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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- 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
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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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