A Course in Mathematical Analysis Volume 1 Edouard Goursat 9781144188199 Books
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A Course in Mathematical Analysis Volume 1 Edouard Goursat 9781144188199 Books
This text is another older (1904) gem of mathematical exposition. Goursat's First Volume is about as unique as one is able to find at this level.That level, by the way, would be--for an American student--at, or beyond (better) the Calculus Three sequence. Thus, if you've already assimilated
Calculus (single and multi-variable) at a rudimentary level, then Goursat's publication will prove a delight of further excursion into these topics.
Now, having made the mistake of purchasing the text while in high-school, I foolishly attempted to devour its contents. Not until many years later,
and many mathematics courses later, was I fully able to comprehend and appreciate what it is that Goursat is attempting. What would that be ?
Volume One (Derivatives, Integrals, Series, Geometry) reconsiders much of Single and Multi-Variable Calculus in a geometric setting. In fact,
I believe Goursat provides preliminary geometric conceptions useful as an aid to a full-fledged differential geometry text (say, of Barrett O'Neil).
And, too, let us not forget mathematician Einar Hille's words: "...differentials are one of the diffuse concepts...and have caused endless confusion and misunderstanding. It is only for functions of several variables that the concept is really important." (Analysis, Volume One, 1964, Page 212).
And, too, where Whittaker and Watson, in their Modern Analysis tome (1902) are decidedly much more advanced (and, rigorous), Goursat, on the other hand, actually writes in a more pedagogic vein. Theorems are first described in words. Take Note: Closed Intervals denoted by parentheses,
that is, modern textbooks utilize brackets for closed intervals. So, preliminaries aside, let us take a quick tour:
(1) A nice, brief exposition of the Logarithm Function, and its relationship to Jacobians, presented Page 57.
(2) Potential Equation, in curvilinear coordinates, explicitly constructed, Pages 80-83.
(3) Taylor Series--Fifty Pages, Chapter Three--goes far beyond the usual introductory accounts from elementary Calculus.
(4) Integration, another fine exposition, Chapter Four. Highlights: Dedekind Cuts (Page 141), Intermediate Value Theorem (Page 146).
Page 156 alerts the student to "...a paradox, of which the explanation later in the study of definite integrals taken between imaginary limits."
Quite an interesting discussion of arc length (Pages 161-164) and a beautiful discussion of the transcendental nature of e (Page 171).
Differentiation under the integral sign, here an exposition as lucid as any in modern dress (Page 192-194). And, for a first-rate elaboration
of approximation evaluation of Integrals, one can hardly surpass Goursat's lucid account (Pages 196-203).
(5) Fifth Chapter--Indefinite Integrals--is sure to pave the way for the more analytical Whittaker and Watson. An introduction to Rational
Functions, Hyperbolic Functions (Page 219) and Elliptic Integrals (Page 231). Geometrical considerations kept to the fore (e.g., Page 220).
(6) Sixth Chapter, double integrals and Green. Goursat describes most Theorems in descriptive words, first. Analogy is invoked (Page 261).
Again, Geometrical considerations kept to the fore ( Elemental Surface Area--Pages 275-277). Euler Gamma Function, too (Pages 279-280).
Again, nice exposition of differentiation under the integral sign (Pages 287-289) and, lovely derivation of the Stirling Approximation (Page 290).
(7) Multiple Integration is continued to the next chapter (seven). Analogy, again, invoked. (Page 309). Divergence Theorem (not, however,
in vector notation garb) on Pages 309-310. And, for those with knowledge of differential forms, Goursat provides much preliminary.
(8) Infinite Series. Always a favorite topic, arrive a bit late in the game for my tastes. Happily, they are given leisurely and extensive consideration.
That is, about one-hundred pages which retraces that which you may have learned in an elementary Calculus course, now--much better !
(9) Finally, the text culminates with applications (again and again throughout) to Geometry. Curves to Curvature and much more besides.
A lovely presentation of basics--see Page 469 for a 'derivation' of Radius of Curvature formula and later (Page 477) the Frenet Formulae
(though not in vectorial notation--all is spelled out).
There you have it. A quick tour of the contents--though, I have left much out. Needless to say, get hold of the book (in any format).
It will make for wonderful excursions in Calculus (advanced, but not too much so!).
Before I forget, the Exercises for student solution are sometimes easy, sometimes not. But, all of them are fascinating.
Enjoy this text at your leisure. Also, historical considerations (footnotes) prove enlightening.
1902: publication of Whittaker and Watson's Modern Analysis,
1904: publication of Goursat's Course Of Mathematical Analysis,
1908: publication of Hardy's Course of Pure Mathematics.
Apparently, the early decade of 1900 was an important one for Mathematical Analysis.
All three, above mentioned, are Highly Recommended.
