# Monthly Archives: February 2011

## Homology 2: Simplexes and Simplicial Complexes

Definition. A 0-simplex $\langle p_0\rangle$ is a point or a vertex. A 1-simplex $\langle p_0p_1\rangle$ is a line or an edge. A 2-simplex $\langle p_0p_1p_2\rangle$ is a triangle with its interior included. A 3-simplex $\langle p_0p_1p_2p_3\rangle$ is a solid tetrahedron. … Continue reading

## Homology 1: Free Abelian Groups

Before we discuss homology groups, we review some basics of abelian group theory. The group operation for an abelian group is denoted by $+$. The unit element is denoted by $0$. Let $G_1$ and $G_2$ be abalian groups. A map … Continue reading

## Derivatives

In this lecture, I am going to introduce you a new idea, which was discovered by Sir Issac Newton and Gottfried Leibiz, to find the slope of a tangent line. This is in fact a quite ingenious idea as you … Continue reading

Posted in Calculus | 4 Comments

## Finding a Line Tangent to a Curve

Let us consider a simple geometry problem. Given a curve $y=f(x)$, we want to find a line tangent to the graph of $y=f(x)$ at $x=a$ (meaning the line meets the graph of $y=f(x)$ exactly at a point $(a,f(a))$ on a … Continue reading

Posted in Calculus | 1 Comment

## Limits involving Infinity and Asymptotes

So far we have mainly studied finite limits. Here we would like to discuss infinite limits. You may wonder why  we need to study infinite limits. They in fact do have important applications. One immediate application is that it provides  … Continue reading

## Continuity

Intuitively speaking, we say a function is continuous at a point if its graph has no separation, i.e. there is no hole or breakage, at that point. Such notion of continuity can be defined explicitly as follows. Definition: A function … Continue reading

## Some Important Formulas from Algebra and Trigonometry

I think it would be a good idea to review some important formulas from algebra and trigonometry before we get into serious stuff in calculus. Expansion of Polynomials $(a+b)^2=a^2+2ab+b^2$ $(a-b)^2=a^2-2ab+b^2$ $(a+b)^3=a^3+3a^2b+3ab^2+b^3$ $(a-b)^3=a^3-3a^2b+3ab^2-b^3$ Factorization of Polynomials $a^2-b^2=(a+b)(a-b)$ $a^3-b^3=(a-b)(a^2+ab+b^2)$ $a^3+b^3=(a+b)(a^2-ab+b^2)$ Trigonometric Identities … Continue reading

In this posting, we discuss limits of trigonometric functions. The most basic trigonometric functions are of course $y=\sin x$ and $y=\cos x$. They have the following limit properties. Theorem 5. For any $a\in\mathbb R$, \[\lim_{x\to a}\sin x=\sin a,\ \lim_{x\to a}\cos … Continue reading