Abstract
In this chapter, we address the fundamental problem of extending the differential calculus to manifolds. To understand the problem we are faced with, consider a C 1 vector field Y(t) assigned along the curve x(t) on the manifold V n . We recall that on an arbitrary manifold the components Y i(t) of Y(t) are evaluated with respect to the local natural bases of local charts (U, x i), U⊂V n . Consequently, when we try to define the derivative of Y along x(t), we must compare the vector \({\bf{Y}}(t + \Delta t) \in {T}_{x(t+\Delta t)}{V }_{n}\), referred to the local basis e i (t+Δt), with the vector Y(t)∈T x(t) V n , referred to the local basis e i (t). Since we do not know how to relate the basis e i (t+Δt) to the basis e i (t), we are not in a position to compare the two preceding vectors; consequently, we cannot assign a meaning to the derivative of the vector field Y(t) along the curve x(t) of V n .
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Abraham, R., Marsden, J.E.: Foundation of Mechanics. Benjamin Cummings, Reading, Mass. (1978)
Arnold, V.L.: Equazioni Differenziali Ordinarie. Mir, Moscow (1979)
Arnold, V.L.: Metodi Matematici della Meccanica Classica. Editori Riuniti, Rome (1979)
Barger, V., Olsson, M.: Classical Mechanics: A Modern Perspective. McGraw-Hill, New York (1995)
Bellomo, N., Preziosi, L., Romano, A.: Mechanics and Dynamical Systems with Mathematica. Birkhäuser, Basel (2000)
Birkhoff, G., Rota, G.C.: Ordinary Differential Equations. Wiley, New York (1989)
Bishop, R.L., Goldberg, S.I.: Tensor Analysis on Manifolds. MacMillan, New York (1968)
Boothby, W.M.: An Introduction to Differential Manifolds and Riemannian Geometry. Academic, San Diego (1975)
Borrelli, R.L., Coleman, C.S.: Differential Equations: A Modeling Approach. Prentice-Hall, Englewood Cliffs, NJ (1987)
Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Diluite gases. Springer, Berlin Heidelberg New York (1994)
Choquet-Bruhat, Y.: Gèometrie Diffèrentielle et Systèmes Extèrieurs. Dunod, Paris (1968)
Chow, T.L.: Classical Mechanics. Wiley, New York (1995)
Crampin, M., Pirani, F.: Applicable Differential Geometry. Cambridge University Press, Cambridge (1968)
de Carmo, Manfredo P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs, NJ (1976)
Dixon, W.G.: Special Relativity: The Foundations of Macroscopic Physics. Cambridge University Press, Cambridge (1978)
Dubrovin, B.A., Novikov, S.P., Fomenko, A.T.: Geometria delle Superfici, dei Gruppi di Trasformazioni e dei Campi. Editori Riuniti, Rome (1987)
Dubrovin, B.A., Novikov, S.P., Fomenko, A.T.: Geometria e Topologia delle Varietà. Editori Riuniti, Rome (1988)
Dubrovin, B.A., Novikov, S.P., Fomenko, A.T.: Geometria Contemporanea. Editori Riuniti, Rome (1989)
Fasano, A., Marmi, S.: Meccanica Analitica. Bollati Boringhieri, Turin (1994)
Fawles, G., Cassidy, G.: Analytical Dynamics. Saunders, Ft. Worth, TX (1999)
Flanders, H.: Differential Forms with Applications to the Physical Sciences. Academic, New York (1963)
Fock, V.: The Theory of Space, Time and Gravitation, 2nd edn. Pergamon, Oxford (1964)
Goldstein, H.: Classical Mechanics, 2nd edn. Addison-Wesley, Reading, MA (1981)
Goldstein, H., Poole, C., Safko, J.: Classical Mechanics, 3rd edn. Addison Wesley, Reading, MA (2002)
Hale, J., Koçak, H.: Dynamics and Bifurcations. Springer, Berlin Heidelberg New York (1991)
Hill, R.: Principles of Dynamics. Pergamon, Oxford (1964)
