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» » Applied Differential Geometry
Applied Differential Geometry


William L. Burke


Applied Differential Geometry


Science & Math

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Cambridge University Press; 1 edition (May 31, 1985)







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Applied Differential Geometry by William L. Burke

This is a self-contained introductory textbook on the calculus of differential forms and modern differential geometry. The intended audience is physicists, so the author emphasises applications and geometrical reasoning in order to give results and concepts a precise but intuitive meaning without getting bogged down in analysis. The large number of diagrams helps elucidate the fundamental ideas. Mathematical topics covered include differentiable manifolds, differential forms and twisted forms, the Hodge star operator, exterior differential systems and symplectic geometry. All of the mathematics is motivated and illustrated by useful physical examples.
One might consider this book to be "Part One" of his two-part opus:
His earlier (1980) text entitled Spacetime,Geometry,Cosmology is the ideal supplement to this (1985) text !
That being said, his earlier book is easier than the later book, but that really depends on one's perspective.
This text is a goldmine of thought. From the frontispiece we read Burke's real motive behind it all:
"To all those who, like me, have wondered how in hell you can change dq/dt without changing q."
(Ah, that is a question that has bugged me for many a year, not simply because Goldstein adds little clarity
to said query, but one that apparently professors never feel the need to discuss--at least, not one I have encountered !).
Then again, reading Burke's motives : "...helps us to avoid the formal symbol manipulations that so often lead to the
horrible calculus of variations manipulations and mistakes in Goldstein..." (Preface).
In any event, Burke here offers his unique perspective on a subject which--at the time,1985--was hardly in prominence.
That subject being applications of differential forms to many a facet of theoretical physics; applications including:
Dynamics of Particles and Fields,Thermodynamics,Electrodynamics,Special and General Relativity.
If your preparation is (at least) a standard undergraduate mathematics curriculum along with some physics, then the text
should be within compass (partial derivatives, simple differential equations, a bit of vector analysis should be sufficient).
Diagrams are copious. Preface: "...often they can be substituted for the verbal hints that sometimes constitute a proof..."
There are innovations of Pedagogy:
(1) Uses of Covariance: here, a one-two-three build-up, beginning with Doppler Shift formula beautifully justified.
Follow through as, step-by-step, Burke proceeds from that shift to aberration to dihedral products to spacetime products.
(2) Leibniz rule for derivatives of products alongside Taylor expansions lead effortlessly to Lie Derivatives:
We read: "...there is a vector concealed in the second derivatives..." (Page 69).
(3) Remember the rotating disc ? Sometimes in use to justify going from special to general relativity.
Here, Burke will show how the 'velocity-space' is isometric to the 3-pseudo-sphere previously defined (Pages 75-77).
Reading: "This metric is curved. It is not a spacetime metric, however, and its curvature has no more to do with
Einstein's field equations than does the curvature of the velocity-space metric."
(4) Bundles are everywhere. You will delight in the discussion of cotangent bundles, fiber bundles, tangent bundles.
The tour de force is discussion of Thermodynamics,read : "Thermodynamics is a contact bundle." (Page 109).
(5) Calculation of Lie derivative of a vector field is accomplished twice, and, as Burke states:
"I will first calculate in a totally pedestrian manner. A more elegant calculation will then follow." (Pages 124-125).
(6) Contact Transformations are put to great use: transform nonlinear partial differential equation into a linear one.
Again, beautiful applications to special relativity (Page 140) and thermodynamics (Page 141) follow.
(7) The meat of the mathematics is Chapter Four: Differential Forms.
We read: "The antisymmetry of forms is important and useful." (Page 148); "Tensors are Linear Operators." (Page 155),
We learn of Laplace's Equation, Linear Wave Equations, where "exterior calculus provides a natural language" (Page 166).
The highlight of this chapter is the discussion of block indices, segueing to beautiful exposition of determinants.
Also, we learn of Weyl's introduction of twisted tensors. This leads into the discussion of twisted differential forms.
Integration, in Section 29, particularly interesting. It concludes with discussion of cohomology (All too brief, I fear).
What is learned in Chapter Four is placed into service in Chapter Five:
(8) Applications. And, what a wonderful excursion. Hard to overestimate the efficacy of this chapter.
Burke: "It is a bit of physics lore that to every symmetry there corresponds a conservation law and
vice versa....a conservation law does always lead to a potential, however." (Page 264).
Also, we read, concluding the chapter: "Use the right tool for the job, even if it is encrusted with indices."
(9) Electrodynamics, next. Here, c is set equal to one and Gaussian units utilized.
(Note: A typo in the last equation, second term, Page 278, which is then rectified on next line of Page 279 !).
Burke: "Note how important it is that the inhomogeneous equations involve twisted forms." (Page 282).
(10) Classical Mechanics,next. Lagrange and Hamilton. "The first step is to get into the right space."
We read: "One goal of this book is to eliminate all such funny deltas." That is, to compare Burke to Goldstein !
(Note Page 313 and 314, where Burke says, again: "Goldstein is wrong," regards anholonomic constraints).
We read: "It is often said that Lagrange's equations are n second-order equations, whereas Hamilton's
equations are 2n first-order equations. This is nonsense....."(Page 320).
The chapter concludes with exposition of wave-packet dynamics. More about that can be found in his earlier book.
(11) A favorite, Calculus on Fiber Bundles: "...covariant differentiation...that name can be misleading." (Page 347).
From Linear Connection to Parallel Transport, then to Curvature. All prelude to final chapter on Gravitation.
The final two chapters, then, form somewhat of a concerted sequence. (As with chapters four and five).
And, if the final two chapters leave you wanting more, proceed to his earlier--even more inspiring--1980 text.
I found few typographic errors. None of any import--none that would deter from the efficacy of the book.
Read with paper and pencil, fill in the intermediate steps. Derivations are often left to the reader.
A few (yet, informative !) problems for student involvement are scattered throughout the prose:
"Are there any conformal symmetries of the heat equation ? (Page 213).
"What conservation law comes from the field equation with source terms ? (Page 269).
"Where is the angular momentum in a circularly polarized plane electromagnetic wave ? (Page 293).
"Repeat the discussion of the conducting fluid in a magnetic field using differential forms." (Page 304).
Burke offers insight on every page: whether a pithy statement or well-drawn geometric figure.
Highly recommended for collateral enrichment.
pro: good introductory book to physicist. The words are far from rigorous in the mathematical sense, yet provide good intuition. Strongly recommend this book you are just curious about using differential geometry as a tool, without diving too much into the mathematics.
con: Sometimes the the writes mentions with just one sentence an important property or intuition. But if you are not already familiar with the fact he is trying to say, you need quite an amount of effort to figure it out.
A unique book. Changes the way one thinks about geometry. The concepts and tools become second nature. I strongly recommend it for engineers who need differential geometry in their research (they do, whether they know it or not).

