Entries by fresh_42

Series in Mathematics: From Zeno to Quantum Theory

Introduction Series play a decisive role in many branches of mathematics. They accompanied mathematical developments from Zeno of Elea (##5##-th century BC) and Archimedes of Syracuse (##3##-th century BC), to the fundamental building blocks of calculus from the ##17##-th century on, up to modern Lie theory which is crucial for our understanding of quantum theory….

Epsilontic – Limits and Continuity

Abstract I remember that I had some difficulties moving from school mathematics to university mathematics. From what I read on PF through the years, I think I’m not the only one who struggled at that point. We mainly learned algorithms at school, i.e. how things are calculated. At university, I soon met a quantity called…

Introduction to the World of Algebras

Abstract Richard Pierce describes the intention of his book [2] about associative algebras as his attempt to prove that there is algebra after Galois theory. Whereas Galois theory might not really be on the agenda of physicists, many algebras are: from tensor algebras as the gown for infinitesimal coordinates over Graßmann and Banach algebras for…

The Art of Integration

Abstract My school teacher used to say “Everybody can differentiate, but it takes an artist to integrate.” The mathematical reason behind this phrase is, that differentiation is the calculation of a limit $$ f'(x)=\lim_{v\to 0} g(v) $$ for which we have many rules and theorems at hand. And if nothing else helps, we still can…

An Overview of Complex Differentiation and Integration

Abstract I want to shed some light on complex analysis without getting all the technical details in the way which are necessary for the precise treatments that can be found in many excellent standard textbooks. Analysis is about differentiation. Hence, complex differentiation will be my starting point. It is simultaneously my finish line because its…

When Lie Groups Became Physics

Abstract We explain by simple examples (one-parameter Lie groups), partly in the original language, and along the historical papers of Sophus Lie, Abraham Cohen, and Emmy Noether how Lie groups became a central topic in physics. Physics, in contrast to mathematics, didn’t experience the Bourbakian transition so the language of for example differential geometry didn’t…

Classification of Mathematics by 42 Branches

  I often read questions about our classification scheme that we use on physicsforums.com to sort posts by science fields and subjects, what has to be studied first in order to learn something else, what is a good way through physics or mathematics in self-study or simply about the desire to understand, e.g. general relativity…

Évariste Galois and His Theory

  * Oct. 25th, 1811  † May 31st, 1832 … or why squaring the circle is doomed. Galois died in a duel at the age of twenty. Yet, he gave us what we now call Galois theory. It decides all three ancient classical problems, squaring the circle, doubling the cube, and partitioning angles into three…

Yardsticks to Metric Tensor Fields

I asked myself why different scientists understand the same thing seemingly differently, especially the concept of a metric tensor. If we ask a topologist, a classical geometer, an algebraist, a differential geometer, and a physicist “What is a metric?” then we get five different answers. I mean it is all about distances, isn’t it? “Yes”…

The History and Importance of the Riemann Hypothesis

Riemann Hypothesis History The Riemann Hypothesis is one of the most famous and long-standing unsolved problems in mathematics, specifically in the field of number theory. It’s named after the German mathematician Bernhard Riemann, who introduced the hypothesis in 1859. RH: All non-trivial zeros of the Riemannian zeta-function lie on the critical line. ERH: All zeros…

The Extended Riemann Hypothesis and Ramanujan’s Sum

Riemann Hypothesis and Ramanujan’s Sum Explanation RH: All non-trivial zeros of the Riemannian zeta-function lie on the critical line. ERH: All zeros of L-functions to complex Dirichlet characters of finite cyclic groups within the critical strip lie on the critical line. Related Article: The History and Importance of the Riemann Hypothesis The goal of this…

The Amazing Relationship Between Integration And Euler’s Number

We use integration to measure lengths, areas, or volumes. This is a geometrical interpretation, but we want to examine an analytical interpretation that leads us to Integration reverses differentiation. Hence let us start with differentiation. Weierstraß Definition of Derivatives ##f## is differentiable at ##x## if there is a linear map ##D_{x}f##, such that \begin{equation*} \underbrace{D_{x}(f)}_{\text{Derivative}}\cdot…

How to Write a Math Proof and Their Structure

Proofs in mathematics are what mathematics is all about. They are subject to entire books, created entire theories like Fermat’s last theorem, are hard to understand like currently Mochizuki’s proof of the ABC conjecture, or need computer assistance like the 4-color-theorem. They are sometimes even missing, although everybody believes in the statement like the Riemann…

Lie Algebras: A Walkthrough The Representations

  Part III: Representations   10. Sums and Products. Frobenius began in ##1896## to generalize Weber’s group characters and soon investigated homomorphisms from finite groups into general linear groups ##GL(V)##, supported by earlier considerations from Dedekind. Representation theory was born, and it developed fast in the following decades. The basic object of interest, however, has…

