EULER'S
This marks the completion of part 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. one
third of Euler's monumental Foundations
of Differential Calculus : this is not really much help to me, as I would
have to refer readers to texts that might not be available to them, and even
initially, I have had to delve into Ch.18 of Part II of the latter book to
obtain explanations of the formulae used in the first chapter of the
integration, which was our main concern here initially. 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.s.a.p. 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. These works are available in the
public domain on the Euler Archive website and from Google Books ; I have made the corrections suggested from
time to time by the editors in the Opera
Omnia edition, to all of whom I express my thanks. The work is divided as
in the first edition and in the Opera Omnia into 3 volumes. All the chapters
presented here are in the books of Euler's original treatise, which corresponds
to Series I volumes 11,12 & 13 of the O.O.
edition. 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 Investigation of Functions of
Two Variables from a Relation of the Differentials of the First Order.
Click here for the 1st chapter : Concerning the nature of differential equations, from which functions of two variables are determined in general. Euler establishes a criterion for the total derivative of functions of two variables using the idea of a multiplier, which is quite ingenious, and which can be shown to be equivalent to a well-known vector identity. 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. Euler establishes the solution to a number of equations in which p = (dz/dx) is the partial derivative first of a constant, then a function of x, then a function of x and y, and finally a function of x, y, and z. 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. The historian of mathematics will be interested to know that the arbitrary form of the function f : x related initially to the shape of a stretched string, as described here.
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. Perhaps they are equal, depend on each other in a linear manner, or there is some other simple relation between p and q, etc.
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.e. considered. 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.e.
Click here for the 5th chapter : Concerning the resolution of equations in which a relation is given between the two differential formulas (dz/dx) , (dz/dy) and any two of the three variables x, y, and z. This chapter is a continuation of the methods introduced in ch.4 above. 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.
Click here for the 6th chapter : Concerning the resolution of equations in which some relation is given between the two differential formulas (dz/dx) , (dz/dy) and all three variables x, y, and z. 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.
Volume III, Section II.
The Investigation of Functions of
Two Variables from a Relation of the Differentials of the Second Order.
Click here for
the 1st chapter : Concerning Differential Formulas of the
Second Order in General.
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!
Click here for
the 2nd chapter : In which a Single Formula of the Second
Order Differential is given in terms of some other remaining quantities.
This is a reasonably straight-forwards chapter in which techniques employed before are given new territory ; essentially the equation d2z/dx2 , d2z/dxdy , or d2z/dy2 = some function of dz/dx and other possible functions of x and y are integrated in general, with the customary examples.
Click here for
the 3rd chapter : In which two or all of the Formulas of the
Second Order are given in terms of some
other remaining quantities.
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.
Click here for
the 4th chapter : A Special Method of Integrating Equations of
this kind in another way.
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.
Click here for
the 5th chapter : A Special Method of Integrating Equations of
this kind in another way.
This is also a very long but very interesting chapter wherein Euler develops a transformation, initially of the first order differentials, whereby the solution of a second differential equation in terms of the unknown z can be found from the solution of a similar equation in the variable v, related by the form z = (Mdv/dx) + Nv. The main part of the chapter is taken up finding appropriate values for s = M/N. The method is extended to forms involving the second degree. This is the last chapter in this section.
Volume III, Section III.
The Investigation of Functions of
Two Variables from a Relation of the Differentials of the Third of Higher Order.
Click here for
the 1st chapter : Concerning the Resolution of the Simplest
Differential Formulas involving a Differential Formula.
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.
Click here for
the 2nd chapter : Concerning the Integration of Higher Order Differential
Equations by Reduction to Lower Orders.
This is a another short chapter in which Euler explores relatively easily solved differential equations by using a transformation based on z(x, y) = eaxv(x, y).
Click here for
the 3rd chapter : Concerning the Integration of Homogeneous
Equations where the Individual Terms contain Differential Formulas of the same
Order.
This is another short chapter, as Euler reaches the bounds of his knowledge. Here initially he investigates the solution of the equation A(ddz/dx2) + B(ddz/dxdy) + C (ddz/dy2) =0 , where he finds a primitive first order form and an algebraic equation. The method is extended to more general cases.
Volume III, Section IV.
The Investigation of Functions of
Three Variables from a Given Relation of the Differentials.
Click here for
the 1st chapter : Concerning Differential Formulas Involving
Functions of Three Variables.
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.
Click here for
the 2nd chapter : Concerning the Finding of Functions of Three
Variables from the Value of a Certain Differential Formula.
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.
Click here for
the 3rd chapter : Concerning the Resolution of Differential
Equations of the First Order.
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.
Click here for
the 4th chapter : Concerning the Resolution of Homogeneous
Differential Equations.
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 1793 ran to four volumes, where additions to most chapters were put in place in volume IV.
Volume III, APPENDIX.
THE CALCULUS OF VARIATIONS
Click
here for the 1st chapter : Concerning the Calculus of
Variations in General.
In this chapter there are no formulas : Euler devotes considerable space to explaining the nature of variations, which are to be applied both to the differential as well as to the integral calculus in the chapters that follow.
Click
here for the 2nd chapter : Concerning the Variations
of Differential Formulas involving two Variables.
In this chapter Euler sets out the ground work for his treatment, where a single order of the variation (delta) δ is considered. The differential operator d selects neighbouring points on a line or surface, and may run to any order we please, while (delta) δ selects points on neighbouring lines or surfaces of the same kind; it is shown initially that d and delta commute, and the variations of p, q, r, etc. , defined already can be investigated in terms of differentials.
Click
here for the 3rd chapter : Concerning the Variations
of Simple Integral Formulas involving two Variables.
In this chapter the groundwork is established for calculating the variation of integrals involving two variables. This follows on from above, and is extended to some intriguing results regarding the integrability of integrals.
Click
here for the 4th chapter : Concerning the Variations
of Complicated Integral Formulas involving two Variables.
In this chapter the method produced above is applied to more complicated cases, where one or more additional intermediate integrals are inserted in evaluating the variation of the integral Vdx, so that V becomes a function of an intermediate variable also expressed by an integral.
Click
here for the 5th chapter : Concerning the Variations
of Integral Formulas involving three Variables and including two relations.
In this chapter the method of the previous chapters is extended further to include two independent functions y and z of x, and the variations thereof; this is further extended to the case of an extra variable v defined via an integral. The final two chapters follow the same sort of procedures.
Click
here for the 6th chapter : Concerning the Variations
of Integral Formulas involving three Variables the relation of which is
contained by a single relation.
Click
here for the 7th chapter : Concerning the Variations
of Integral Formulas involving three Variables in which one is considered as a
function of the other two.
Volume III, APPENDIX.
SUPPLEMENT ON THE INTEGRATION OF CERTAIN SPECIAL DIFFERENTIAL
EQUATIONS
Click
here for the supplement.
In this final supplement, Euler expands on his method of integrating by the use of multipliers, which he extends to wider and wider classed of functions of two variables; one can see the tentative origins of transforms introduced as Euler grapples with this problem, in his attempt to reduce integrals to known forms for which he can evaluate the complete integral.
Ian
Bruce. August 16,
2010, latest revision. Copyright : I
reserve the right to publish this translated work in book form. However, if you
are a student, teacher, or just someone with an interest, you can copy part or
all of the work for legitimate personal or educational uses. Please feel free
to contact me if you wish by clicking on my name, especially if you have any
relevant comments or concerns.