Differential Calculus

This book is addressed to all those who, after finishing the high school, wish a practical initiation in the domain of differential calculus. This is the second volume of the series Mathematics for future engineers. It has its origins in a course written for the first-year students of the Technical University of Constructions of Bucharest.To provide useful tools for future engineers and for specialists, in general, we emphasized physical and/or geometrical interpretation of analysis formulas, as well as their applications to numerical calculus or error estimation (e.g., Lagrange's and Taylor's formula). We made connections between mathematical formulae, on the one hand, and mechanical or physical phenomena, on the other hand (e.g., examples of functions of several real variables, constrained extrema, directional derivative of a functions, etc.). We tried to make the involved mathematics as attractive as possible, by simplifying the presentation without loosing the mathematical rigor of the profs and results. To increase accessibility and to encourage the reader to get a technical know-how about differential calculus, we provided for each newly introduced notion a series of applications and solved problems: each chapter ends by a section containing exercises and problems, each one of these being accompanied by hints and answers.The sections marked by asterisks can be omitted, as well as some proofs; we introduced them, though, for the sake of a unitary and logical presentation.We developed a new special technic to get partial derivatives preventing the beginners to mix up variables and also an easy procedure to apply changes of variables to linear and quasi-linear operators, based on formal computation. As examples, we applied standard changes of variables to the wave and Laplace equations, well known in mechanics.The mathematical concepts presented here require only elementary notions, usually learnt at the high school, such as the notion of a function of one real variable, its derivative, derivatives of elementary functions, etc.The references contain, along with books, some links with useful sites, which can be helpful for the reader.