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This is the start of a large project that will take a year or two to complete : yet I feel that someone should do it in its entirety, since Euler's calculus works are interconnected in so many ways, as one might expect, and Euler had a habit of returning to earlier ideas and making improvements.
John D. Blanton has already translated Euler's Introduction to Analysis and approx. 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.
Euler's abilities seemed to know no end, and in these texts well ordered formulas march from page to page according to some grand design. 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. Click here for some introductory material , in which Euler defines integration as the inverse process of differentiation.
A large part of Ch. This is now available below in its entirety. Euler finds ways of transforming irrational functions into rational functions which can then be integrated. He makes extensive use of differentiation by parts to reduce the power of the variable in the integrand. Click here for the 3 rd Chapter : Concerning the integration of differential formulas by infinite series.
Particular simple cases involving inverse trigonometric functions and logarithms are presented first. Following which a more general form of differential expression is integrated, applicable to numerous cases, which gives rise to an iterative expression for the coefficients of successive powers of the independent variable. Finally, series are presented for the sine and cosine of an angle by this method.
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 no difficulty in the geometric situations we have looked at previously, as in his Mechanica. Click here for the 4 th Chapter : Concerning the integration of differential formulas involving logarithmic and exponential functions.
Particular simple cases involving logarithmic functions are presented first; the work involves integration by parts, which can be performed in two ways if needed. Progressively more difficult differentials are tackled, which often can be integrated by an infinite series expansion. A new kind of transcendental function arises here. Those who delight in such things can see the exponential function set out as we know it, and various integrations performed, including the derivation of some very cute series, as Euler himself notes in so many words.
Click here for the 5 th Chapter : Concerning the integration of differential formulas involving angles or the sines of angles. Again, particular simple cases involving sines or powers of sines and another function in a product are integrated in two ways by the product rule for integrals.
This leads to the listing of numerous integrals, on continuing the partial integrations until simple integrals are arrived at; the chapter culminates with the sine and cosine function being linked to an exponential function of the angle ; the case where such an exponent disappears on summing to infinite is considered. Click here for the 6 th Chapter : Concerning the development of integrals in series progressing according to multiple angles of the sine or cosine.
Much labour is involved in creating the coefficients of the cosines of the multiple angles. This chapter is thus heavy in formulas; recursive relations of the second order are considered; means of evaluating the coefficients from infinite sums are considered; all in all a rather heady chapter, some parts of which I have just presented, and leave for the enthusiast to ponder over.
Click here for the 7 th Chapter : A general method by which integrals can be found approximately. This chapter starts by considering the integral as the sum of infinitesimal strips of width dx, from which Euler forms upper and lower sums or bounds on the integral, for a dissection of the domain of integration into sections. This lead to an improved method involving successive integration by parts, applied to each of the sections, and leading to a form of the Taylor expansion, where the derivatives of the integrand are evaluated at the upper ends of the intervals.
This method is applied to a number of examples, including the log function. Various cases where the integral diverges are considered, and where the divergence may be removed by transforming the integrand. The even powers depend on the quadrature of the unit circle while the odd powers are algebraic.
Products of the two kinds are considered, and the integrands are expanded as infinite series in certain ways. Click here for the 9 th Chapter : Concerning the development of integrals as infinite products.
Euler proceeds to investigate a wide class of integral of this form, relating these to the Wallis product, etc. Eventually he devises a shorthand way of writing such infinite products or their integrals, and investigates their properties on this basis.
One might presume that this was the first extensive investigation of infinite products. This chapter ends the First Section of Book I. Click here for the 1 st Chapter : Concerning the separation of variables. The focus now moves from evaluating integrals treated above to the solution of first order differential equations.
You should find most of the material in this chapter to be straightforward. Euler finds to his chagrin that there is to be no magic bullet arising from the separation of the variables approach, and he presents an assortment of methods depending on special transformations for particular families of first order differential equations; he obviously spent a great deal of time examining such cases and this chapter is a testimony to these trials. Click here for the 2 nd Chapter : Concerning the integration of differential equations by the aid of multipliers.
Euler now sets out his new method, which involves finding a suitable multiplier which allows a differential equation to become an exact differential and so be integrated. This chapter relies to some extend on Ch. Euler refers to such differential equations as integral by themselves; examples are chosen for which an integrating factor can be found, and he produces a number of examples already treated by the separation of variables technique, to try to find some common characteristic that enables such equations to be integrated without first separating the variables.
This task is to be continued in the next chapter. Click here for the 3 rd Chapter : Concerning the investigation of differential equations which are rendered integrable by multipliers of a given form. Euler moves away from homogeneous equations and establishes the integration factors for a number of general first order differential equations.
The technique is to produce a complete or exact differential, and this is shown in several ways. For example, the d. A general method of analyzing integrating factors in terms of consecutive powers equated to zero is presented.
There is much material and food for thought in this Chapter. Click here for the 4 th Chapter : Concerning the particular integration of differential equations. Euler declares that while the complete integral includes an unspecified constant: the particular integrals to be defined and investigated here may relate to the existence of solutions where the values of the added constant is zero or infinity, and in which cases the solution, perhaps found by inspection, degenerates into an asymptotic line, in which no added constant is apparent.
Other situations to be shown arise in which an asymptotic line is evident as a solution, while some solutions may not be valid. A number of situations are examined for certain differential equations, and rules are set out for the evaluation of particular integrals.
This is a most interesting chapter, in which Euler cheats a little and writes down a biquadratic equation, from which he derives a general differential equation for such transcendental functions. From the general form established, he is able after some effort, to derive results amongst other things, relating to the inverse sine, cosine, and the log.
More general differential equations of the form discussed are gradually introduced. This is a continuation of the previous chapter, in which the mathematics is more elaborate, and on which Euler clearly spent some time.
It seems best to quote the lad himself at this point, as he put it far better than I, in the following Scholium :. Now here the use of this method, which we have arrived at by working backwards from a finite equation to a differential equation, is clearly evident. Whereby we shall set out this argument more carefully.
Click here for the 7 th Chapter : Concerning the approximate integration of differential equations. In this final chapter of this part, a number of techniques are examined for the approximation of a first order differential equation; this is in addition to that elaborated on above in Section I, CH.
Click here for the single chapter : Concerning the resolution of more complicated differential equations. The resolution of differential equations of the second order only. Click here for the 1 st chapter : Concerning the integration of simple differential formulas of the second order.
At this stage the exclusive use of the constant differential dx , which can be seen in the earlier work of Euler via Newton is abandoned, so that ddx need not be zero, and there are now four variables available in solving second order equations : p, q, x , and y. Euler admits that this is a more powerful method than the separation of variables in finding solutions to such equations, where some differential quantity is kept constant. Click here for the 2 nd chapter : Concerning second order differential equations in which one of the variables is absent.
The idea of solving such equations in a step—like manner is introduced; most of the equations tackled have some other significance, such as relating to the radius of curvature of some curve, etc.
Click here for the 3 rd chapter : Concerning homogeneous second order differential equations, and those which can be reduced to that form. In these examples a finite equation is obtained between some of the variables, as x disappears. Click here for the 4 th chapter : Concerning second order differential equations in which the other variable y has a single dimension. A lot of familiar material is uncovered here, perhaps in an unusual manner : for example, we see the origin of the particular integral and complementary function for integrals of this kind.
Click here for the 5 th chapter : Concerning the integration by factors of second order differential equations in which the other variable y has a single dimension. Click here for the 6 th chapter : Concerning the integration of other second order differential equations by putting in place suitable multipliers.
This is a harder chapter to master, and more has been written by way of notes by me, though some parts have been left for you to discover for yourself. The methods used are clear enough, but one wonders at the insights and originality of parts of the work.
The use of more complicated integrating factors is considered in depth for various kinds of second order differential equations. How much of this material is available or even hinted at in current texts I would not know; it seems to be heading towards integral transforms, where the integral of the transformed equation can be evaluated, and then the inverse transform effected : but this latter operation is not attempted here. This chapter is rather labour intensive as regards the number of formulas to be typed out; however, modern computing makes even this task easier.
The relatively easy task of setting up an infinite series for the integral chosen is accomplished; after which considerable attention is paid to series that end abruptly due to the introduction of a zero term in the iteration, thus providing algebraic solutions.
Euler had evidently spent a great deal of time investigating such series solutions of integrals, and again one wonders at his remarkable industry. Recall that this book was meant as a teaching manual for integration, and this task it performed admirably, though no thought was given to convergence, a charge often laid. Click here for the 8 th chapter : Concerning the resolution of other second order differential equation by infinite series.
This chapter is also rather labour intensive as regards the number of formulas to be typed out; here a more general second order differential equation is set up and integrated by a series expansion. The emphasis is now on degenerate cases, which arise when the roots of the indicial equation are equal or imaginary, and the ln function is introduced as a multiplier of one of the series; there is a desire to obtain the complete integral for these more trying cases.
Click here for the 9 th chapter : Concerning the resolution of other second order differential equation of the form. This is a most interesting chapter, in which other second order equations are transformed in various ways into other like equations that may or may not be integrable. It builds on the previous chapter to some extent, and ends with some remarks on double integrals, or the solving of such differential equations essentially by double integrals, a process which was evidently still under development at this time.
Click here for the 10 th chapter : On the construction of second order differential equations from the quadrature of curves.
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