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The arc length of one branch between x = x 1 and x = x 2 is a ln y 1 / y 2 . The area between the tractrix and its asymptote is π a 2 / 2 , which can be found using integration or Mamikon's theorem. The envelope of the normals of the tractrix (that is, the evolute of the tractrix) is the catenary (or chain curve) given by y = a ...
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a module also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers. [1]
The angle θ is taken in the positive sense and must lie in the interval 0 < θ ≤ π (radian measure). The chord function can be related to the modern sine function, by taking one of the points to be (1,0), and the other point to be (cos θ, sin θ), and then using the Pythagorean theorem to calculate the chord length: [2]
A transversal produces 8 angles, as shown in the graph at the above left: 4 with each of the two lines, namely α, β, γ and δ and then α 1, β 1, γ 1 and δ 1; and; 4 of which are interior (between the two lines), namely α, β, γ 1 and δ 1 and 4 of which are exterior, namely α 1, β 1, γ and δ.
The circle inversion map is anticonformal, which means that at every point it preserves angles and reverses orientation (a map is called conformal if it preserves oriented angles). Algebraically, a map is anticonformal if at every point the Jacobian is a scalar times an orthogonal matrix with negative determinant: in two dimensions the Jacobian ...
The measure f ∗ (λ) might also be called "arc length measure" or "angle measure", since the f ∗ (λ)-measure of an arc in S 1 is precisely its arc length (or, equivalently, the angle that it subtends at the centre of the circle.) The previous example extends nicely to give a natural "Lebesgue measure" on the n-dimensional torus T n.
As a special case, note that if F is a linear form (or (0,1)-tensor) on W, so that F is an element of W ∗, the dual space of W, then Φ ∗ F is an element of V ∗, and so pullback by Φ defines a linear map between dual spaces which acts in the opposite direction to the linear map Φ itself:
In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain.