Differential and Integral Calculus (Implicit Functions, Stieltjes Integrals and Curvilinear Integrals)

The book contains six chapters. In Chapter 1, the set Rn is defined. n-dimensional ball, sphere, and rectangular neighborhood of a point are defined. Sequences in Rn are defined by their properties. Open, closed and compact sets in Rn are investigated. The Heine-Borel theorem is formulated and proved. In Chapter 2, functions of several variables are introduced. Limits of functions of several variables are defined with their properties are deducted. Continuous functions of several variables are investigated. Uniform continuity is introduced and explored. In Chapter 3, partial derivatives of higher order and differentials for functions of several variables and their properties are introduced. Criteria for the differentiability of functions of several variables are deducted. Gradient of a function explored. Directional derivatives are defined and investigated. In Chapter 4, higher order partial derivatives of functions of several variables are investigated. The minimum and maximum of functions of several variables are introduced and investigated. Implicit functions are defined and explored. The method of Lagrange multipliers is introduced. The book contains proofs, numerous examples, and exercises with solutions in the two last chapters.