Functions of Several Variables
by Wendell Fleming, W. H. Fleming · first published 1963
80% recommend20% neutral
This new edition of FUNCTIONS OF SEVERAL VARIABLES is an extensive revision. Like the first edition it presents a thorough introduction to differential and integral calculus, including the integration of differential forms on manifolds. A new chapter on elementary topology makes the book more complete as an advanced calculus text; sections have been added introducing physical applications in thermodynamics, fluid dynamics, and classical rigid body mechanics.
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…much better. That second pass was slow, careful, where I chewed on the proofs to try to understand why they worked, took the time to develop intuition, etc. Then I took a careful pass through Fleming, Functions of Several Variables and, in my career, studied a wide variety of related topics. Then on my Ph.D. analysis qualifying exam, those passes through Rudin and Fleming got me the best score in the department. Lesson: That…
on “Princeton University Math Major Course Guide” ↗…enjoy it or even get through it. Hint: Get a really good teacher! Then for more, popular is Spivak, 'Calculus on Manifolds', mostly because it is short. Actually, it's too short. I prefer Fleming, 'Functions of Several Variables' until get to the exterior algebra chapter at which time, if care, can now get the thin Henri Cartan, {\it Differential Forms,\/} ISBN 0-486-45010-4, Dover, Mineola, NY, 2006.\ \ in English. Between…
on “Ask HN: I want to learn statistics and data mining” ↗…Rao, {\it Linear Statistical Inference and Its Applications:\ \ Second Edition,\/} ISBN 0-471-70823-2, John Wiley and Sons, New York, 1967.\ \ A good start on convexity is Wendell H.\ Fleming, {\it Functions of Several Variables,\/} Addison-Wesley, Reading, Massachusetts, 1965.\ \ Total, fun, dessert ice cream on convexity is Jensen's inequality; right away can use it to prove a lot of classic inequalities. Gee, look on the…
on “Machine Learning Crash Course” ↗…a lot of independent learning from some of the best math texts, e.g., I took a second pass through Rudin's 'Principles', went carefully through Halmos's 'Finite Dimensional Vector Spaces', Fleming's 'Functions of Several Variables', the math parts of von Neumann's 'Quantum Mechanics', and much more, a big stack more. When I went for my Ph.D., what I had learned before I entered was nearly enough for the course work. For the research,…
on “Obsession” ↗…local, non-linear generalization of what you will see really easily for linear transformations in linear algebra via, right, just Gauss elimination. For a source? There is a good treatment in W. Fleming, Functions of Several Variables. IIRC, there is also a cute proof based on contractive mappings. So, by then you will have a good start on both continuity and linearity.
on “Ask HN: Resources to learn real analysis?” ↗…here to see during the course some clarity in what is going on, how, and why; (iii) also study or glance at one or two other competitive or similar books, e.g., Spivak, Calculus on Manifolds, Fleming, Functions of Several Variables, Kolmogorov and Fomin, Apostol, and more, and (iv) take a second pass through Baby Rudin. Net, Baby Rudin has discouraged a lot of students, too many. To many students, Baby Rudin looks forbiddingly…
on “Algebra, Topology, Differential Calculus, and Optimization Theory for CS and ML [pdf]” ↗…dimensional introduction to von Neumann's Hilbert space. For more, see the books by R. Bellman or R. Horn. Work carefully through W. Rudin, 'Principles of Mathematical Analysis.' Supplement that with Fleming, 'Functions of Several Variables', Buck, 'Advanced Calculus', etc. As a crucial start on optimization, work carefully through, say, Chvatal, 'Linear Programming'. Somewhere get a good theorem proving course in ordinary differential…
on “Thinking of Doing a PhD? Take this simple test” ↗…how to get the eigen value and orthogonal eigen vector results for self-adjoint matrices from the inverse and implicit (these two go together like ice cream and cake) function theorems is in Fleming, Functions of Several Variables. Then you will be off and running on factor analysis, principle components, the polar decomposition, the singular value decomposition, and more.
on “Ask HN: Older textbooks/papers you consider classics still worth studying today?” ↗…Institute for Advanced Study. I had Rudin's book in college but later rushed to work carefully through both Halmos and Spivak ASAP after college. Instead of Spivak, I preferred: Wendell H. Fleming, 'Functions of Several Variables', Addison-Wesley, Reading, Massachusetts. Might also consider: Lynn H. Loomis and Shlomo Sternberg, 'Advanced Calculus', ISBN 0-201-04305-X, Addison-Wesley, Reading, Massachusetts. For exterior algebra,…
on “Teachers write their own online textbook, save district $175,000” ↗…metric spaces, and that work applies fully to vector spaces. ---------------------------------------- A nice place to get comfortable doing mathematics in several dimensions is Wendell H. Fleming, 'Functions of Several Variables', Addison-Wesley, Reading, Massachusetts, 1965. Some of the material here is also good for optimization. ---------------------------------------- Another place to get comfortable doing mathematics in…
on “Linear algebra tutorial in four pages” ↗…can do basic AC circuit theory like eating ice cream. Advanced calculus from one or several more traditional books, e.g., the old MIT favorite Hildebrand, Advanced Calculus for Applications, Fleming, Functions of Several Variables, Buck, Advanced Calculus -- can now look at Maxwell's equations and understand at least the math. And can work with the gradient for steepest descent in the maximum likelihood approach to machine learning.…
on “Ask HN: How or where to begin learning mathematics from first principles?” ↗…Vector Spaces. E.g., at one time, Halmos, Rudin, and Spivak, Calculus on Manifolds were the three main texts for Harvard's famous Math 55. Get good with proving the theorems. I also recommend Fleming, Functions of Several Variables. Then, sure, Royden, Real Analysis. Couldn't be prettier. If not in a hurry, then the real half of Rudin's Real and Complex Analysis. Especially if you like Fourier theory! Then of the probability…
on “Foundations of probability theory” ↗…numerical methods and applications. You can do linear programming, non-linear programming, group representation theory, multi-variate Newton iteration, differential geometry. Do look at W. Fleming, Functions of Several Variables and there the inverse and implicit function theorems and their applications to Lagrange multipliers and the eigenvalues of symmetric or Hermitian matrices. The inverse and implicit function theorems…
on “Ask HN: How do I learn math/physics in my thirties?” ↗…eigenvalues. The polar decomposition is also the source of the powerful singular value decomposition, principle components analysis, factor analysis, analysis of saddle points in optimization (see W. Fleming, Functions of Several Variables), the matrix condition number in the numerical analysis of Gauss elimination, and much more in theory and applications. Hope this helps.
on “The Calculus Trap (2005)” ↗…too seriously since are going to cover all of it and more again later. I can recommend the beginning sections on vector spaces, convexity, and the inverse and implicit function theorems in Fleming, Functions of Several Variables Fleming was long at the Brown University Division of Applied Math. The later chapters are on measure theory, the Lebesgue integral, and the exterior algebra of differential forms, and there are better…
on “Ask HN: Recommend a maths book for a teenager?” ↗