7. Show that for a stationary hydraulic jump in a rectangular channel, the upstream Froude number , and the downstream Froude number , are related by

8. Consider a simply-connected volume whose boundary is the surface . Suppose that contains an incompressible fluid whose motion is irrotational. Let the velocity potential be constant over . Prove that has the same constant value throughout . [Hint: Consider the identity .]

9. In Exercise 8, suppose that, instead of taking a constant value on the boundary, the normal ve

for projective geometry

+LAGRANGIAN N ]]>

DEDEKIND’s terminology might seem strange, but from his definitions it is clear that ‘divisor’ and ‘multiple’ mean the two sides of an inclusion; ‘greatest common divisor’ designs the intersection; and ‘least common multiple’ refers to the union, in the sense of the smallest structure that contains both. The selection of these expressions is easy to understand if we consider that DEDEKIND was dealing with algebraic number theory. In the case of Z, the inclusion of principal ideals corresponds strictly to the divisibility of their generators; this led him to employ the analogy between inclusion and division throughout the supplement on ideal theory. Moreover, his intention was to establish terminology in such a way that theorems could be formulated with exactly the same wording as in elementary number theory.

There exists evidence that DEDEKIND’s terminology was understood by at least one outstanding mathematician as related to the general notion of set. CANTOR used DEDEKIND’s terminology for inclusion, union and intersection of fields in his decisive series of papers ‘Über unendliche, lineare Punktmannigfaltigkeiten’, from 1880 to 1884. This is particularly striking because that terminology is rather inappropriate in a general set-theoretical setting such as the one to which CANTOR applied it─DEDEKIND himself replaced it in such a context (see below).

In the text on fields, DEDEKIND’s treatment of ‘substitutions’ or mappings, and particularly those which transfer structure, was equally a model. The fact that he is considering morphisms which may not be injective, is made clear by the comment on zero images: when he says that the numbers b constitute a field “if not all of them are zero”, it means that his notion of mapping allows for the trivial case in which all elements of the original have 0 as their image. (We have seen that non-injective maps were already present in his 1856-58 ‘Gruppen-Studien’.) Moreover, the clarity with which the reflexive, symmetric and transitive properties of field ‘conjugation’ are set forth in the text deserves attention.

The example we have considered here is only that of fields, discussed by DEDEKIND in the first paragraph of his 1871 ideal theory, but the whole exposition insisted on the importance of set-structures for algebraic number theory. The very viewpoint adopted involved a constant exercise in the translation of problems stated in terms of numbers to new and more abstract set-formulations: DEDEKIND replaced KUMMER’s ideal numbers with the kind of sets of numbers that he called ‘ideals’ (cf. section 4.2).

4.1.2. Sets and maps in a general setting. The year after the publication of this first version of ideal theory, DEDEKIND began to write the draft for his later book Was sind und was sollen die Zahlen?. The 1872 draft begins with definitions of some set operations (DUGAC 1976, 293-294); he presented them in a general setting, abstracted from the structural restrictions that are necessary in an algebraic context. The word ‘system’ was defined in a way that undeniably pointed to the extensional notion of set, and the most characteristic part of DEDEKIND’s algebraic terminology changed in accordance with the new abstract framework: the word ‘divisor’ was replaced by ‘part’ [Theil], meaning subset. Meanwhile, he maintained the expression ‘least common multiple’ for the union set, although he also wrote ‘compound system’, which was to become his final choice.

Nevertheless, set operations were by no means his primary interest. They were completely clear in themselves, and DEDEKIND had long been accustomed to them. The whole draft was primarily devoted to a study of the notions of mapping and chain, the second being based on the first. The term ‘mapping’ [Abbildung] appears also at the very beginning of the draft (DUGAC 1976, 294), and is defined in its general sense. We will return to this essential part of Was sind und was sollen die Zahlen? later on.

The translation of set-structural operations to a general set-theoretical framework, which DEDEKIND sketched in his 1872 draft, reveals clearly what I have already stated: that DEDEKIND’s mature set theory has its roots in his algebraic ideas, that emerged during the late 1850s.

It would be interesting to know about DEDEKIND’s motivation for writing the 1872-78 draft. It probably was one or two of the following factors. In 1872, he published his theory of the real number system after learning for the first time about two other theories, those of WEIERSTRASS and CANTOR. The knowledge that some mathematicians had developed ideas similar to his own might have motivated him to publish his more general reflections on the foundations of the natural numbers, and arithmetic as a whole, since they established the general framework for his theory of the real numbers. CANTOR had also developed the notion of derived set (see CANTOR 1872), which perhaps was even closer to DEDEKIND’s viewpoint, since it used the notion of set. On the other hand, it was

In any event, the Dedekind cuts form a complete ordered field. The additive identity in the Dedekind model is the open interval from minus infinity to 0. The multiplicative identity is the open interval from minus infinity to 1. More generally, each rational number r corresponds to the cut (– ∞, r), and this correspondence allows us to identify the rational numbers with a subfield of the Dedekind model for the real numbers.

COMPLETION by cauchy sequences

The most far-reaching method for constructing the real numbers is due independently to Charles Meray (1869, 1872) and Georg Cantor (1872, 1883). Again one begins with the rational numbers. One considers the set of all sequences {xn} of rational numbers such that xn-xm tends to zero as n and m tend to infinity. Such sequences are called Cauchy sequences. We introduce an equivalence relation in the set of Cauchy sequences by declaring two Cauchy sequences {xn} and {yn} to be equivalent if xn – yn tends to zero as n tends to infinity. The real numbers are then defined to be the set of equivalence classes of Cauchy sequences. Addition and multiplication are easy to define. The sum of the equivalence classes represented by two such sequences {xn} and {yn} is defined to be the equivalence class of {xn + yn}, and similarly for the product. It is straightforward to verify the axioms of an ordered field, and a little more complicated to verify the completion axiom. The main disadvantage of the method is the excess labor and the level of sophistication required for working with equivalence classes rather than just sequences. The advantage of the method is that it can be used in a fairly general context to embed metric spaces in “complete” spaces. (A metric space can be embedded as a dense subset of a complete metric space, which is essentially unique.)

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