Mathematical Tools

quanto erit maius aediﬁcium,

tanto altius fodit fundamentum

tanto altius fodit fundamentum

Aurelius Augustinus

While the problems in scientiﬁc computing always have a footing in a speciﬁc problem domain, such as one of the disciplines of physics, the algorithms used are often the same in very diﬀerent contexts. This, however, raises the question, what the uniting elements of scientiﬁc computing are, which prevents it from fracturing into separate parts, which are then considered more as a part of the application domain instead of scientiﬁc computing. As already hinted at, algorithms may be applied to problems with greatly diﬀering semantics. As such algorithms and solution techniques may be regarded as a ﬁrst step of identifying common items, which can be identiﬁed to belong to the ﬁeld of scientiﬁc computing and which are deployed as tools in the variously diﬀerent ﬁelds of application. Thus, the algorithms require a suﬃcient level of abstraction to describe them in an application agnostic fashion.

While many implementations of algorithms directly utilize a very low level of abstraction, it is desirable to provide higher levels of abstraction, in order to make broader use more easy to recognize. Following a generic programming approach does not preclude to also utilize specializations which are often already well established, while providing a more general and top level view.

The task of formulating algorithms on a high level of abstraction is eased in case the problem descriptions are of similarly high level. Thus the introduction of mathematical formalisms has a particular focus on terms encountered in the ﬁeld of dynamics.

Mathematics and its strict and formal deﬁnitions are employed to formalize the modelling of algorithms. Hence, it is unavoidable to be familiar with a basic set of mathematical terms and deﬁnitions. Whithout an understanding of the setting (the terrain) a problem is much more diﬃcult to solve [63 ].

Therefore, it is necessary to provide solid foundations concerning the terms used and the concepts connected to them, before proceeding to the analysis or synthesis of systems of non-trivial complexity. To address at least the barest of minima of requirements, a short aggregation of terms is provided in the following to provide an attempt at consistent nomenclature for the remaining chapters, thus making the terrain easier to navigate. The collection of terms draws upon several distinct sources [64][65][66][67][68][69][70][71][72], which should be consulted in the case that this limited and short compilation of terms and deﬁnitions, which is in this context unprecedented in both thoroughness as well as extent, thus acting not only as a backbone but also represents a contribution in its own right, is found to be insuﬃcient. It should be noted, however that the deﬁnitions in the given literature are not necessarily compatible to amongst each other when attempting to simply combine several sources, so that great care needs to be taken, not to miss small, but crucial diﬀerences in the deﬁnitions. Therefore, considerable eﬀort has been invested to make the deﬁnitions as self-contained and consistent as possible.

Another venue, not requiring the procurement of books of any kind, is also available by simply turning to Wikipedia [73], which also contains a surprising treasure of mathematical knowledge. A greater caveat applies to that source, however, since not only are the entries often applying terms inconsistently, where every book at least aims to be consistent within itself. Not only has no such uniting aspect in this regard yet surfaced in the context of Wikipedia, but some of the entries may even be outright incorrect.

The diﬀerent wording and points of view provided by diﬀerent sources may provide easier insights to diﬀerent backgrounds and motivations. However, the uniting goal should always be to develop an understanding which is consistent and hopefully has more depth and covers at least a bit more than the problem at hand demands in order to allow for the exploration and investigation, as should be the goal for any and all scientiﬁc endeavours.

4.1 Bare Basics

4.2 Algebraic Structures

4.3 Mappings

4.4 Topology

4.5 Fibers

4.6 Diﬀerential Geometry

4.6.1 Tensor Coordinates

4.6.2 Bra-Ket Notation

4.6.3 Beyond 1-Forms

4.6.4 Fields

4.6.5 Derivatives

4.6.6 Special Spaces

4.6.7 Peculiarities in Special Spaces

4.7 Integration

4.8 Distances

4.9 Integral Equations

4.10 Probability

4.2 Algebraic Structures

4.3 Mappings

4.4 Topology

4.5 Fibers

4.6 Diﬀerential Geometry

4.6.1 Tensor Coordinates

4.6.2 Bra-Ket Notation

4.6.3 Beyond 1-Forms

4.6.4 Fields

4.6.5 Derivatives

4.6.6 Special Spaces

4.6.7 Peculiarities in Special Spaces

4.7 Integration

4.8 Distances

4.9 Integral Equations

4.10 Probability