Advanced Calculus

Pietro-Luciano Buono

• Publisher: De Gruyter
• Year: 2016
• ISBN:
  9783110429114
• Edition: 1
• Topic: Matemáticas
• Subtopic: Cálculo

This textbook offers a high-level introduction to multi-variable differential calculus. Differential forms are introduced incrementally in the narrative, eventually leading to a unified treatment of Green's, Stokes' and Gauss' theorems. Furthermore, the presentation offers a natural route to differential geometry.

Contents:
Calculus of Vector Functions
Tangent Spaces and 1-forms
Line Integrals
Differential Calculus of Mappings
Applications of Differential Calculus
Double and Triple Integrals
Wedge Products and Exterior Derivatives
Integration of Forms
Stokes' Theorem and Applications

1Introduction

This chapter gives an overview of several of the topics necessary for the remainder of the book. The first sections on Set Theory and Linear Algebra are review sections.

1.1Review of Set Theory

We begin with a quick review of basic concepts from set theory with a focus on real numbers ℝ. A set of real numbers is an unordered collection of real numbers. One denotes sets inside brackets in enumerative style as follows

The symbol ∈ means “element of” and denotes the belonging of an element to a set. We can also describe a set using a defining condition

which is read as:

x is the placeholder for elements of ℝ such that x is greater than 2.

In the defining condition notation, to verify whether a number belongs to a given set one has to check if the condition is satisfied. Is −3 ∈ B? Let x = −3, then −3 > 2 is false; therefore, −3 is not an element of B and it is denoted: −3 ∉ B. Sets

Fig. 1.1. In bold, the set B.

can have a finite number of elements as A or an infinite number of elements as B. An important type of sets on the real line are the intervals, defined as follows: Let a, b ∈ ℝ and a < b

If a = −∞ or b = ∞ then we always use “(a” or “b)”.

Fig. 1.2. From left to right, the intervals [a, b], (a, b) and (a, b].

A set E is a subset of a set F if every element of E is also an element of F . We then write E ⊂ F . Another notation is E ⊆ F which allows for E and F to have

exactly the same elements, that is E = F .

Example 1.1.1. Let

and F be the set of even integers. Is E ⊂ F ? To see this, one has to make sure that every element of E is an even integer. But, E contains an infinite number of elements and so we use the defining condition instead. At this point, if one believes that E ⊂ F , then we can proceed to verify it properly as follows: let x = 4n for an arbitrary natural number n ∈ ℕ, but x = 2...