Questions tagged [motivation]
For questions about the motivation behind mathematical concepts and results. These are often "why" questions.
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How do Algebraic functions happen to be important in the study of Riemann surfaces and arise naturally?
I am a Research Scholar and I am trying to explore the connection between Algebraic functions and Branched coverings in the study of Riemann surface.Since,I am a beginner,I would like to have some ...
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A consideration of the mathematical tools that contributed to the birth of Fubini's theorem
I posted a question about the history of Fubini's theorem on the History of Science and Mathematics Stack Exchange. The answer to this question was also really helpful to me, but I wanted to dig a ...
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How to Motivate the Gauss Divergence Theorem in a simplest possible setup in multi D?
I usually motivate the Fundamental Theorem of Calculus by breaking down the difference $F(b) - F(a)$ into a sum of small differences, then using the Mean Value Theorem to show how it connects to the ...
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"Raison d'être" for Separable Spaces
A naive question about separable spaces:
Are there deeper reasons from viewpoint of functional analysis making these interesting? So far I know only only set theoretical advantages (=better "size ...
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What motivated Democritus to discover the volume of a cone?
In the works of Archimedes, it is mentioned that Democritus was the first to discover that the volume of a cone is 1/3 that of the cylinder with the same base and same height.
I am curious to know ...
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The beginning of deeper mathematical abstraction [closed]
In the past, Mathematicians first created a geometric space and then thought about functions on it. For example, we defined a vector space and then thought about a linear map, defined a topological ...
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What is the identity of this zeta function?
There are a Riemann zeta function, a Hurwitz zeta function, and many different types of zeta functions. However, I saw the zeta function below in a Japanese blog.
$$\zeta(s)=\frac{1}{1-2^{1-s}}\sum_{m=...
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The meaning/motivation of "generator" of a $C_0$ semigroup
I am currently studying $C_0$ semigroups and have come across the term infinitisimal generator. Now I'm just wondering why it's called a generator and what it generates? Can we get to the semi-flow $T$...
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Why are prime ideals proper?
As children we all learn this erroneous definition of a prime number: “a number $n\in \Bbb N$ is prime iff it’s only divided by one and itself”. Well that’s fine until the teacher asked us for ...
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Why is studying upper bounds for $|I_\delta(\mathcal P,\mathcal L)|$ useful?
A natural problem in incidence geometry is counting the number of incidences of points and lines. For example, if $\mathcal P$ is a collection of points in $\Bbb R^d$, and $\mathcal L$ is a collection ...
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Motivation behind the defination of scalar multiplication of a Vectorspace over a field
In school, we studied physical notations, such as forces, velocities, and accelerations involving both magnitude and direction. We also called any such entity involving magnitude and direction a "...
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What is the motivation of topos theory?
What is the motivation of topos theory?
https://www.youtube.com/watch?v=gKYpvyQPhZo&list=PL4FD0wu2mjWM3ZSxXBj4LRNsNKWZYaT7k&index=1
So from my understanding, the motivation of a topos is to ...
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Did the Descartes-Euler conjecture influence Poincaré's theory of homology?
What is the Descartes-Euler Conjecture?
all simply-connected polyhedra with simply-connected faces are Eulerian $V-E+F=2$.
Additional explanation: The lemma which was falsified by the ring-shaped ...
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Back to basics: Why do we care about symmetry of functions?
I am teaching a high school class on basic properties of functions, and like to motivate each of the properties with an example of why we even care to look at these properties. E.g. monotonicity ...
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Most natural definition of a product of smooth manifolds with a smooth manifold structure.
In this question,I want to ask and clarify(for myself) some points regarding the definition of product manifolds,so that I can appreciate the definition better.
The definition of any product structure(...
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Defiition of discrete/continous random variables and motivation of them. [closed]
I'm a first year math degree student and I'm taking a probability course. We defined what a random variable is and when it is discrete/continuous, but the professor didn't explain in a rigorous way ...
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Understanding the theorem of Cellular Homology through intuition.
