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Questions tagged [motivation]

For questions about the motivation behind mathematical concepts and results. These are often "why" questions.

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Motivation behind point set topology [closed]

Why should I study point-set topology? What initially interested me in topology was the pop-sci rubber sheet stuff or coffee cup-donut stuff or proving fundamental theorem of algebra using curves but ...
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What's the geometric interpretation of this “vector cross product”?

This answer on StackOverflow answers a question about intersection of two segments. Right at the beginning, it introduces a “vector cross product”. Define the 2-dimensional vector cross product $\...
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What is the link between topology and graphs, if one exists?

In spirit, topology and graph theory seems fairly similar - you have points/vertices, and a notion of "how they are connected", loosely. However, it's not obvious how these fields relate, despite ...
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Motivation for defining filter convergence

I've just learned about a filter converging in a topological space, but I just can't understand what's the motivation to define such a thing... I get that it is a generalization of a sequence ...
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What is the usage of the fact that $Prob(X)$ is Polish if $X$ is Polish?

Let $X$ be a Polish space and $Prob(X)$ be the set of Borel probability measures on $X$, and let $Prob(X)$ be equipped with the weak-* topology (So that a sequence $\mu_m$ converges to $\mu$ in $Prob(...
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Kernel and image of matrix: What are they? Why do they exist?

I've been trying to get an understanding of the Kernel of image of matrices. I'm studying them in college right now, but the problem is, while I can find a ton of resources on how to find them given a ...
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Motivation of the Logarithm

Suppose that someone wants to calculate approximately the product of 101,123,958,959,055 and 342,234,234,234,236 without using a computer. Since these numbers are so long, completely carrying out ...
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Interpretation of Fourier Transformation - what is it?

What exactly is Fourier Transformation? For functions on the Schwartz Space $S(\Bbb R^n)$, we may define, $$ \hat{u}(\xi) := \int e^{-ix\xi} u(x) \, dx $$ This formula seems to come out of no where ...
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Why is the polarization identity important, intuitively?

The polarization identity states, roughly, that a norm satisfying the parallelogram law induces a vector space inner product (and vice versa). This has many nice applications, such as a simple ...
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Brownian motion : what is it exactly and why is it so important?

My question is simple : what is it exactly a Brownian motion and why is it so important ? So, I read the the wiki page of the Brownian motion, and the definition is : continuous stochastic process ...
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A function of Laurent and the prime counting function $\pi(n)$.

The following result is apparently due to Laurent. If $$F_n(z)=\prod_{m=1}^{n-1}\prod_{l=1}^{n-1}(1-z^{ml}),$$ we can show that the series $$f(z)=-\sum_{n=2}^{\infty}\frac{F_n(z/n)}{\{(z/n)^n-1\}n^{n-...
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Purpose of Cycle and Edge-Cut Vector Spaces?

I'm currently studying out of Dummit and Foote's Graph Theory and Its Applications (2nd ed.), and am having trouble understanding the significance of one of the sections. More specifically, section 4....
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Should one study algebraic geometry if not interested in algebraic curves?

I'm currently enrolled in a fundamental mathematics program, where a lot of math is covered. In particular, many of the students follow the Algebraic Geometry program, which is almost the core of the ...
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Why is there only one point at infinity in the extended complex plane, but one in each direction in the real projective plane?

It is a question that my friends recently discussed. My opinion is that, by using one single point at infinity to form $\hat{\mathbb{C}}$, the behavior of functions such as $f(z)=z^2$ is just like ...
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Purpose of Metrizability

What is an example of when metrizability is used to prove a result? Or is metrizability of a more philosophical nature?
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Intuitively speaking why use geodesics to capture the idea of completeness?

