THE FIRST QUADRATIC FORM OF A SURFACE AND ITS APPLICATIONS (Multidisciplinary Journal of Science and Technology, 2026)
This article discusses in detail the concept of the first quadratic form of a surface, its analytical expression and geometric content. The issue of determining the length of curves when the surface is given by parametric equations is considered, and the importance of the first quadratic form in this process is justified. The physical and geometric interpretation of the coefficients E, F, G, and the condition of orthogonality of coordinate lines are also analyzed. The first quadratic form is calculated for various surfaces (plane, sphere, surface of revolution, catenoid, helicoid, etc.) and their properties are shown through examples.