|
In mathematics, an invariant is a property, held by a class of mathematical objects, which remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas of mathematics such as geometry, topology and algebra. Some important classes of transformations are defined by an invariant they leave unchanged, for example conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important step in the process of classifying mathematical objects. == Simple examples == The most fundamental example of invariance is expressed in our ability to count. For a finite collection of objects of any kind, there appears to be a number to which we invariably arrive, regardless of how we count the objects in the set. The quantity—a cardinal number—is associated with the set, and is invariant under the process of counting. An identity is an equation that remains true for all values of its variables. There are also inequalities that remain true when the values of their variables change. Another simple example of invariance is that the distance between two points on a number line is not changed by adding the same quantity to both numbers. On the other hand, multiplication does not have this property, so distance is not invariant under multiplication. Angles and ratios of distances are invariant under scalings, rotations, translations and reflections. These transformations produce similar shapes, which is the basis of trigonometry. All circles are similar. Therefore, they can be transformed into each other and the ratio of the circumference to the diameter is invariant and equal to pi. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Invariant (mathematics)」の詳細全文を読む スポンサード リンク
|