Abstract
Although the theory of partial differential equations (PDEs) is not a mere generalization of the theory of ordinary differential equations (ODEs), there are many points of contact between both theories.
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Notes
- 1.
The dot (or inner) product is not needed at this stage, although it is naturally available. Notice, incidentally, that the dot product is not always physically meaningful. For example, the 4-dimensional classical space-time has no natural inner product.
- 2.
A luxury that we cannot afford on something like the surface of a sphere, for obvious reasons.
- 3.
This notation is justified by the geometric theory of integration of differential forms, which lies beyond the scope of these notes.
- 4.
- 5.
This domain may be the whole of \({\mathbb R}^n\).
References
Arnold VI (1973) Ordinary differential equations. MIT Press, Cambridge
Arnold VI (1978) Mathematical methods of classical mechanics. Springer, Berlin
Chern SS, Chen WH, Lam KS (2000) Lectures on differential geometry. World Scientific, Singapore
Courant R (1949) Differential and integral calculus, vol II. Blackie and Son Ltd, Glasgow
Marsden JE, Tromba AJ (1981) Vector calculus, 2nd edn. W. H. Freeman, San Francisco
Spivak M (1971) Calculus on manifolds: a modern approach to classical theorems of advanced calculus. Westview Press, Boulder
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Epstein, M. (2017). Vector Fields and Ordinary Differential Equations. In: Partial Differential Equations. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-55212-5_1
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DOI: https://doi.org/10.1007/978-3-319-55212-5_1
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