Department of Mathematics, Statistics and Actuarial Sciences
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Item An Introduction to Differential Geometry: The Theory of Surfaces(Science Publishing Group, 2017) Gikonyo, Kuria Joseph; Kinyua, Kande DicksonFrom a mathematical perspective, a surface is a generalization of a plane which does not necessarily require being flat, that is, the curvature is not necessarily zero. Often, a surface is defined by equations that are satisfied by some coordinates of its points. A surface may also be defined as the image, in some space of dimensions at least three, of a continuous function of two variables (some further conditions are required to insure that the image is not a curve). In this case, one says that one has a parametric surface, which is parametrized by these two variables, called parameters. Parametric equations of surfaces are often irregular at some points. This is formalized by the concept of manifold: in the context of manifolds, typically in topology and differential geometry, a surface is a manifold of dimension two; this means that a surface is a topological space such that every point has a neighborhood which is homeomorphic to an open subset of the Euclidean plane. A parametric surface is the image of an open subset of the Euclidean plane by a continuous function, in a topological space, generally a Euclidean space of dimension at least three. The paper aims at giving an introduction to the theory of surfaces from differential geometry perspective.Item Differential Geometry: An Introduction to the Theory of Curves(Science publishing group, 2017) Gikonyo, Kuria Joseph; Kinyua, Kande DicksonDifferential 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.