I'm fond of baby Rudin: elegant presentation, "clever" proofs,certain terseness, difficult exercises, etc. I'm working up my enthusiasm to seek similar textbooks in other areas of math, e.g. linear algebra, abstract algebra, ODE, number theory. Below are some books I want to exclude:
Fundamentals Of Linear Algebra Katsumi Nomizu Pdf 46
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Linear Algebra -- Lang's Linear Algebra is often overshadowed by his much more popular book "Algebra", but it is an amazingly written text on linear algebra of arbitrary vector spaces. I learned linear algebra from the 2nd edition, and it was unbelievably well written and rigorous. It is not as difficult as his book on abstract algebra, but it is by no means easy. (Note: The later editions changed the writing style up a bit so I would recommend looking at one of the first few editions).Also, Greub's Linear Algebra (the predecessor of his Multilinear Algebra book discussed below) is an extremely comprehensive, and tersely written, treatment, which is accompanied by difficult exercises
This inexpensive text covers the basics of linear algebra and serves as an excellent introduction to proof-based mathematics. This is an essential read for scientists of all disciplines; the applications are immediate and uniform across the sciences.
Differential geometry is a general and powerful psychological aid to not only physics in curved space, but calculus in flat space with arbitrary curving coordinates. Its utility is apparent during Landau 2, when justifying the formulae for changes of coordinates of tensor quantities. The first chapter serves as an advanced mini course on multilinear algebra, extending the material from Shilov. The remainder of the book constructs geometric quantities from space rather than numerically, and produces a theory of integration, differentiation, optimization which re-instantiate into familiar calculus when represented locally with coordinates. The material then shifts to consider lie groups and their basic uses for spatial structure and calculus, to be regarded as the polytopes of the continuous differentiable world.
There are essentially three books here. Although there never seems to be a good time to start algebra, it quickly becomes needed either for simplification or explanation of naturally arising structures everywhere. When your need for algebra is strong, consider picking the appropriate and helpfully self-contained sections. The basic objects of algebra, algebraic equations, representation theory of finite groups and related linear algebra. Applications are numerous and deserve a separate blog post.
Algebraic Geometry is foremost about the geometries of solutions to algebraic equations and other locally ringed topological spaces whose functions also behave like such solutions. The subject is most easily studied in analogy to differential geometry, and insights cross feed since both kinds of spaces (manifolds and schemes) can be constructed as dual to certain kinds of rings. One fundamental analogy worth noting is that both classes of geometries have a notion of fundamental groups and covering spaces, where in the algebraic case, fundamental groups are a general case of Galois groups in algebra. At the center of the abstract geometric formalism are still concrete cases of solutions to equations over algebraically closed fields like the complex numbers in linear/affine or projective space called varieties. The later parts of the book concern themselves with historical problems on elementary and easy to see examples using the powerful sheaf and cohomology formalism built in the intermediate chapters. It is also worth noting that algebraic and differential structures can and have been combined, and are a modern topic of study stemming from quintic threefolds such as Calabi-Yau.
Written primarily for students who have completed the standard first courses in calculus and linear algebra, ELEMENTARY DIFFERENTIAL GEOMETRY, REVISED SECOND EDITION, provides an introduction to the geometry of curves and surfaces. 2ff7e9595c
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