Differential Geometry: An Introduction to the Theory of Curves

Kande Dickson Kinyua; Kuria Joseph Gikonyo

doi:doi:10.11648/j.ijtam.20170306.18

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Differential Geometry: An Introduction to the Theory of Curves

Kande Dickson Kinyua, Kuria Joseph Gikonyo

Published in International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 6)

Received: 4 December 2016 Accepted: 18 January 2017 Published: 10 January 2018

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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.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 6)
DOI	10.11648/j.ijtam.20170306.18
Page(s)	225-228
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Curvature, Curves, Differential Geometry, Manifolds, Parametrized

References

[1]	K. D. Kinyua. An Introduction to Differentiable Manifolds, Mathematics Letters. Vol. 2, No. 5, 2016, pp. 32-35.
[2]	Lang, Serge, Introduction to Differentiable Manifolds, 2nd ed. Springer-Verlag New York. ISBN 0-387-95477-5, 2002.
[3]	M. Deserno, Notes on Difierential Geometry with special emphasis on surfaces in R³, Los Angeles, USA, 2004.
[4]	M. P. do-Carmo, Differential Geometry of curves and surfaces, Prentice-Hall, Inc., Englewood Cliffs, New Zealand, USA, 1976.
[5]	M. Raussen, Elementary Differential Geometry: Curves and Surfaces, Aalborg University, Denmark, 2008.
[6]	M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1, Third Edition, Publish or Perish Inc., Houston, USA, 1999.
[7]	R. Palais, A Modern Course on Curves and Surfaces, 2003.
[8]	T. Shifrin, Differential Geometry: A First Course in Curves and Surfaces, Preliminary Version, University of Georgia, 2016.
[9]	V. G. Ivancevic and T. T. Ivancevic Applied Differential Geometry: A Modern Introduction, World Scientific Publishing Co. Pte. Ltd., Toh Tuck Link, Singapore, 2007.
[10]	W. Zhang, Geometry of Curves and Surfaces, Mathematics Institute, University of Warwick, 2014.

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Kande Dickson Kinyua, Kuria Joseph Gikonyo. (2018). Differential Geometry: An Introduction to the Theory of Curves. International Journal of Theoretical and Applied Mathematics, 3(6), 225-228. https://doi.org/10.11648/j.ijtam.20170306.18

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ACS Style

Kande Dickson Kinyua; Kuria Joseph Gikonyo. Differential Geometry: An Introduction to the Theory of Curves. Int. J. Theor. Appl. Math. 2018, 3(6), 225-228. doi: 10.11648/j.ijtam.20170306.18

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AMA Style

Kande Dickson Kinyua, Kuria Joseph Gikonyo. Differential Geometry: An Introduction to the Theory of Curves. Int J Theor Appl Math. 2018;3(6):225-228. doi: 10.11648/j.ijtam.20170306.18

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@article{10.11648/j.ijtam.20170306.18,
  author = {Kande Dickson Kinyua and Kuria Joseph Gikonyo},
  title = {Differential Geometry: An Introduction to the Theory of Curves},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {3},
  number = {6},
  pages = {225-228},
  doi = {10.11648/j.ijtam.20170306.18},
  url = {https://doi.org/10.11648/j.ijtam.20170306.18},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20170306.18},
  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.},
 year = {2018}
}

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TY  - JOUR
T1  - Differential Geometry: An Introduction to the Theory of Curves
AU  - Kande Dickson Kinyua
AU  - Kuria Joseph Gikonyo
Y1  - 2018/01/10
PY  - 2018
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DO  - 10.11648/j.ijtam.20170306.18
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
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AB  - 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.
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Author Information

Kande Dickson Kinyua

Department of Mathematics, Moi University, Eldoret, Kenya
Kuria Joseph Gikonyo

Department of Mathematics, Karatina University, Karatina, Kenya

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APA Style

Kande Dickson Kinyua, Kuria Joseph Gikonyo. (2018). Differential Geometry: An Introduction to the Theory of Curves. International Journal of Theoretical and Applied Mathematics, 3(6), 225-228. https://doi.org/10.11648/j.ijtam.20170306.18

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ACS Style

Kande Dickson Kinyua; Kuria Joseph Gikonyo. Differential Geometry: An Introduction to the Theory of Curves. Int. J. Theor. Appl. Math. 2018, 3(6), 225-228. doi: 10.11648/j.ijtam.20170306.18

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AMA Style

Kande Dickson Kinyua, Kuria Joseph Gikonyo. Differential Geometry: An Introduction to the Theory of Curves. Int J Theor Appl Math. 2018;3(6):225-228. doi: 10.11648/j.ijtam.20170306.18

Copy | Download

@article{10.11648/j.ijtam.20170306.18,
  author = {Kande Dickson Kinyua and Kuria Joseph Gikonyo},
  title = {Differential Geometry: An Introduction to the Theory of Curves},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {3},
  number = {6},
  pages = {225-228},
  doi = {10.11648/j.ijtam.20170306.18},
  url = {https://doi.org/10.11648/j.ijtam.20170306.18},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20170306.18},
  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.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Differential Geometry: An Introduction to the Theory of Curves
AU  - Kande Dickson Kinyua
AU  - Kuria Joseph Gikonyo
Y1  - 2018/01/10
PY  - 2018
N1  - https://doi.org/10.11648/j.ijtam.20170306.18
DO  - 10.11648/j.ijtam.20170306.18
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 225
EP  - 228
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20170306.18
AB  - 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.
VL  - 3
IS  - 6
ER  -

Copy | Download