Reciprocal Eigenvalues
The title of this course uses two technical terms and draws attention from people familiar with them.
The reciprocation of a number, to produce a multiplicative inverse, is an algebraic operation that is singular at zero.
Eigenvalues are properties of certain matrices in linear algebra. They are associated with eigenvectors v. If matrix T operates on a row vector v to produce v T = a v, then the number a is an eigenvalue for T. It means that for a line {x v : x in R} in a vector space, T acts as a magnification if a > 1, and as a contraction if 0 < a < 1. The negative a cases mean that T reflects the line through the origin (zero vector).
In this course two dimensions suffice, so there can be two eigenvalues, in this case reciprocals of one another. Then T can be written as a diagonal matrix
For example, (1, 1) T = (a, 1/a). At the origin there is a square at (1,1) and a rectangle at (a, 1/a). The rectangle, having length and width as reciprocals, has the same area as does the square. In a perfectly elastic plane, the operation of T can be called a squeeze of parameter a.
Stable level curves
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Given any constant c > 0, there is a hyperbola
The application of a squeeze, of whatever parameter a, to H(c) leaves the hyperbola stable:
- (x,y) in H(c) implies (a x, y/a) in H(c).
Given any c > 0, H(c) can be called a level curve of parameter c. Use Q to represent the quadrant with x > 0, y > 0. For any c, the region contained by the asymptotes and H(c) is stable under squeezes.
Now take c = 1, call H(1) the standard hyperbola, and consider the region it bounds with the asymptotes. A subset of the region is the descending staircase of steps of height y = 1/n over the interval [n−1, n]. The sum of the areas under the stairs is called "the harmonic series". A student must learn to show that this area is unbounded.
When the squeeze parameter is taken as a variable, its various actions on Q can be viewed with each H(c) as a streamline in a corner flow. With a > 1 the flow descends and veers right. With 0 < a < 1 the flow reverses.
Two points on an H(c) and the radial lines to them determine a hyperbolic sector. Such a sector is mapped to another sector of equal area by a squeeze. One might ask, fpr the standard hyperbola, what x makes the sector between (1,1) and (x, 1/x) have unit area ? w:Leonard Euler found the answer to be e = 2.718281828 approximately. The number cannot be expressed as a rational fraction of integers, nor as a solution to an algebraic equation (It is called "transcendental").
Though area is preserved by a squeeze, shape is distorted and Euclidean distances changed under squeezing. For example, a sector near (1,1) has a broader shape than its image when a >> 1. These transformed sectors are so narrow that they appear as lines in Q.
To standardize area measure of sectors, a sector of one unit is one wing. The rays defining a sector can be viewed as a hyperbolic angle. For Euler number e, the angle between (1,1) and (e, 1/e) has area equal to one wing. Squeezing now with a = e, the image of the above sector is between (e, 1/e) and (e2, 1/ e2), which has another wing of area. As every pair (en, 1/ en) and (en+1, 1/ en+1) contributes a wing to the total area, there is no upper bound on the measure of the area of a hyperbolic sector or of the size of a hyperbolic angle.
Proponents of angle as area
[edit | edit source]The notion of angle size being related to area measurement has had proponents and detractors. As has been shown, hyperbolic angle depends on area for its definition, but what of circular angle? In the fourth century w:Theon of Alexandria wrote "the area of sectors of a circle are proportional to their angles at the center."
The idea of unifying the circular and hyperbolic angle by reference to sector areas was propounded by w:Robert Baldwin Hayward in 1892. The following year w: Alexander Macfarlane proposed this unification in a paper "On the definition of the trigonometric functions" which he submitted for consideration at the mathematical conference held in Chicago in connection with the Columbia Exposition. For some reason the paper was withdrawn and published later in his Essays on Space Analysis.
An exposition of the unification of angles through the notion of sector areas has been contributed to the Wikibook Geometry in the chapter Unified Angles.
Dented trapezoids
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Consider a hyperbolic sector that extends from (1,1) to (a, 1/a). To develop the calculus of a single variable, another view of this sector area is taken by addition and subtraction of triangles of area one-half. First consider the right triangle with base [0, a] and altitude 1/a. Join this triangle to the sector, then take away the right triangle on base [0,1] of altitude 1. The remaining region has base [1, a], parallel sides at x=1 and x=a, and a concave top determined by y=1/x. This region will be called a dented trapezoid, and its area is equal to the area of the hyperbolic sector. For students of calculus this area is familiar as an expression of the natural logarithm of a. Except for the Wikibook b:Calculus, the integration of the function f(x) = 1/x over an interval [1, a] is introduced without mention of hyperbolic sectors. Evidently the steps given above for approaching the dented trapezoid have been obviated by other calculus texts. Nevertheless, the area of the sector is log a. When 0 < a < 1, the logarithm is negative, so evidently the area of the sector between [a, 1/a] and [1, 1] is taken as negative area.
Lemma: The area of a dented trapezoid over [a,b] depends only on the ratio b/a.
- proof: Squeeze mapping with parameter c moves the sector determined by [a,b] to the sector of [ca, cb].
Theorem: Log ab = log a + log b.
- proof: Log a + log b represents the sum of the areas of trapezoids over [1,a] and [1,b]. By the lemma, the area over [1,b] equals the area over [a, ab]. Joining the trapezoids at x=a yields one over [1, ab] which represents log ab.
The measure of dented trapezoids over [a,1] where 0 < a < 1 follows from the symmetry of the standard hyperbola with respect to the line y = x . A reflection of a sector to (a, 1/a) produces the sector to (1/a, a) which is considered negative. Algebraically, ab = 1 means log ab = 0 since the dented trapezoid at 1 collapses to a segment which has measure zero. Thus log (1/a) = − log a .
Corollary: Log (b/a) = log b − log a.
Readings
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