Hutter, K., van der Ven, A.A., Ursescu, A.: Electromagnetic Field Matter Interactions in Thermoelastic Solids and Viscous Fluids. Springer, Berlin Heidelberg New York (2006)
Jackson, E.A.: Perspectives of Nonlinear Dynamic, vol. I. Cambrige University Press, Cambridge (1990)
Jackson, E.A:, Perspectives of Nonlinear Dynamic, vol. II. Cambrige University Press, Cambridge (1990)
Josè, J.V., Saletan, E.: Classical Dynamics: A Contemporary Approach. Cambridge University Press, Cambridge (1998)
Khinchin, A.I.: Mathematical Foundations of Statistical Mechanics. Dover, New York (1949)
Landau, L., Lifshitz, E.: Mechanics, 3rd edn. Pergamon, Oxford (1976)
MacMillan, W.D.: Dynamics of Rigid Bodies. Dover, New York (1936)
Marasco, A.: Lindstedt-Poincarè method and mathematica applied to the motion of a solid with a fixed point. Int. J. Comput. Math. Appl. 40, 333 (2000)
Marasco, A., Romano, A.: Scientific Computing with Mathematica. Birkhauser, Basel (2001)
Möller, C.: The Theory of Relativity, 2nd edn. Clarendon, Gloucestershire, UK (1972)
Norton, J.D.: General covariance and the foundations of general relativity: eight decades of dispute. Rep. Prog. Phys. 56, 791–858 (1993)
Panowsky, W., Phyllips, M.: Classical Electricity and Magnetism. Addison-Wesley, Reading, MA (1962)
Penfield, P., Haus, H.: Electrodynamics of Moving Media. MIT Press, Cambridge, MA (1967)
Petrovski, I.G.: Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs, NJ (1966)
Resnick, R.: Introduction to Special Relativity. Wiley, New York (1971)
Rindler, W.: Introduction to Special Relativity. Clarendon Press, Gloucestershire, UK (1991)
Romano, A., Lancellotta, R., Marasco, A.: Continuum Mechanics using Mathematica, Fundamentals, Applications, and Scientific Computing. Birkhauser, Basel (2006)
Romano, A., Marasco, A.: Continuum Mechanics, Advanced Topics and Research Trends. Birkhauser, Basel (2010)
Saletan, E.J., Cromer, A.H.: Theoretical Mechanics. Wiley, New York (1971)
Scheck, F.: Mechanics: From Newton’s Laws to Deterministic Chaos. Springer, Berlin Heidelberg New York (1990)
Stratton, J.A.: Electromagnetic Theory. McGraw-Hill, New York (1952)
Synge, J.: Relativity: The Special Theory, 2nd edn. North-Holland, Amsterdam (1964) (Reprint 1972)
Synge, J.L., Griffith, B.A.: Principles of Mechanics. McGraw-Hill, New York (1959)
Thomson, C.J.: Mathematical Statistical Mechanics. Princeton University Press, Princeton, NJ (1972)
Truesdell, C., Noll, W.: The Nonlinear Field Theories of Mechanics. Handbuch der Physik, vol. III/3. Springer, Berlin Heidelberg New York (1965)
von Westenholz, C.: Differential Forms in Mathematical Physics. North-Holland, New York (1981)
Wang, K.: Statistical Mechanics, 2nd edn. Wiley, New York (1987)
Whittaker, E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge (1989)
Zhang, W-B.: Differential Equations, Bifurcations, and Chaos in Economy. World Scientific, Singapore (2005)
Zhong Zhang, Y.: Special Relativity and Its Experimental Foundations. Advanced Series on Theoretical Physical Science, vol. 4. World Scientific, Singapore (1997)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media New York
About this chapter
Cite this chapter
Romano, A. (2012). Absolute Differential Calculus. In: Classical Mechanics with Mathematica®. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8352-8_9
Download citation
DOI: https://doi.org/10.1007/978-0-8176-8352-8_9
Published:
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-0-8176-8351-1
Online ISBN: 978-0-8176-8352-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)