ABOUT PLANE TRANSFORMATIONS THAT PRESERVE THE AREAS OF FIGURES

Abstract

In this paper are studied the coordinate plane transformations that preserve the areas of bounded quadrable figures. Such transformations are called isoarcs. First, it is shown that the area of any bounded quadrable figure is equal to the sum of the areas of at most countable manifold of certain triangles disjoint two by two and located in this figure. Each of these triangles has a base parallel to the axis Ox. An example of a nonisometric isoaric transformation is shown. Consequently, the formula for the area of the figure bounded by an ellipse is derived without using integral calculus. The plane transformations, examined in this article, are essentisl to both fields, mathematics and economics. For a smooth transformation of the coordinate plane, is given a transformation represented by two functions of two variables with continuous first-order partial derivatives on the whole plane, so as a criterion is proved according to which this transformation is isoaric.

Keywords

• transformation
• quadrable
• isoaric
• figure
• function
• derivative