I read the first volume of Wuest's serie; now I am approaching the second. I would state at first two premisses:
1. I am not German; although I know German, it is for me a foreign lanuguage
2. I have not studied it, but only read
3. I am only a PhD in Logic and Epistemology
This book is really a masterwork! There is not a simple succession of theorems, lemmas and so on, in an order or in an other. Wuest has a unitary vision of mathematics. He reveals the unity underlying result apparent very far each other. So we reckon that integrals, dual-vector, delta-function (and I suggest, Skolem functions) are three example of the same concept. He is as Wuest composed some variations on the theme of physical mathematics, there in any one is it easily to reckon the main theme (as in Mozart's variations).
For example, he doesn't devote a section to the trigonometric functions, but he shows that they are solution of particular differential equations. And firstly, the hyperbolic-functions, then the usual sin, cos, ecc. So all this area is subsumed under the hat of the differential equations. Or I think to his treatment of Dirac's functional which he finds as a special case of a particular class of function: the functionals, in the theory of distribution.
His treatment of linear algebra is superb. No flaw, according to my opinion. Wuest books are not books of mathematics but ON mathematics; they reveal the inner soul of this discipline, in a rigorous style, clean and clear. In other terms, he insists of the profound harmony living in the realm of mathematics. It is not an affair to remember laws, functions, so on. But it is poetry. Sublime poetry.
His proof are very fine and elegant. Incidentally, I would note that Wuest introduce the Gram-Schmidt Verfahren in order to prove that if a space has a basis, then it has also a orto-normal basis. The proof is, obviously, by recursion on the length of the basis. And here it intervenes the GS-Verfahren.
This aspect is absolutely obscure in 'Jaenich, Mathematik I, geschrieben fuer Physiker'. He ends a section with this process and a drawing, but he doesn't explain his role and his functioning.
In two words: I suggest Wuest's serie of books for the Wiley, has an introductory approach to mathematical physics. Perhaps, at firts sight it could be seem too arid and fearful. But it is not so. It is self consistent. You don't need to search for other books. Everything is explained and there is no symbol without a precise definition
Jaenich could be sound more appealing. I think that there is often too much slang and few proofs. The proofs are necessary, because they justify the complex, by the simple. Some times, a theorem could be sound obscure, or counter-intuitive. Then, in the proof we are convinced of the contrary.
A proof has also a rethorical component: it must justify one thing and persuade us of its correctness. Proofs are the ground on which the tree of mathematics has its roots.
Until now, the best book on mathematics I have yet read. It is not only mathematics, but philosophy of mathematics. Philosophy is not opposite to mathematics. Since Pitagora there is a profound link. And, to make philosophy doesn't mean avoiding formulas. In some cases, they speak more clearfully than the reflections of the mathematician himself.
Thanks Rainer!
