Vector Reaal

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Vector calculus is the fundamental language of mathematical physics. It pro­ vides a way to describe physical quantities in three-dimensional space and the way in which these quantities vary. Many topics in the physical sciences can be analysed mathematically using the techniques of vector calculus. These top­ ics include fluid dynamics, solid mechanics and electromagnetism, all of which involve a description of vector and scalar quantities in three dimensions.

As emphasized by J. calculus primarily deals with the approximation (in a neighborhood of some point) of given mappings of vector spaces by linear mappings. The approximating linear map has to be a "good" approximation in some precise sense: it has to be "tangent" to the given map. A very useful notion of "tangent" can easily be introduced for maps between normed vector spaces; it leads to the notion of mappings and gives, in particular for Banach spaces, a very satisfactory theory (cf. Chap. VIII of [3]).