Learn Lie Algebras: A Walkthrough The Structures

  Part II: Structures 5. Decompositions. Lie algebra theory is to a large extend the classification of the semisimple Lie algebras which are direct sums of the simple algebras listed in the previous paragraph, i.e. to show that those are all simple Lie algebras there are. Their counterpart are solvable Lie algebras, e.g. the Heisenberg algebra ##\mathfrak{H}=\langle…

Learn Lie Algebras: A Walkthrough The Basics

  Part I: Basics 1. Introduction. This article is meant to provide a quick reference guide to Lie algebras: the terminology, important theorems, and a brief overview of the subject. Physicists usually call the elements of Lie algebras generators, as for them they are merely differentials of trajectories, tangent vector fields generated by some operators. Thus…

Match the Scientist with the Story Quiz

Among the most famous people are often geniuses. It’s hard to tell whether this is the reason for the many anecdotes which are told about them, or whether this is just incidentally true. Doubts are allowed, since most scientists are quite ordinary people. But some of them are cranky and all kind of stories circulate…

Learn the Basics of Hilbert Spaces and Their Relatives: Operators

  Operators. The Maze Of Definitions. We will use the conventions of part I (Basics), which are ##\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}##, ##z \mapsto \overline{z}## for the complex conjugate, ##\tau## for transposing matrices or vectors, which we interpret as written in a column if given a basis, and ##\dagger## for the combination of conjugation and transposition, the adjoint…

A Journey to The Manifold SU(2): Representations

Part 1   Representations Image source: [23]   6. Some useful bases of ##\mathfrak{su}(2,\mathbb{C})## Notations can differ from author to author: the numbering of the Pauli matrices ##(\text{I 4}), (\ref{Pauli-I})##, the linear combinations of them in the definition to basis vectors ##\mathfrak{B}## of ##\mathfrak{su}(2,\mathbb{C}) \; (\text{I 5}), (\ref{Pauli-II}), (\ref{Pauli-III})##, the embedding of the orthogonal groups ##(\text{I…

A Journey to The Manifold SU(2): Differentiation, Spheres, and Fiber Bundles

Part 2   Differentiation, Spheres, and Fiber Bundles Image source: [24] The special unitary groups play a significant role in the standard model in physics. Why? An elaborate answer would likely involve a lot of technical terms as Lie groups, Riemannian manifolds or Hilbert spaces, wave functions, generators, Casimir elements, or irreps. This already reveals…

10 Math Tips to Save Time and Avoid Mistakes

Exam situations are always situations of stress. It comes with our endeavor to be as good as possible together with our fear of failure. Some students handle these situations better than others. But there are some tricks I encountered over the years tutoring young students. I’m sure everybody has developed their own ways to get…

What Is a Tensor? The mathematical point of view.

Key points The author begins by questioning the nature of numbers and their deeper significance. Numbers can represent various mathematical constructs, from scalars to linear mappings of one-dimensional vector spaces. These numerical representations can be described as elements of fields, dual spaces, and matrices. The concept of tensors is introduced, which are multi-dimensional arrays of…

The Pantheon of Derivatives – Part V

  Important Theorems – biased, of course Implicit Function Theorem [1] Jacobi Matrix (Chain Rule). Let ## (x_0,y_0 ) ## be a point in $$U_1 \times U_2 = \{x \in \mathbb{R}^k\,\vert \,||x-x_0||< \varepsilon_1 \} \times \{y \in \mathbb{R}^m\,\vert \,||y-y_0||< \varepsilon_2\}$$ and ## f: U_1 \times U_2 \rightarrow \mathbb{R}^m ## a function with  ##f(x_0,y_0)=0## which is totally…

The Pantheon of Derivatives – Part IV

  Lie Derivatives A Lie derivative is in general the differentiation of a tensor field along a vector field. This allows several applications since a tensor field includes a variety of instances, e.g. vectors, functions, or differential forms. In the case of vector fields, we additionally get a Lie algebra structure. This is, although formulated…

The Pantheon of Derivatives – Part III

  Some Topology Whereas the terminology of vector fields, trajectories, and flows almost by itself suggests its origins and physical relevance, the general treatment of vector fields, however, requires some abstractions. The following might appear to be purely mathematical constructions, and I will restrict myself to a minimum, but they actually occur in modern physics:…

The Pantheon of Derivatives – Part II

  Generalizations Beyond ##\mathbb{R}## and ##\mathbb{C}## As mentioned in the section on complex functions (The Pantheon of Derivatives – Part I), the main parts of defining a differentiation process are a norm and a direction. So to extend the differentiation concepts on normed vector spaces seems to be the obvious thing to do. Fréchet Derivative…

The Pantheon of Derivatives – 5 Part Series

  Differentiation in a Nutshell I want to gather the various concepts in one place, to reveal the similarities between them, as they are often hidden by the serial nature of a curriculum. There are many terms and special cases, which deal with the process of differentiation. The basic idea, however, is the same in…