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An explanation of the mathematics needed as a foundation for a deep
understanding of general relativity or quantum field theory.
Physics is naturally expressed in mathematical language. Students
new to the subject must simultaneously learn an idiomatic
mathematical language and the content that is expressed in that
language. It is as if they were asked to read Les Miserables while
struggling with French grammar. This book offers an innovative way
to learn the differential geometry needed as a foundation for a
deep understanding of general relativity or quantum field theory as
taught at the college level. The approach taken by the authors (and
used in their classes at MIT for many years) differs from the
conventional one in several ways, including an emphasis on the
development of the covariant derivative and an avoidance of the use
of traditional index notation for tensors in favor of a
semantically richer language of vector fields and differential
forms. But the biggest single difference is the authors'
integration of computer programming into their explanations. By
programming a computer to interpret a formula, the student soon
learns whether or not a formula is correct. Students are led to
improve their program, and as a result improve their understanding.
The new edition of a classic text that concentrates on developing
general methods for studying the behavior of classical systems,
with extensive use of computation. We now know that there is much
more to classical mechanics than previously suspected. Derivations
of the equations of motion, the focus of traditional presentations
of mechanics, are just the beginning. This innovative textbook, now
in its second edition, concentrates on developing general methods
for studying the behavior of classical systems, whether or not they
have a symbolic solution. It focuses on the phenomenon of motion
and makes extensive use of computer simulation in its explorations
of the topic. It weaves recent discoveries in nonlinear dynamics
throughout the text, rather than presenting them as an
afterthought. Explorations of phenomena such as the transition to
chaos, nonlinear resonances, and resonance overlap to help the
student develop appropriate analytic tools for understanding. The
book uses computation to constrain notation, to capture and
formalize methods, and for simulation and symbolic analysis. The
requirement that the computer be able to interpret any expression
provides the student with strict and immediate feedback about
whether an expression is correctly formulated. This second edition
has been updated throughout, with revisions that reflect insights
gained by the authors from using the text every year at MIT. In
addition, because of substantial software improvements, this
edition provides algebraic proofs of more generality than those in
the previous edition; this improvement permeates the new edition.
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