Differential Geometry: An Introduction to the Theory of Curves
International Journal of Theoretical and Applied Mathematics
Volume 3, Issue 6, December 2017, Pages: 225-228
Received: Dec. 4, 2016;
Accepted: Jan. 18, 2017;
Published: Jan. 10, 2018
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Kande Dickson Kinyua, Department of Mathematics, Moi University, Eldoret, Kenya; Department of Mathematics, Karatina University, Karatina, Kenya
Kuria Joseph Gikonyo, Department of Mathematics, Karatina University, Karatina, Kenya
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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.
Curvature, Curves, Differential Geometry, Manifolds, Parametrized
To cite this article
Kande Dickson Kinyua,
Kuria Joseph Gikonyo,
Differential Geometry: An Introduction to the Theory of Curves, International Journal of Theoretical and Applied Mathematics.
Vol. 3, No. 6,
2017, pp. 225-228.
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
K. D. Kinyua. An Introduction to Differentiable Manifolds, Mathematics Letters. Vol. 2, No. 5, 2016, pp. 32-35.
Lang, Serge, Introduction to Differentiable Manifolds, 2nd ed. Springer-Verlag New York. ISBN 0-387-95477-5, 2002.
M. Deserno, Notes on Difierential Geometry with special emphasis on surfaces in R3, Los Angeles, USA, 2004.
M. P. do-Carmo, Differential Geometry of curves and surfaces, Prentice-Hall, Inc., Englewood Cliffs, New Zealand, USA, 1976.
M. Raussen, Elementary Differential Geometry: Curves and Surfaces, Aalborg University, Denmark, 2008.
M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1, Third Edition, Publish or Perish Inc., Houston, USA, 1999.
R. Palais, A Modern Course on Curves and Surfaces, 2003.
T. Shifrin, Differential Geometry: A First Course in Curves and Surfaces, Preliminary Version, University of Georgia, 2016.
V. G. Ivancevic and T. T. Ivancevic Applied Differential Geometry: A Modern Introduction, World Scientific Publishing Co. Pte. Ltd., Toh Tuck Link, Singapore, 2007.
W. Zhang, Geometry of Curves and Surfaces, Mathematics Institute, University of Warwick, 2014.