When we start reading cellular homology,we begin with this basic but important theorem:
Theorem: Let $X$ be a CW-complex and $X^m$ denote the $m$-th skeleton of its CW structure.Then we have the ...
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When does $-(-v)=v$ not hold?
I am going through some simple questions in Axler's Linear Algebra Done Right and for each I'm trying to come up with "motivation" for why these things must be proved.
Examples:
We want to ...
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The physical motivation of simplex
I read that homology, cohomology, and simplex emerged due to physical motivation on our country's blog. However, I cannot attach a link because my country is not an English-speaking country. For ...
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Why we consider the dual space when defining tensors
My question is very simple: A type $(m,n)$ tensor is an element of $V^{\otimes m}\otimes (V^*)^{\otimes n}$. Is there a reason/motivation, beyond more general definitions, to consider the dual space ...
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Motivation about "Analysis Situs"
I read Poincaré's paper called Analysis Situs. And here's the thing about chain complex.
(Page 104 in this file)
That being given, let ${ε}^q_{i,j}$ be a number which is equal to zero if ${a}^{q−1}...
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What is the motivation for this method of finding the general formula of a sequence?
I'm reading an online note by a teacher in my country on finding the general formula of different types of sequences. The note starts by discussing basic sequences like the arithmetic sequence and the ...
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A real-world example for a centralizer property
Context: I'm re-studying basic group theory and looking for "real-world" examples/puzzles that can be translated into abstract group theoretic statements. By real-world I mean not something ...
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What's the motivation or intuition of defining the regularity of measures in Measure Theory?
In Measure Theory, the regularity of measures is defined as:
In any topological space $ \left(\Omega,\tau\right) $ , $ \mu $ is a measure on it; $ \mu $ is inner regular:
$$ \forall A\subseteq\Omega, \...
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Since we can identify functions with vectors for the dual space, why couldn't we also use a regular norm as opposed to the dual norm?
The dual space of a vector space $\mathbb{V}$ is the set of linear functionals $L: \mathbb{V} \to \mathbb{R}.$
By certain important theorem, since every linear functional has the form $L(x) = \langle ...
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Stochastic Calculus : Dangers of Incorrectly Calculating Derivatives
I am trying to understand the importance of Ito's Lemma in Stochastic Calculus.
When I learn about some mathematical technique for the first time, I always like to ask questions such as : "Is ...
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Motivation for/behind the use of $\delta$ function in the concept of euclidean domain
$\color{Green}{Background:}$
The Division Algorithm was a key tool in analyzing the arithmetic of both $\mathbb{Z}$ and $F[x].$ So we now look at domains that have some kind of analogue of the ...
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foliations subdivided into maximally distinct classes that preserve some global metric?
It's simple to construct an analytic codimension one foliation of $\Bbb R^2_{\gt 0}$ with one class of functions. I wonder if there are pros and or cons of using more distinct classes of functions for ...
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Pulling back a differential form.
In a typical differential geometry course,when we study integration over manifolds,then we define first something called the pullback of a differential-$k$ form.Suppose $f:U\subset \mathbb R^n\to V\...
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Motivation of operators
Let $V$ and $W$ be linear spaces, $D$ be a subspace of $V$, and $f:D\to W$ be a linear map.
Why do functional analysts call $f$ an operator on $V$ instead of a linear map on $D$?
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What are applications of changing limit and differentiation/integration?
I know the following theorems but don’t know their usefulness.
If a series $\{f_n\}$ of Riemann integrable functions on $[a, b]$ uniformly converges to $f$, $f$ is Riemann integrable and $\lim\limits_{...
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What is the motivation to introduce Tate-cohomology groups?
What is the motivation to introduce Tate-cohomology groups ?
Let $G$ be a Galois group and $M$ be a $G-$module.
Let $H^n(G,M)$ be usual Galois cohomology.
In group cohomology theory, we often ...
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How to motivate the teaching of differential equations?
I will have to teach a first course in differential equations. A good motivator might be to promulgate modelling with differential equations but I have seen some teachers have made polemic against ...
