Abstract
Differential geometry is a discipline of mathematics that uses the techniques of calculus and linear algebra to study problems in geometry. The theory of plane, curves and surfaces in the Euclidean space formed the basis for development of differential geometry during the 18th and the 19th century. The core idea of both differential geometry and modern geometrical dynamics lies under the concept of manifold. A manifold is an abstract mathematical space, which locally resembles the spaces described by Euclidean geometry, but which globally may have a more complicated structure. The purpose of this paper is to give an elaborate introduction to the theory of curves, and those are, in general, curved. Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and in the Euclidean space by applying the concept of differential and integral calculus. The curves are represented in parametrized form and then their geometric properties and various quantities associated with them, such as curvature and arc length expressed via derivatives and integrals using the idea of vector calculus.
Highlights
Differential geometry is a discipline of mathematics that uses the techniques of calculus and linear algebra to study problems in geometry
The theory of plane, curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th and the 19th century
Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds, [2, 4, 6] Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations
Summary
Differential geometry is a discipline of mathematics that uses the techniques of calculus and linear algebra to study problems in geometry. The theory of plane, curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th and the 19th century. Our main tools to understand and analyze these curved objects are (tangent) lines and planes and the way those change along a curve, respective surface. This is why we start with a brief chapter assembling prerequisites from linear geometry and algebra. The study tangents of curves, and the concept of curvature will be introduced and International Journal of Theoretical and Applied Mathematics 2017; 3(6): 225-228 defined through differentiation of the parametrization, and related to first and second derivatives, respectively. The culmination is a famous theorem of Gauss, which shows that the so-called Gauss curvature of a surface can be calculated directly from quantities which can be measured on the surface itself, without any reference to the surrounding three dimensional space, [5, 10]
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