Mathematical Analysis II (UNITEXT, Volume 85) (2nd Edition)

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Limits of functions Continuity. Mean value theorem Rolle's theorem.

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Differentiation notation Second derivative Third derivative Change of variables Implicit differentiation Related rates Taylor's theorem. Fractional Malliavin Stochastic Variations. Glossary of calculus. The first and second derivatives of y with respect to x , in the Leibniz notation.

Main article: Leibniz's notation. The single and double indefinite integrals of y with respect to x , in the Leibniz notation. For functions of 2 or more variables, see Multiple integral. A function f of x , differentiated once and twice in Lagrange's notation. The single and double indefinite integrals of f with respect to x , in the Lagrange notation. The x derivative of y and the second derivative of f , Euler notation. The x antiderivative of y and the second antiderivative of f , Euler notation.

The first and second antiderivatives of x , in one of Newton's notations. A function f differentiated against x , then against x and y. Archived from the original on Retrieved CS1 maint: archived copy as title link. Whiteside, , pp. New York. Engelsman has given more strict definitions in Families of Curves and the Origins of Partial Differentiation , pp. CS1 maint: archived copy as title link 1st to 3rd integrals: Method of Fluxions Newton , , pp.

Cajori, The n th integral notation is deducted from the n th derivative.

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Differential equations. Difference discrete analogue Stochastic Stochastic partial Delay. Inspection Separation of variables Method of undetermined coefficients Variation of parameters Integrating factor Integral transforms Euler method Finite difference method Crank—Nicolson method Runge—Kutta methods Finite element method Finite volume method Galerkin method Perturbation theory. List of named differential equations. Categories : Differential calculus Mathematical notation.

Hidden categories: CS1 maint: archived copy as title Articles needing additional references from November All articles needing additional references.

Mathematical Analysis II

Namespaces Article Talk. The dual Schur complement system is defined accordingly, and so are the preconditioners. For instance, the inverse of a FETI preconditioner can be obtained from the right- hand side of 6. The contents are organised to appeal especially to Engineering, Physics and Computer Science students, all areas in which mathematical tools play a crucial role.

Basic notions and methods of differential and integral calculus for functions of one real variable are presented in a manner that elicits critical reading and prompts a hands-on approach to con Canuto , Alfio Quarteroni , Thomas A. Zang , M. The authors of that book have incorporated into this new edition the many improvements in the algorithms and the theory of spectral methods that have been made since then. This latest book retains the tight integration between the Canuto , M. Yousuff Hussaini , Alfio M.

Mathematics,Probability and Statistics,Applied Mathematics

Quarteroni , Thomas A. Spectral methods, particularly in their multidomain version, have become firmly established as a mainstream tool for scientific and engineering computation. Canuto : Boundary conditions in Legendre and Chebyshev methods. SIAM J. Canuto : Spectral methods and a maximum principle.

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Canuto : Stabilization of spectral methods by finite element bubble functions. Book and De rebus D. Book Originally, i. There is no mention of this in Const. Deo auctore , but there is three years later in Const. Whereas he does not mention him explicitly in the passage from Tanta cited above — even boasting, as he does with the Digest , that the Institutes were carefully composed from almost all the elementary textbooks the ancient jurists had composed 63 —, he does mention Gaius by name in the Const.

Imperatoriam as the primary source of the Institutes Since we purport to study the composition of the Digest and its partes , rather than the Institutes , we will leave the latter alone and concentrate on the prima pars legum that was to be the follow-up of the Institutes in the first year of the new curriculum Since the readings from these four monographs were considered indispensable, room had to be found for them elsewhere in the curriculum. It was available in the course of the second year.

Only two partes of the readings ad edictum — De iudiciis and De rebus — were left for the second and third year, amounting to what is substantively a course in property law De iudiciis and the law of obligations De rebus creditis. The compilers surely the law professors among them decided to discontinue the practice of consecutive reading of these two partes. It seemed obvious to make a clear distinction and to have readings on property law De iudiciis and the law of obligations De rebus creditis in the second and third year of the curriculum alternatively, leaving room for extra readings of other sources in the second and third year each.

The original four libri singulares — De re uxoria and De tutelis and De testamentis and De legatis — were replaced by their for educational purposes most suitable counterparts in the Digest : Book 23 De sponsalibus and Book 26 De Tutelis from the Quarta pars Digestorum and Book 28 De testamentis liber primus and Book 30 De legatis et fideicommissis liber primus from the Quinta pars Digestorum. Each of these four books from the Digest was the first book of four more extensive volumes compositiones incorporated in the fourth and fifth pars of the Digest 69 : Book 23 is the first book of the tripartite volume tripertitum volumen De dotibus D.