Let $(M,g)$ be a Riemannian manifold. We would like to define one notion of completeness which captures the idea of "missing points". For example $\mathbb{R}^n\setminus \{0\}$ should be incomplete in ...
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Motivation and definition of square functions in harmonic analysis

Let $f \in L^1_{loc}([0,1))$. Let $h_I$ denote the Haar function supported on the dyadic interval $I$. The dyadic Littlewood-Paley square function is defined by $$Sf(x) := \left(\sum_I \frac{|\...
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How does this nursery rhyme pertain to power series: “There was a little girl Who had a little curl Right in the middle of her forehead…"

This is from The Way of Analysis by Strichartz, chapter $7$, section $7.4$, page $276$. He writes "In discussing power series it is good to recall a nursery rhyme:" "There was a little girl ...
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Dynamical systems with large number of attractors and their dependence on the parameters?

It is much important to study the attractors in a dynamical system as these indicate how the system behaves once the initial transients are discarded. Also, the study of systems with many numbers of ...
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What does this exercise from field theory really tell us?

Let $\phi: F\to K$ be a field homomorphism then there exist a field $L$ containing $F$ and a field homomorphism $\Phi: K \to L$ such that $\Phi \phi=$ id. Is the above exercise a particular case of ...
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$4^{\text{th}}$ order Runge-Kutta method

I would like to know the motivation behind the choice of numbers or coefficients in front of $k_1$, $k_2$, $k_3$ and $k_4$ in $4^{\text{th}}$ order Runge-Kutta method. There are many choices of the ...
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Geometrical interpretation of the trace? [duplicate]

A while back, I learnt about the geometrical interpretation of the determinant of a linear map, that given the exterior algebra of vector spaces $V$, and a linear map $\phi: V \to W$, then we can ...
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Motivation behind using complex numbers in combinatorics

There's a common technique in couting/tiling problems in Olympiad which is to use complex numbers. I'm giving some examples: (IMO 1995 P6) Let $p > 2$ be a prime number and let $A = \{1, \cdots, ...
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Why Set Theory as Foundation? [closed]

Set Theory, particularly ZFC, is the most widely accepted foundation for mathematics. Why is that? Why is Set Theory (and, in particular, ZFC) the (in some sense) "best" foundation we can come up ...
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Are there interesting example of pseudo-Riemannian manifold other than spacetime manifold?

The 4 dimensional spacetime manifold is a typical example of pseudo-Riemannian manifold. Are there other mathematically or physically interesting example of it?
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Applications of linear algebra other than Euclidean vector spaces.

A typical example of finite-dimensional vector space is Euclidean space $\mathbb{R}^n$, but there are other type of it. For example, the space of polynomials whose order is less than $n$, the space ...
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4answers
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What problems can we solve using linear algebra? [closed]

I thought that linear algebra is a tool for solving systems of linear equations, but this can be done without most of linear algebra. That is, we just have to know matrix and the gaussian elimination ...
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2answers
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Mortivation for the definition of measurable function?

1) We say that $f:\mathbb R^n\to \mathbb R$ is measurable if $$\{x\in\mathbb R^n\mid f(x)<\alpha \},\tag{D}$$ is measurable for all $\alpha $. What is the motivation for such definition ? We can ...
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1answer
52 views

How to introduce quadratic residues?

What is the most motivating way to introduce quadratic residues? I would like some concrete examples which have an impact. This is for first-year undergraduates doing an elementary number theory ...
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Why is the derivative important? [duplicate]

Derivatives, both ordinary and partial, appear often in my mathematics courses. However, my teachers have never really given a good example of why the derivative is useful. My questions: Other than ...
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*WHY* are half-integral weight modular forms defined on congruence subgroups of level 4N?

The standard definition for a half-integral weight meromorphic modular form is a meromorphic function that obeys that following functional equation for all matrices $\begin{bmatrix} a & b \\ c &...
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Why is the inner measure problematic?

Currently the Lebesgue measure is defined by the outer measure $\lambda^*(A)$ by the criterion of Carathéodory: A set $A$ is Lebesgue measurable iff for every set $B$ we have $\lambda^*(B)=\lambda^*(B\...
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What's the Intuition behind divisors?

