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This textbook proposes an informal access to the most important issues of multidimensional differential and integral calculus. The traditional style―characterized by listing definitions, theorems, and proofs―is replaced by a conversational approach, primarily oriented to applications. The topics covered, developing along the usual path of a textbook for undergraduate courses, are always introduced by thoroughly carried out examples. This drives the reader in building the capacity of properly use the theoretical tools to model and solve practical problems. To situate the contents within a historical perspective, the book is accompanied by a number of links to the biographies of all scientists mentioned as leading actors in the development of the theory.
Preface.
Basic Concepts and Parametrisation of Curves.
Differential and Geometric Properties of Curves.
Curves in Space: The Frenet Frame.
Functions of a Vector Variable.
Continuity and Differentiability of Functions of a Vector Variable.
Partial Derivatives.
Sequences of Functions.
Series of Functions.
Taylor Series for Functions of Several Variables.
Applications of the Taylor Series.
Integration of Functions of Two Variables.
Samples of Two-Dimensional Integration and Change of Variables.
Two-Dimensional Integration and Area of a Surface.
Vector Functions of Vector Variables.
Line Integral and Flux of Vector Functions.
Triple Integrals and Coordinate Changes.
Green’s Formulae for the Integral Calculus.
Application of Green’s Formulae.
Gauss and Stokes Theorems.
Partial Differential Equations.
Quasi-Linear Second Order Partial Differential Equations.
Parabolic and Hyperbolic Partial Differential Equations.
Solution of Poisson’s Equation.
References.
Index