To give an example from page 134: "Vector fields that do not commute are called anholonomic. If two transformations commute, then the system would never leave a 2-surface. This obvious results is called the Frobenius Theorem."

Now after reading about the Frobenius Theorem elsewhere, few people would call in "obvious." Nonetheless, when you read Burke, you will agree. (Granted, it will not happen at first reading unless you are already familiar with the material. So you will read the book several times, which only adds to the pleasure.) Afterwards, you will be happy to consult the proof elsewhere.

Caveat: this book is not the place to go for a formal presentation. It may cause conniptions in the more ideological bourbakistes. Nothing should prevent one from also reading some of the excellent texts that present the material in a precise way, for instance those by Manfredo Perdig√£o do Carmo, Spivak, or Lang. Nonetheless, Burke is the one to go for the intuition.
Swift Summer
"Although William Burke left this world - albeit prematurely - his book is still with us, today as a solid teacher of Differential Geometry. Few books have the depth and clarity required as an introduction.

Burke was one of the few that both understood the value of twisted tensors and was deft at teaching Differential Geometry using forms with torsion.

Few have been able to simplify Schouten forms for undergrads.

Perhaps that was William Burke's greatest academic achievement - bringing Differential Geometry mana from the heavens to students unaware of the benefits.

Burke's other tome 'Spacetime, Geometry, Cosmology' is also suitable as a reference for undergrads. ADG, here, is very flexible as a reference, and grad student primer.

He was also a godfather to the Chaos Cabal and fostered the Eudaemons: [...]
The previous review is amazingly perceptive into Bill Burke's personality and thinking. He was not the most discplined writer or lecturer, (I had no less than 4 courses from him) but his insight and intuition could be amazing. I would recommend this book as a companion to something more traditional. If you are interested in General Relativity, which is what the book was suppose to be a precursor for, get Schutz or Misner, Thorne and Wheeler, or Wald.
Also, if you do want this book, get the errata from Burke's webpage, quite helpful.
I would also hearitly recommend Burke's best book: Geometry, Spacetime and Cosmology which is out of print. It is much physical and the examples are clearer. He taught english majors and theater students general relativity with that book.
The book does not make any sense to me. Explanation is fuzzy and the text is full of typos.
very good

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