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- Introduction to Topology 3: Limit Points, Boundary Points, and Sequential Limits
- Introduction to Topology 2: Bases and Subbases
- Introduction to Topology 1: Open and Closed Sets
- Parallel Transport, Holonomy, and Curvature
- Sections of a Line Bundle II: Gauge Potential, Gauge Transformation, and Field Strength
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- Introduction to Topology 3: Limit Points, Boundary Points, and Sequential Limits | SouthernMiss Math Archives on Introduction to Topology 1: Open and Closed Sets
- Parallel Transport, Holonomy, and Curvature | SouthernMiss Math Archives on Line Bundles
- Sections of a Line Bundle II: Gauge Potential, Gauge Transformation, and Field Strength | SouthernMiss Math Archives on Sections of a Line Bundle I
- Sections of a Line Bundle I | SouthernMiss Math Archives on Sections of a Line Bundle II: Gauge Potential, Gauge Transformation, and Field Strength
- Lie Brackets (for $ntimes n$ Matrices) | SouthernMiss Math Archives on The Lie Algebra of the Orthogonal Group $\mathrm{O}(n)\ (\mathrm{SO}(n))$
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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
Posted in Algebraic Topology, Homology
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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
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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
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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
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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
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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
Posted in Precalculus, Trigonometry
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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
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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
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