Analysis

Introduction Series play a decisive role in many branches of mathematics. They accompanied mathematical developments from Zeno of Elea (##5##-th century…

Introduction I am convinced students learn Calculus far too late.   In my view, there has never been a good reason for this.In the US, they go through…

Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say…

This is not meant as a full introduction to recursion relations but it should suffice for just about any level of the student.Most of us remember recursion…

Introduction A frequent concern among students is how to carry out higher order partial derivatives where a change of variables and the chain rule are…

Definition of real numbers Real numbers are the set of all values that can appear on the continuous number line. They include rational numbers (fractions…

Introduction This article will attempt to take the mystery out of setting up surface integrals. It will explain the basic ideas underlying surface integration…

  Basics Language first: There is no such thing as the Hilbert space.Hilbert spaces can look rather different, and which one is used in…

Introduction: In this Insight article,  Parseval's theorem will be applied to a sinusoidal signal that lasts a finite period of time.  It will be shown…

  Differentiation in a Nutshell I want to gather the various concepts in one place, to reveal the similarities between them, as they are often…

Differentiation is usually taught quite well. Perhaps that's because it is the first introduction to calculus, which is considered a big step in a student's…

Prerequisites If you wish to follow this guide, you should be familiar with analysis on ##\mathbb{R}## and ##\mathbb{R}^n##. See my previous insight for…

At the first sight, there are many paradoxes in complex number theory. Here are some nice examples of things that don't seem to work:Example A [itex]-1=i^2=\sqrt{-1}\cdot\sqrt{-1}=\sqrt{(-1)(-1)}=\sqrt{1}=1[/itex]Example…

Why do people say 1 and 0.999... are equal? Aren't they two different numbers? No — 1 and 0.999... really are the same number, although that can feel…

Introduction Understanding the behavior of infinity is one of the major accomplishments of mathematics. However, the infinite is often misunderstood and…