I am currently studying Algebraic Geometry (by Hartshorne), for the first time, and had attended/am attending to some lectures related to it. (Commutative Algebra, Complex manifolds, ...) As I learn ...
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Why do PDE's seem so unnatural? [closed]

First let me preface by saying that I'm highly aware of the fact that plenty a math topic seems unnatural upon first learning. But PDEs seem to have a special place in my "unnatural" category of ...
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What is the motivation of the identity $x\circ(a\circ b)=(((x\circ b)\circ a )\circ b$?

A symmetric set (also called an involutary quandle or a kei) is a set $A$ with a binary operation $\circ$ satisfying the following conditions for all $a,b,x\in A$: $a\circ a =a$; $(x\circ a)\circ a=x$...
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About study of sub group [closed]

Consider the group G(under operation +) i take subset H of G. I proved that H itself become the group( under same operation + )Now how study of H helps us to study the structure of G. In other words ...
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How does one introduce characteristic classes

How do you introduce or how are you introduced to characteristic classes. I am assuming the student is comfortable with principal bundles and connections on principal bundles.
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Why does linear regression involve the $y$-coordinate error?

When dealing with linear regression, we are concerned about how far away a given point's $y$ component is from the "best fitting line". My question: why do we choose the $y$ component instead of the $...
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Why must the squared error function be at its minimum?

In this Khan Academy video series Khan goes through the derivation of the formula for the linear regression line for some data points. The only part I do not understand is the one I've given a link ...
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Why are we interested in harmonic maps between manifolds?

This is a serious question. The notion of harmonic maps $M \to N$ between general Riemannian manifolds has two important special cases, which are obviously interesting, for many reasons: $M=\mathbb{R}...
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Why is the fact that a quotient group is a group relevant?

I'm studying the basics of quotient groups. I understand that if you build a quotient set from cosets of a group and the subgroup you are using to build them is normal then you end up with a group. I ...
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What is the motivation behind the (metric spaces) definition of an open set?

As far as I know, the standard definition of an open set is that the set $A$ is called open if $A \subseteq X$ for some set X and if $A \cap \partial A=\emptyset$ where $\partial A$ is the set of ...
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Every well-ordering is isomorphic to the set of all lesser ordinals under ordinal comparison

Wikipedia states that Every well-ordered set (S,<) is order isomorphic to the set of ordinals less than one specific ordinal number [the order type of (S,<)] under their natural ordering. I'...
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1answer
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How to focus for long hours? [closed]

I'm currently working on my A levels and would like to know about how to focus for longer hours and stay motivated? I'm also working on STEP Support program for entry into Cambridge and would like to ...
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Motivation of Splines

What is the motivation of splines, in particular cubic splines. For example, why does it matter that they have any type of smoothness at the knots.
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Importance of integral extension

I am studying basic algebraic number theory these days and I am curious if the concept of “integral extension” is important in purely number theoretic sense. Of course, integral extension is a ...
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Discovering Quadratic Reciprocity

Is there anything similar to this (page written by Field Medalist Timothy Gowers) for quadratic reciprocity ? I mean, the link there explains how you can figure out the solution of cubic equation by ...
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163 views

Divergence formula on manifolds using a physical approach

In rectangular coordinates, one can define the divergence of a vector field $\vec{V}$ as the following limit $$ \text{Div } V = \lim_{\epsilon \rightarrow 0} \frac{1}{\epsilon^3} \oint_{\partial C_{\...
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Motivating Number Theory/Combinatorics questions leading to finite field

I want to learn about Finite Field, but don't want to learn but by starting from the memorizing the axioms of finite field, I wanna learn it by solving a few problems (good if NT /Combinatorics), ...
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Why should I care about proving a polynomial to be irreducible?

Are there any number theoretic/combinatorial/other applications of proving that a polynomial is irreducible over integers/any other field ?(But integers are preferred)