Justinian describes the whole operation as follows:.

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  5. After having thus secured the continuity of traditional legal education, there remained the room left in the third year for additional readings after the readings from the mandatory edictal part De iudiciis or De rebus in that year were finished. It was the reason why, under the old curriculum, third-year students were designated as Papinianistae , since it was in their third year that they were introduced to his writings As a matter of course, this tradition had to be preserved in the new curriculum as well.

    In honour of Papinian, Justinian instructed his drafting committee to change the usual distribution of fragments from the writings of the jurists over the titles of each book in the case of the six titles of Book 20 ad formulam hypothecariam by putting a fragment from Papinian at the beginning of each of the titles of which it was composed The professors had to read D.

    Students had heard introductory classes in their second year on De sponsalibus and De tutelis from the Quarta pars and on De testamentis and De legatis from the Quinta pars , but only the first books of these four compositiones. Now, in their fourth year, they had to read all of it without the learned assistance of their professors.

    Justinian circumspectly prescribes which of the left over books from these partes had to be read in this year. He is careful to point out that the sum of all the volumina , the greater units within each pars , contained in the Umbilicus and the pars De testamentis is seventeen After having read the three first books Book of Umbilicus in the previous third year, there remained fourteen books to be read from what Justinian called the compositio quattuordecim librorum 79 , being the units De dotibus three books and De tutelis et curationibus two books from Umbilicus and De testamentis two books and De legatis et fideicommissis seven books from the pars De testamentis.

    Four books of this compositio had already been read in the second year Book 23; 26; 28 and 30 , which leaves ten books to be studied by the students in their fourth year. In this way, says Justinian, all 17 books of Umbilicus and the pars De testamentis together will have been studied My good friend, the mathematician Theo de Vries, has made a study of it, which I am happy to add at this point Consider table 2, showing the distribution of books over the seven partes Digestorum. It is obvious that there is a discrepancy between the number of books attributed to the pars prima and the distribution of books over the remaing six partes.

    It was, therefore, a predetermined number, but it confronted the mastermind behind the project Tribonian with a considerable mathematical problem: how to distribute the remaining 46 books over the residual six partes? The number fifty was very dear to Justinian. Be this as it may, the total number of 50 books for the Digest as a whole was preordained, as was the number of four books to be attributed to the first pars.

    So, how to distribute the remaining 46 over the residual 6 partes in a way that is harmonious and symmetrical as well? The obvious solution is in table 3. However, this is not what Tribonian did, due to the constraints implied by the implementation of the academic curriculum in the composition of the Digest. Justinian had not only preordained that the Digest as a whole should number exactly 50 books, but also that the compulsory readings of the Digest public, as well as private should be concluded at the end of the Quinta pars Book This complicated the problem of an even distribution of all the 46 books over all the remaining six partes even more, since Tribonian had to conceive of a distribution of books concluding with book 36 at the end of the Quinta pars.

    Consequently, the Quinta pars was predestined to contain more books than the Quarta pars. There is no way to divide 17 books in two equal parts of whole numbers and so the Quinta pars was to be artificially extended, which explains the curious phenomenon of no less than three homonymous books De legatis et fideicommissis Book in that pars. Since, for reasons of balance and symmetry, the distribution of books over all the remaining six partes had to come as close as possible to the mean 7 or 8 for each pars , the amount of nine books for the Quinta pars was imperative.

    The allotment of eight books to the Sexta pars and the remaining six to the Septima must have been predetermined by this mathematical exercise, since Justinian explicitly states in Const. Exercises like this — how to divide a given number 46 in 6 whole numbers creating a domelike structure — were inspired by the mathematics of Diophantus of Alexandria fl. It was Constantinople, rather than Rome, that saved the Greek mathematical tradition from oblivion; Anthemius of Tralles, for example, one of the architects of Hagia Sophia, was an accomplished mathematician.

    The Quarta and Quinta pars Digestorum were conceived of as one continuous compositio , consisting of 17 libri singulares students should have studied in order to become perfecti , having read 36 books of the Digest The Quinta pars concluded the sections of the Digest that were composed on the basis of a scheme dictated by the legal curriculum.

    Consequently, there is a clear break in the composition of the Digest at the end of Book It is plainly visible in table 4, when one considers the distribution of titles over the books contained in these partes The distribution of titles over the first five partes is fairly balanced between 40 and 52 , whereas the number of titles in the sixth and seventh pars explodes: no less than in the sixth pars and 93 in the seventh.