Abstract
Lake physics cannot be described let alone understood without tailoring the statements in mathematical expressions and deducing results from these. We now wish to lay down the mathematical prerequisites that are indispensable to reach quantitative results. A systematic presentation will not be given because it is assumed that the reader is (or once has been) familiar with the subjects and only needs to be reminded of knowledge that may be somewhat dormant. Let us begin with mathematics.
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Notes
- 1.
The mathematical prerequisites and the knowledge of physics required to follow the ensuing developments are those of a basic education in engineering or natural sciences at universities that is generally acquired in two or three semesters of calculus, linear algebra, differential equations and vector and matrix calculus. However, while the elements of these subjects are taught, they do not lie in the centre of the syllabi of the mentioned fields of study. Likewise, as far as the background of physics is concerned, only the fundamentals of classical physics are needed which are generally taught at universities to engineers and natural scientists in a one or two semester course during their basic education. To lay a common ground of this knowledge and to outline the ‘language’ used we shall repeat subsequently elements of both fields. There are many books on basic physics and on calculus, and each university seems to teach these subjects from its own lecture notes. A very popular set of physics books are The Feynman Lecture Notes [8]. Well-known calculus books are [2, 12, 15–17]. Readers familiar with the calculus of vectors and tensors – here only the Cartesian tensor notation is used – may omit a careful reading of this chapter and directly pass to Chap. 3. Nevertheless, a quick glance through this section may be helpful, since the notation used throughout the entire text is introduced.
- 2.
We restrict here considerations to real numbers. If \(\textbf{\textit{x}} \cdot \textbf{\textit{y}}\) can be a complex number, then P 1 is defined as \(\textbf{\textit{x}} \cdot \textbf{\textit{y}} = (\textbf{\textit{y}} \cdot \textbf{\textit{x}})\) *, where * denotes the conjugate complex number.
- 3.
In a general basis, these components can be constructed parallel to the base vectors or orthogonal to them. In a Cartesian system the two different projections coincide.
- 4.
If the thumb, index finger and the middle finger of the right hand are stretched out such that they form a triad of non-co-planar vectors such that thumb, index finger and middle finger are identified with a, b, and c, respectively, then those ‘arrows’ form a right-handed system of vectors.
- 5.
Some readers may feel the desire for complementary reading. Books on tensor analysis are, e.g., Betten [3], Block [4], Bowen and Wang [5] and Klingbeil [11]. Books on continuum mechanics containing chapters on tensors are by Chadwick [7], Gurtin [9], Spencer [18], Hutter and Jöhnk [10] and Liu [13]. This is only a selection of many.
- 6.
We use
$$\begin{aligned}&&\lim_{\varDelta x \to 0}2\left[f\left(x + \frac{\varDelta x}{2}\right) - f(x) \right] = \lim_{\varDelta x \to 0}f'\left(x + \frac{\varDelta x}{4}\right)\varDelta x\\ && =\lim_{\varDelta x \to 0}\left(f'(x)\varDelta x+f''(x)\frac{(\varDelta x)^2}{4}\right).\end{aligned}$$ - 7.
One also says that the components of the ε tensor have value 1 when the indices \(i,j,k\) are evenly permuted between 1, 2 and 3; they have value (\(-1\)) when the indices \(i,j,k\) are odd permutations of 1, 2 and 3; and they have value 0 if \(i,j,k\) are no permutation of 1, 2 and 3.
- 8.
When shrinking this area to zero, it is tacitly assumed that the closed curve does not leave the region of definition of \(\textbf{\textit{v}}(\textbf{\textit{x}},t)\). A region for which this reduction can be done, starting with any curve and reducing it to any point of the region, is called simply connected. An annulus in two dimensions is not simply connected, but the exterior of a cavity in three dimensions is simply connected.
- 9.
Strictly, according to the mean-value theorem, it is a point between the two edge points. The difference to the value at the central point is, however, of higher order small.
- 10.
The symbol for ‘curl’ in continental Europe is ‘rot’.
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Hutter, K., Wang, Y., Chubarenko, I.P. (2011). Mathematical Prerequisites. In: Physics of Lakes. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15178-1_2
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