
Recent Posts
Recent Comments
 Determinants II: Determinants of Order $n$  MathPhys Archive on Determinants I: Determinants of Order 2
 Inverses  MathPhys Archive on The Matrix Associated with a Linear Map
 The Matrix Associated with a Linear Map  MathPhys Archive on Linear Maps
 Introduction to Topology 3: Limit Points, Boundary Points, and Sequential Limits  MathPhys Archive on Introduction to Topology 1: Open and Closed Sets
 Parallel Transport, Holonomy, and Curvature  MathPhys Archive on Line Bundles
Archives
Categories
 Algebraic Topology
 Calculus
 Classical Differential Geometry
 College Algebra
 Differential Equations
 Differential Geometry
 Electromagnetism
 Engineering Mathematics
 Functions of a Complex Variable
 General Topology
 Homology
 Lie Groups and Lie Algebras
 Linear Algebra
 Mathematical Physics
 Partial Differential Equations
 Precalculus
 Quantum Mechanics
 Representation Theory
 Sage
 Trigonometry
 Uncategorized
Meta
Monthly Archives: February 2011
Homology 2: Simplexes and Simplicial Complexes
Definition. A 0simplex $\langle p_0\rangle$ is a point or a vertex. A 1simplex $\langle p_0p_1\rangle$ is a line or an edge. A 2simplex $\langle p_0p_1p_2\rangle$ is a triangle with its interior included. A 3simplex $\langle p_0p_1p_2p_3\rangle$ is a solid tetrahedron. … Continue reading
Posted in Algebraic Topology, Homology
Leave a comment
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
Posted in Algebraic Topology, Homology
Leave a comment
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
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
Posted in Calculus
Leave a comment
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
Posted in Calculus
Leave a comment
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 \((a+b)^2=a^2+2ab+b^2\) \((ab)^2=a^22ab+b^2\) \((a+b)^3=a^3+3a^2b+3ab^2+b^3\) \((ab)^3=a^33a^2b+3ab^2b^3\) Factorization of Polynomials \(a^2b^2=(a+b)(ab)\) \(a^3b^3=(ab)(a^2+ab+b^2)\) \(a^3+b^3=(a+b)(a^2ab+b^2)\) Trigonometric Identities … Continue reading
Posted in Precalculus, Trigonometry
Leave a comment
SouthernMiss Math Forum
Recently I have put up a math forum site, called SouthernMiss Math Forum. This is an online meeting place where math faculty members, undergrad students, and grad students can discuss math outside of classrooms. This forum is also open to … Continue reading
Posted in Uncategorized
Leave a comment
How to Calculate Limits III
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
Posted in Calculus
2 Comments