Advanced Calculus - Fundamentals of Mathematics

This chapter defines the concept of vectors on the real coordinate space ℝ

This chapter focuses on the characterisation of a real-valued function, its graphs, and level surfaces; its limits, continuity, and differentiation. The operators here reviewed are gradient, directional derivatives, and the polynomial approximation to a function named Taylor´s theorem. All of them will be extensively used in the following chapters.

This chapter focuses on the identification of the maximum, minimum, or saddle points located in the domain of the real-valued function in a graph of a function. The first part of this chapter focuses on the open set domain of a function and the second part on the closed set domain. The characterization of these points called critical points" is resolved with the Hessian matrix and the Bordered Hessian matrix. Finally, we will review the Implicit Function Theorem.

This chapter focuses on describing the vector-valued function

This chapter introduces Riemann’s double and triple integrals over integration domains, bounded by real-valued functions with and without the use of mappings that transform the bounded integration domains.

This section reviews the integrals whose domain of integration corresponds to an unbounded region, or a partially unbounded region, on a plane or in space. This type of integrals are called improper integrals.

This chapter introduces the Line and Surface Integrals as a measure of the effect of a vector field or scalar field over oriented curves and surfaces.

This chapter reviews the theorems of Vector Calculus: Green’s theorem, Stokes’ theorem, and Gauss’ theorem. The approach will be done using mappings, to show the operative simplification that can be obtained.

This chapter intends to be a survey on Exterior Algebra. This algebra is attributed to Hermann Grassmann [Die lineare Ausdehnungslehre, ein neuer Zweig der Mathematik, 1842], and it is formed by two operators: the exterior product and inner product. William K. Clifford unified both products under the geometric product operator and David Hestenes improved the geometric and computer aspects of Geometric algebra. Here, we review the main operators and their properties, as well as their application in the representation of lines and planes.

This chapter reviews the main aspects of the Differential forms and their application to facilitate the resolution of the integrals involved in Green’s theorem, Stokes’ theorem, and Gauss’ theorem.

This chapter provides the complete solution to all the exercises pointed out at the end of each chapter of this book. The solutions not only indicate the final result, but the respective procedures are exhibited. It is recommended in all cases that the reader review the solutions to all the exercises.