Is Spivak calculus necessary?
If you want to learn calculus Spivak is a good book, but it is not necessary or sufficient to read it. Most people would read an easier book before Spivak and some other books because Spivak does not cover all topics, covers some topics in a strange way, and it not general enough.
What is a manifold in differential geometry?
Manifolds. A differentiable manifold is a Hausdorff and second countable topological space M, together with a maximal differentiable atlas on M. Much of the basic theory can be developed without the need for the Hausdorff and second countability conditions, although they are vital for much of the advanced theory.
Why is Spivak Calculus good?
Spivak’s Calculus is a classic for a reason. It provides an accessible yet rigorous introduction to analysis. The exercises in Calculus are often a wondrous intersection of insightful and challenging. Moreover, there is a very good set of lecture notes which more or less follow Spivak’s text.
Why do we study manifolds?
Manifolds are important objects in mathematics and physics because they allow more complicated structures to be expressed and understood in terms of the relatively well-understood properties of simpler spaces.
Who discovered manifolds?
Poincaré pioneered the study of three-dimensional manifolds and raised a fundamental question about them, today known as the Poincaré conjecture. After nearly a century, Grigori Perelman proved the Poincaré conjecture (see the Solution of the Poincaré conjecture).
What is @calcalculus on manifolds?
Calculus on Manifolds is a brief monograph on the theory of vector-valued functions of several real variables (f : Rn→Rm) and differentiable manifolds in Euclidean space.
What is the best book to learn calculus on manifolds?
Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (1965) by Michael Spivak is a brief, rigorous, and modern textbook of multivariable calculus, differential forms, and integration on manifolds for advanced undergraduates.
Is Spivak’s analysis on manifolds a preface to calculus on manifold?
Nevertheless, Munkres acknowledges the influence of Spivak’s earlier text in the preface of Analysis on Manifolds. Spivak’s five-volume textbook A Comprehensive Introduction to Differential Geometry states in its preface that Calculus on Manifolds serves as a prerequisite for a course based on this text.
Is calculus on manifolds a prerequisite for differential geometry?
Spivak’s five-volume textbook A Comprehensive Introduction to Differential Geometry states in its preface that Calculus on Manifolds serves as a prerequisite for a course based on this text. In fact, several of the concepts introduced in Calculus on Manifolds reappear in the first volume of this classic work in more sophisticated settings.