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A Course in Mathematical Analysis Volume 1 Edouard Goursat 9781144188199 Books Reviews
This is a classic analysis text from french mathematician edouard goursat. This books covers topics such as integration, differential equation and multiple integral and etc. The proof are rigorous, and the development of proofs are much more make sense than today's delta-epsilon proofs. You could see the theorems in the book are proved in a much more natural and intellectual way. Of course delta-epsilon could bring you a "rigorous" proof too, but somethimes the development of the proof is just so awkward.
very poor facsimile. Much better to buy an old copy. The book itself is a classic.
This text is another older (1904) gem of mathematical exposition. Goursat's First Volume is about as unique as one is able to find at this level.
That level, by the way, would be--for an American student--at, or beyond (better) the Calculus Three sequence. Thus, if you've already assimilated
Calculus (single and multi-variable) at a rudimentary level, then Goursat's publication will prove a delight of further excursion into these topics.
Now, having made the mistake of purchasing the text while in high-school, I foolishly attempted to devour its contents. Not until many years later,
and many mathematics courses later, was I fully able to comprehend and appreciate what it is that Goursat is attempting. What would that be ?
Volume One (Derivatives, Integrals, Series, Geometry) reconsiders much of Single and Multi-Variable Calculus in a geometric setting. In fact,
I believe Goursat provides preliminary geometric conceptions useful as an aid to a full-fledged differential geometry text (say, of Barrett O'Neil).
And, too, let us not forget mathematician Einar Hille's words "...differentials are one of the diffuse concepts...and have caused endless confusion and misunderstanding. It is only for functions of several variables that the concept is really important." (Analysis, Volume One, 1964, Page 212).
And, too, where Whittaker and Watson, in their Modern Analysis tome (1902) are decidedly much more advanced (and, rigorous), Goursat, on the other hand, actually writes in a more pedagogic vein. Theorems are first described in words. Take Note Closed Intervals denoted by parentheses,
that is, modern textbooks utilize brackets for closed intervals. So, preliminaries aside, let us take a quick tour
(1) A nice, brief exposition of the Logarithm Function, and its relationship to Jacobians, presented Page 57.
(2) Potential Equation, in curvilinear coordinates, explicitly constructed, Pages 80-83.
(3) Taylor Series--Fifty Pages, Chapter Three--goes far beyond the usual introductory accounts from elementary Calculus.
(4) Integration, another fine exposition, Chapter Four. Highlights Dedekind Cuts (Page 141), Intermediate Value Theorem (Page 146).
Page 156 alerts the student to "...a paradox, of which the explanation later in the study of definite integrals taken between imaginary limits."
Quite an interesting discussion of arc length (Pages 161-164) and a beautiful discussion of the transcendental nature of e (Page 171).
Differentiation under the integral sign, here an exposition as lucid as any in modern dress (Page 192-194). And, for a first-rate elaboration
of approximation evaluation of Integrals, one can hardly surpass Goursat's lucid account (Pages 196-203).
(5) Fifth Chapter--Indefinite Integrals--is sure to pave the way for the more analytical Whittaker and Watson. An introduction to Rational
Functions, Hyperbolic Functions (Page 219) and Elliptic Integrals (Page 231). Geometrical considerations kept to the fore (e.g., Page 220).
(6) Sixth Chapter, double integrals and Green. Goursat describes most Theorems in descriptive words, first. Analogy is invoked (Page 261).
Again, Geometrical considerations kept to the fore ( Elemental Surface Area--Pages 275-277). Euler Gamma Function, too (Pages 279-280).
Again, nice exposition of differentiation under the integral sign (Pages 287-289) and, lovely derivation of the Stirling Approximation (Page 290).
(7) Multiple Integration is continued to the next chapter (seven). Analogy, again, invoked. (Page 309). Divergence Theorem (not, however,
in vector notation garb) on Pages 309-310. And, for those with knowledge of differential forms, Goursat provides much preliminary.
(8) Infinite Series. Always a favorite topic, arrive a bit late in the game for my tastes. Happily, they are given leisurely and extensive consideration.
That is, about one-hundred pages which retraces that which you may have learned in an elementary Calculus course, now--much better !
(9) Finally, the text culminates with applications (again and again throughout) to Geometry. Curves to Curvature and much more besides.
A lovely presentation of basics--see Page 469 for a 'derivation' of Radius of Curvature formula and later (Page 477) the Frenet Formulae
(though not in vectorial notation--all is spelled out).
There you have it. A quick tour of the contents--though, I have left much out. Needless to say, get hold of the book (in any format).
It will make for wonderful excursions in Calculus (advanced, but not too much so!).
Before I forget, the Exercises for student solution are sometimes easy, sometimes not. But, all of them are fascinating.
Enjoy this text at your leisure. Also, historical considerations (footnotes) prove enlightening.
1902 publication of Whittaker and Watson's Modern Analysis,
1904 publication of Goursat's Course Of Mathematical Analysis,
1908 publication of Hardy's Course of Pure Mathematics.
Apparently, the early decade of 1900 was an important one for Mathematical Analysis.
All three, above mentioned, are Highly Recommended.
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