References for the DifferentialGeometry package
DescriptionTextbooks on Differential Geometry Textbooks on the Calculus of VariationsTextbooks on Lie Algebras and Lie GroupsTextbooks on Jet Spaces and Their Applications to the Calculus of Variations and to Differential EquationsTextbooks on General Relativity ThesesPapers and Conference Proceedings
<Text-field style="Heading 2" layout="Heading 2" bookmark="info">Description</Text-field>
This page provides a partial list of textbooks and papers which contain in-depth presentations of the mathematics covered by the DifferentialGeometry package and its various subpackages.
<Text-field style="Heading 2" layout="Heading 2" bookmark="bkmrk0">Textbooks on Differential Geometry </Text-field>
Bishop, R. L., and Goldberg, S. Tensor Analysis on Manifolds. New York: Dover, 1980.
Boothby, W. M. An Introduction to Differentiable Manifolds and Riemannian Geometry. 2nd ed. Academic Press, 1986.
Flanders, H. Differential Forms with Applications to the Physical Sciences. New York: Dover, 1989.
Lovelock, D., and Rund, H. Tensors, Differential Forms, and Variational Principles. New York: Dover, 1989.
Spivak, M. A Comprehensive Introduction to Differential Geometry. 2nd ed. Vol. 1. Houston: Publish or Perish, 1979.
Spivak, M. A Comprehensive Introduction to Differential Geometry. 2nd ed. Vol. 5. Houston: Publish or Perish, 1979.
Wolf, J. Spaces of Constant Curvature. New York: McGraw-Hill, 1967.
<Text-field style="Heading 2" layout="Heading 2" bookmark="bkmrk1">Textbooks on the Calculus of Variations</Text-field>
Gelfand, I. M., and Fomin, S. V. Calculus of Variations. New York: Dover.
<Text-field style="Heading 2" layout="Heading 2" bookmark="bkmrk2">Textbooks on Lie Algebras and Lie Groups</Text-field>
Curtis, M. L. Matrix Groups. 2nd ed. New York: Springer-Verlag.
Fuks, D. B. Cohomology of Infinite Dimensional Lie Algebras. New York: Plenum.
Fulton, W. F., and Harris, J. Representation Theory, A First Course. Graduate Texts in Mathematics. New York: Springer, 1991.
Humphreys, J. E. Introduction to Lie Algebras and Representation Theory. Revised ed. New York: Springer-Verlag, 1980.
Varadarajan, V. S. Lie Groups, Lie Algebras and Their Representations. Graduate Texts in Mathematics. New York: Springer-Verlag, 1984.
<Text-field style="Heading 2" layout="Heading 2" bookmark="bkmrk3">Textbooks on Jet Spaces and Their Applications to the Calculus of Variations and to Differential Equations</Text-field>
Bluman, G. W., and Kumei, S. Symmetries and Differential Equations. Applied Mathematical Sciences, no. 81. New York: Springer, 1989.
Olver, P. J. Applications of Lie Groups to Differential Equations. 2nd ed. New York: Springer-Verlag, 1993.
Olver, P. J. Equivalence, Invariants and Symmetry. Cambridge: Cambridge University Press, 1995.
<Text-field style="Heading 2" layout="Heading 2" bookmark="bkmrk4">Textbooks on General Relativity </Text-field>
Penrose, R. and Rindler, W. Spinors and Spacetime. Cambridge University Press, 1986.
Petrov, A. Z. Einstein Spaces. Oxford: Pergamon, 1969.
Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; and Herlt, E. Exact Solutions to Einstein's Field Equations. 2nd ed. Cambridge Monographs on Mathematical Physics, 2003.
Stewart, J. Advanced General Relativity. Cambridge University Press, 1991
<Text-field style="Heading 2" layout="Heading 2" bookmark="bkmrk5">Theses</Text-field>
Delong, R. P. Killing tensors and the Hamiltonian-Jacobi equation. Univ. of Minnesota, 1982.
Hillyard, C. A Maple Package for the Variational Calculus. MSc. Thesis, Utah State University, 1992.
Miller, C. Vessiot: A Maple Package for Variational and Tensor Calculus in Multiple Coordinate Frames. MSc. Thesis, Utah State University, 1999.
<Text-field style="Heading 2" layout="Heading 2" bookmark="bkmrk6">Papers and Conference Proceedings</Text-field>
Anderson, Ian M. "Introduction to the Variational Bicomplex" in Mathematical Aspects of Classical Field Theory, pp. 51-73. Contemporary Mathematics, no. 132, edited by M. Gotay, J. Marsden, and V. Moncrief. 1992.
Anderson, Ian M. "Maple Packages and Java Applets" in Foundations of Computational Mathematics. London Mathematical Society Lecture Notes Series, no. 312, edited by F. Cucker, R. DeVore, P. Olver, and E. Suli. Minneapolis: Cambridge University Press, 2002.
Fels, M. E., and Olver, P. J. "Moving coframes I: A practical algorithm." Acta. Appl. Math, Vol. 55, (1999): 127-208.
Hochschild, G., and Serre, J. P. "Cohomology of Lie Algebras." Annals of Math, Vol. 57, (1953): 591-603.
Koszul, J. L. "Homologie and cohomologie des algebras de Lie." Bull. Soc. Math. France, Vol. 78, (1950): 65-127.
Newman, E. and Penrose, R. "An Approach to Gravitational Radiation by a Method of Spin Coefficients", Journal of Mathematical Physics, Vol. 3, (1962): 566-578.
Patera; Winternitz; and Zassenhaus. "On the Identification of a Lie Algebra Given by its Structure Constants I: Direct Decompositions, Levi Decompositions, and Nilradicals. Journal of Linear Algebra and its Applications, Vol. 109, (1988): 197-246.
Rand, W. D. "Pascal programs for the identification of Lie algebras I." Comput. Phys. Comm., Vol. 41, (1986): 105-125.
Woodhouse, N. M. J. "Killing Tensors and Separation of the Hamilton-Jacobi equation." Comm. Math. Physics., Vol. 44, (1975): 9 -38.
See AlsoDifferentialGeometry