ b \ | \ dot \ gamma (t) \ | \, \ mathrm dt. $

Rectifiable paths in any metric space

Let $ (X, d) $ be a metric space and $ \ gamma \ colon [0,1] \ to X $ a path in $ X $. Then $ \ gamma $rectifiablewhen the supremum

$ L (\ gamma) = \ sup \ left \ {\ left. \ Sum_ {i = 0} ^ {k-1} d (\ gamma (t_i), \ gamma (t_ {i + 1})) \ right | k \ in \ mathbb N, 0 = t_0

is finite. In this case one calls $ L (\ gamma) $ die length of the path $ \ gamma $.

The length of a rectifiable path is therefore the supremum of the lengths of all approximations of the path by means of segments. For the differentiable paths considered above, the two definitions of length agree.

There are steady paths that cannot be rectified, for example the Koch curve or other fractals, space-filling curves, and almost certainly the paths of a Wiener process.

The word rectify or rectification means just do it, that means taking the curve (the thread) at the ends and pulling it apart, stretching it out so that you get a line whose length you can measure directly. Nowadays this word mainly appears in the term rectifiable on.

Lengths of curves

Definition of the length of a curve

The image set $ \ Gamma = \ gamma ([a, b]) \, $ belonging to a path $ \ gamma: [a, b] \ to X $ is called a curve (also track of the path $ \ gamma \, $). The path $ \ gamma \, $ is also called Parametric representation or Parameterization the curve $ \ Gamma \, $. Two different paths can have the same picture, so the same curve can be parameterized through different paths. It is obvious to define the length of a curve as the length of an associated path; but this assumes that the length supplies the same value for each parameterization. This is graphically clear, and it can actually be shown for injective parameterizations. In particular:

Let $ \ gamma_1: [a_1, b_1] \ to \ R ^ n $ and $ \ gamma_2: [a_2, b_2] \ to \ R ^ n $ be two injective parameterizations of the same curve $ \ Gamma \, $, i.e. $ \ gamma_1 ([a_1, b_1]) = \ gamma_2 ([a_2, b_2]) = \ Gamma \, $. Then: $ L \ left (\ gamma_1 \ right) = L \ left (\ gamma_2 \ right) = L \ left (\ Gamma \ right) $.

Parameterization of a curve according to the path length

As already said, there are different parameterizations for a curve. A special parameterization is the Parameterization according to the path length (or Arc length).

Is $ \ Gamma $ a rectifiable curve with the parameterization

$ \ begin {matrix} \ gamma: & [a, b] & \ to & \ R ^ n \ & \ tau & \ mapsto & \ gamma (\ tau) \ end {matrix} $

and $ \ Gamma_t $ for $ t \ in [a, b] $ the partial curve with the parameterization $ \ gamma | [a, t] $, this is how the function is called

$ \ begin {matrix} s: & [a, b] & \ to & \ R \ & t & \ mapsto & L \ left (\ Gamma_t \ right) \ end {matrix} $

as a path length function of $ \ Gamma $. This path length function $ s (t) $ is continuous and monotonically growing, for $ \ gamma $ injectively it even grows strictly monotonically and therefore also bijectively. In this case there is an inverse function $ t (s) $. The function

$ \ begin {matrix} \ hat {\ gamma}: & [0, L (\ gamma)] & \ to & \ R ^ n \ & s & \ mapsto & \ gamma (t (s)) \ end { matrix} $

is referred to as the parameterization of $ \ gamma $ with the arc length as a parameter.

If $ \ gamma $ is continuously differentiable and $ \ dot {\ gamma} (\ tau) \ neq 0 $ for all $ \ tau \ in [a, b] $, the peculiarity of the parameterization according to the arc length is that also $ \ hat {\ gamma} $ is continuously differentiable and for all $ s \ in [0, L (\ Gamma)] $

$ \ left \ | \ frac {\ mathrm {d} \ hat {\ gamma} (s)} {\ mathrm {d} s} \ right \ | = 1 $

applies.

See also

literature

  • Wolfgang Ebeling, Institute for Algebraic Geometry, University of Hanover: Lecture notes Analysis II.[1]