Ian Bruce Introduction. John D. I have decided to start with the integration, as it shows the uses of calculus, and above all it is very interesting and probably quite unlike any calculus text you will have read already. I hope that people will come with me on this great journey : along the way, if you are unhappy with something which you think I have got wrong, please let me know and I will fix the problem a. There are of course, things that Euler got wrong, such as the convergence or not of infinite series; these are put in place as Euler left them, perhaps with a note of the difficulty. The other works mentioned are to follow in a piecemeal manner alongside the integration volumes, at least initially on this web page.
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Ian Bruce Introduction. John D. I have decided to start with the integration, as it shows the uses of calculus, and above all it is very interesting and probably quite unlike any calculus text you will have read already. I hope that people will come with me on this great journey : along the way, if you are unhappy with something which you think I have got wrong, please let me know and I will fix the problem a.
There are of course, things that Euler got wrong, such as the convergence or not of infinite series; these are put in place as Euler left them, perhaps with a note of the difficulty. The other works mentioned are to follow in a piecemeal manner alongside the integration volumes, at least initially on this web page. The work is divided as in the first edition and in the Opera Omnia into 3 volumes.
I have done away with the sections and parts of sections as an irrelevance, and just call these as shown below, which keeps my computer much happier when listing files. The supplements of the posthumous volume IV are attached here following Vol. Click here for the 1st chapter : Concerning the nature of differential equations, from which functions of two variables are determined in general. I have used the word valid in the to indicate such functions, rather than real or actual, as against absurd, which Euler uses.
He then shows how this criterion can be applied to several differential equations to show that they are in fact integrable, other than by using an integrating factor ; this includes a treatment of the normal distribution function.
Click here for the 2nd chapter : Concerning the resolution of equations in which either differential formula is given by some finite quantity. These solutions are found always by initially assuming that y is fixed, an integrating factor is found for the remaining equation, and then the complete solution is found in two ways that must agree.
Click here for the 3rd chapter : Concerning the resolution of equations in both differential formulas are given in terms of each other in some manner. Euler establishes the solution of some differential equations in which there is an easy relation between the two derivatives p and q.
Click here for the 4th chapter : Concerning the resolution of equations in which a relation is proposed between the two differential formulas and a single third variable quantity.
This chapter sees a move towards the generalization of solutions of the first order d. Initially a solution is established from a simple relation, and then it is shown that on integrating by parts another solution also is present. Several examples are treated, and eventually it is shown that any given integration is one of four possible integrations, all of which must be equivalent.
A simple theory of functions is used to show how this comes about; later Euler establishes the conditions necessary for a particular relation to give rise to the required first order d. This chapter is a continuation of the methods introduced in ch. A very neat way is found of introducing integrating factors into the solution of the equations considered, which gradually increase in complexity. All in all a most enjoyable chapter, and one to be recommended for students of differential equations.
This chapter completes the work of this section, in which extensive use is made of the above theoretical developments, and ends with a formula for function of function differentiation. This is a short chapter but in it there is much that is still to be found in calculus books, for here the chain rule connected with the differentiation of functions of functions is introduced. Much frustration is evident from the bulk of the formulas produced as Euler transforms second order equations between sets of variables x, y and t, u.
A lead is given to the Jacobi determinants of a later date that resolved this difficulty. Happy reading! This is a long but interesting chapter similar to the two above, but applied to more complex differential equations; at first an equation resembling that of a vibrating string is investigated, and the general solution found.
Subsequently more complex equations are transformed and by assuming certain parts vanishing due to the form of transformation introduced, general solutions are found eventually. Examples are provided of course.
This is also a long but very interesting chapter wherein Euler develops the solution of general second order equations in two variables, with non-zero first order terms, in terms of series that may be finite or infinite; the coefficients include arbitrary functions of x and y in addition, leading to majestic formulas which are examined in cases of interest — especially the case of vibrating strings where the line density changes, and equations dealing with the propagation of sound.
Euler himself seems to have been impressed with his efforts. The method is extended to forms involving the second degree. This is the last chapter in this section.
This is a short chapter but in it there are some interesting developments. Thus, the distinction is made between repeated integrals and integrals over two or more dimensions. The use of arbitrary functions takes the place of constants in these general integrations considered.
This is another short chapter, as Euler reaches the bounds of his knowledge. The method is extended to more general cases. This is another short chapter, forming a basis for the integration of functions of three variables: differential formulas of higher orders are established at first; however, it soon becomes apparent that this is a far more difficult task, due to the introduction of arbitrary functions of the other variables in the integration.
In this chapter, arbitrary functions of the integration are introduced for the variables not integrated over at the time, and the work relates back to extending results already established for one or two variables. In this chapter some procedures are put in place for the integration of such equations in general, which are then applied to certain cases in which a simple relation exists between the first order partial derivatives.
This is the penultimate chapter. In this chapter, Euler applies his skills to the solution of homogeneous differential equations in two and three dimensions, especially those of orders one, two, and three. By making linear substitution of the coordinate x, y, and z, he is able to derive algebraic equations to be solved in a straight forwards manner, and also to reduce the number of variables to two.
The chapter ends with some hints as to his current research on the theory of fluids, in which such integrations find a place.
This is the final chapter of the original work; an appendix on the Calculus of Variations follows. A later posthumous edition published in ran to four volumes, where additions to most chapters were put in place in volume IV.
File:Euler - Institutionum calculi integralis, 1768 - BEIC 1338320 F.jpg
Aragore Click here for the 2 integrslis chapter: Concerning the resolution of other second order differential equation by infinite series. This is the most beautiful of chapters in this book to date, and one which must have given Euler a great deal of joy ; there is only one thing I suggest you do, and that is to read it. Institutionum calculi integralis — Leonhard Euler — Google Books The method is extended to forms involving the second degree. Here Euler lapses in his discussion of convergence of infinite series; part of the trouble seems to be the lack of an analytic method of approaching a limit, with which he has calvuli difficulty in the geometric situations we have looked at previously, as in his Mechanica.
Institutiones calculi integralis