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The material is integrated to emphasize the role of linearity in all of calculus and the recurring theme of implicit versus explicit that persists in linear algebra and analysis. In the text, the author addresses all of the standard computational material found in the usual linear algebra and multivariable calculus courses, and more , interweaving the material as effectively as possible and also including complete proofs.

Chapter 2. Functions, Limits, and Continuity. Scalar- and Vector-Valued Functions. A Bit of Topology in R n. Limits and Continuity. Chapter 3. The Derivative. Partial Derivatives and Directional Derivatives. Differentiation Rules. The Gradient. Higher-Order Partial Derivatives. Implicit and Explicit Solutions of Linear Systems.

Gaussian Elimination and the Theory of Linear Systems. Elementary Matrices and Calculating Inverse Matrices. Linear Independence, Basis, and Dimension. The Four Fundamental Subspaces.


The Nonlinear Case: Introduction to Manifolds. Chapter 5. Extremum Problems. Compactness and the Maximum Value Theorem. Quadratic Forms and the Second Derivative Test. Lagrange Multipliers. Chapter 6.

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Solving Nonlinear Problems. The Contraction Mapping Principle.

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    Multiple Integrals. Polar, Cylindrical, and Spherical Coordinates. How could I resist?

    Shifrin Math 3510 Day23: Examples using different definitions of manifolds

    I found the little book for sale, bought it, and set to. The book is fairly short, after all! First lesson: when it comes to mathematics books, "short" does not necessarily mean "a quick read". This one isn't. Already on the first page or two, I realized that I'd need to slow down and read carefully.

    But golly, it was neat! It treated Fubini's theorem for Riemann integrals, for one thing, something that had been hand-waved through in my classes.

    Course MA Calculus on manifolds

    It gave a full proof of the change-of-variables theorem for multivariable integration. And then it really got going, defining differentiable manifolds, differential forms, manifolds with border, integration on chains, and getting all the way to the general Stokes Theorem. Quite a ride! Of course, I didn't get through it all. I went back to it as a graduate student, and learned a little more and a little better.

    I went back to it again when I had to teach vector calculus. I have never been able to overcome a certain disdain for the vector versions of the theorems of Gauss, Green, and Stokes, which are just special cases of the main theorem in this book. I even tried to teach future engineers about differential forms, with predictably terrible results.