Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach (5th edition)
Abstract
This 5th edition contains all the things that made the earlier editions different from other textbooks. Among others:
Integrating linear algebra and multivariable calculus
Using effective algorithms to prove the main theorems (Newton's method and the implicit function theorem, for instance)
A new approach to both Riemann integration and Lebesgue integration
Manifolds and a serious introduction to differential geometry
A new way of introducing differential forms and the exterior derivative.
There is also new material, in response to requests from computer science colleagues:
More big matrices! We included the Perron-Frobenius theorem, and its application to Google's PageRank algorithm
More singular values! We included a detailed proof of the singular value decomposition, and show how it applies to facial recognition: "how does Facebook apply names to pictures?"