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Motivation for presentation of group $G$ in Hungerford’s Abstract Algebra
An immediate consequence of Corollary 9.3 and the First Isomorphism Theorem is that any group $G$ is isomorphic to a quotient group $F/N$, where $G =\langle X\rangle$, $F$ is the free group on $X$ and ...
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Origins behind different terms for the same thing in Linear Algebra
It seems that there are many terms in linear algebra that have multiple names. For example, unitary and orthogonal both refer to the same general idea, a Hermitian is essentially a self-adjoint matrix,...
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Conditions on open sets in the definition of topological space
The question on the definition of a topological space has been appeared many times on this site, but I was unable to get answer to a natural question which not only I but a new learner of this subject ...
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Motivation (intuition) about a formal group
A (one-dimensional) formal group over $\mathbb{C}$ is a formal power series $F(x,y)\in\mathbb{C}[[x,y]]$ such that
$$
F(x,y)=x+y + \text{terms of higher order}
$$
$$
F(x,F(y,z))=F(F(x,y),z))
$$
...
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Motivation for integrals over differential forms
I am trying to find some sort of motivation as to why we integrate manifold over differential form and why especially does it in some form corresponds to integrating the surface of the area. I have ...
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What is the motivation behind the definition of Cech cohomology?
From Hartshorne's Algebraic Geometry, chapter 3.4:
The Cech cohomology of a sheaf topological space wrt an open cover is defined as $\frac{ker(d_{i+1})}{im(d_i)}$, where $d$ is some operator involving ...
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Why inner automorphisms; why conjugation? Why not closure under other automorphims?
I've been casually reading up on group theory recently, and I want to get a really solid and motivated understanding of where all the definitions we use come from.
Notions like the center of a group ...
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Motivation for representation theory
Next year I have choose a few optative courses from a big list, but syllabi are not available yet, so I have to make a choice based on names only.
I am thinking about taking one titled "Groups ...
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Profinite completion motivation
I have started galois theory recently, and curiosity quickly leads one to the subject of profinite groups. Although I have yet to be comfortable using these, I get what they are and we define them as ...
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Some intuition behind the properties of intersection number.
In the study of algebraic curves,we define intersection number of $F$ and $G$ at a point $p$ to be $I(F\cap G,p)=\dim_K(\mathcal O_p(\mathbb A^2)/\langle F,G\rangle$.But it is a rather unintuitive ...
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Why is the function $r(n)$ of particular interest in Circle method?
I was reading Goldbach problem(Ternary version) and encountered the Hardy-Littlewood circle method.In this method,we work with a number $r(n)=\sum\limits_{n=n_1+n_2+n_3}\Lambda(n_1)\Lambda(n_2)\Lambda(...
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Motivation for the study of affine semigroups
I have recently been studying affine semigroup (= Semigroups that is commutative, finitely generated, embeddable in a lattice) in the context of toric varieties (Cox, Little & Schenck). I was ...
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Understanding fractional ideals. [closed]
I am having a course in algebraic number theory.Dedekind domain plays an important role in number theory.It is defined as an integral domain all of whose fractional ideals are invertible.So we need to ...
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Why is span defined as the linear combination with *finite* terms?
I understand that, for a vector space $V$ with scalar field $K$, the linear span of a family of vectors $S\subseteq V$ is usually defined as $$ \mathrm{sp}(S):=\left\{ \sum_{i=1}^k a_iv_i \mid k\in \...
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Why are natural transformations the morphisms between functors?
When we have a category $\mathcal C$, it is usual to define the category $\textrm{Ar}(\mathcal C)$ of morphisms of $\mathcal C$ as the one whose objects are the morphisms of $\mathcal C$ and whose ...
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Why are $p$-adic numbers ubiquitous in modern number theory?
I'm currently at a stage where I think I'm quite comfortable with the appearance of local non-archimedean fields in the maths I encounter, having seen a fair bit of technology built upon their ...
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Affine change of coordinates.
I am reading Fulton's book on algebraic curves.In the second chapter they have defined what they call affine coordinate change map between two affine spaces.It is defined in the following manner:
Let ...
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