In this continuation volume, the author discusses fundamental questions of analysis that were postponed in the first volume and completes the topic of differential calculus: Taylor expansion, critical points and Hessian matrix, inverse theorem and implicit functions, extrema under constraints. This is followed by vector analysis and calculus of variations based on Hamilton's principle of classical mechanics. The volume is aimed at physics students in their second semester. Supplements, footnotes, exercises, and many figures assist in working through the material. The content of this second volume is divided into five parts, each consisting of three chapters. The first part (Chapters 23 - 25) is fundamental in nature and leads from the axioms of real numbers to Taylor expansion in multiple variables. The second part addresses the core area of differential calculus in multiple variables, starting from the inverse theorem.