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The Geometry of Physics: An Introduction ペーパーバック – 2001/1/1
'The layout, the typography and the illustrations of this advanced textbook on modern mathematical methods are all very impressive and so are the topics covered in the text.' Zentralblatt für Mathematik und ihre Grenzgebiete
- 出版社 : Cambridge University Press (2001/1/1)
- 発売日 : 2001/1/1
- 言語 : 英語
- ペーパーバック : 678ページ
- ISBN-10 : 0521387531
- ISBN-13 : 978-0521387538
- 寸法 : 2.54 x 17.78 x 25.4 cm
- Amazon 売れ筋ランキング: - 1,157,845位洋書 (の売れ筋ランキングを見る洋書)
- - 792位Differential Geometry
- - 1,341位Topology
- - 1,702位Industrial, Manufacturing & Operational Systems (洋書)
I shall have nothing critical to say about it: only caveats because
no book can satisfy the needs of all readers.
The only part of this book that I read contiguously and in detail was
the last third entitled "Lie Groups, Bundles, and Chern Forms".
The goal was to firm up my rather informal acquaintance with vector bundles
and related concepts such as connections and their curvatures.
Back in the early 1960's when I first learned differential geometry
at the University of Michigan, vector bundles were not even
in the common mathematical lexicon. I had never heard of them,
and I would be surprised to learn a significant number at that place and time
The graduate course that introduced me to differential geometry
would possibly be considered somewhat old-fashioned by modern (2016)
standards, though by 1960's standards it may have been
cutting-edge. The Cartan viewpoint of "moving frames"
which is central to Frankel's treatment of these topics was scarcely mentioned.
Because of this, in order to understand Frankel's treatment of
vector bundles, I had to frequently go back to study his treatment of
classical curvature from the Cartan viewpoint.
Thus I ended up reading most of the book, though not in full detail.
I am now a convert to the Cartan viewpoint. I think it offers
more insight than the way that I originally learned differential geometry,
and I owe this to Frankel's clear exposition.
The whole book seems clearly written, with many
and generally excellent diagrams. I found very few typos.
As a book author myself, I can appreciate and marvel at
the attention to detail and hard work which must
have gone into each of its 600-odd pages. Offhand, the only comparable
book which comes to mind is Misner, Thorne, and Wheeler's "Gravitation".
I imagine that Frankel would be an excellent introduction
to differential geometry, though I didn't read the introductory sections
in detail. However I suspect that many undergraduates may prefer a presentation
which is less abstract and closer to nineteenth century ideas, reserving
more advanced texts such as Frankel for later study.
As a graduate text or for a second course, Frankel would surely
be a reasonable choice, though I should say that I'm not enough of an
expert on differential geometry to judge the suitability
of its choice of topics in this vast field.
The book's title, "The Geometry of Physics", suggests its orientation
but might possibly mislead some. This is a mathematics text,
not a physics text. Some applications to physics are presented to illustrate
the mathematics, but the presentations are usually abbreviated and
often superficial. Physics students hoping for a direct and
seamless connection between Frankel's mathematics and
presentations of similar material in their physics courses may be disappointed.
Regarding the superficiality, there is a short introduction to
general relativity which I did not read because I was already familiar
with the subject. Therefore, I won't attempt to definitively judge
how useful it might be to a beginner, but my guess is that
most beginners would be better off with a more extensive introduction.
This should not be taken as a criticism of the book's treatment,
but instead as a caveat that it may not be what all readers might hope for.
Perhaps Frankel's 1974 book "Gravitational Curvature" (which I haven't read)
might provide a more relaxed introduction.
A summary of the above was already presented in the introductory
paragraph, but for a graceful conclusion I state it once again.
This is a superb book from which I learned much. Inevitably, it will not
meet the needs or hopes of all readers, and I have tried to indicate some
of its limitations. But I do think that almost anyone with an appropriate
background can learn much of value from this book. I doubt that many will
regret adding it to their libraries even if they don't read the whole book.