Questions tagged [motivation]
For questions about the motivation behind mathematical concepts and results. These are often "why" questions.
352
questions
2
votes
1
answer
33
views
intuition behind the construction of the map for showing associativity of $\pi_1(X,x_0)$.
I am studying algebraic topology.I have started with the chapter on fundamental group.Fundamental group at $x_0$ is defined to be the set of all equivalence classes of loops based at $x_0$ together ...
- 2,005
0
votes
1
answer
41
views
Intuition behind the notation of differential operators $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \overline z}$.
I am a graduate student of Mathematics.In this semester I am studying complex analysis.Stein Shakarchi's complex analysis book defines differential operators $\frac{\partial}{\partial z}$ and $\frac{\...
- 2,005
1
vote
0
answers
34
views
Motivation for Derivatives of Measures
Let $\mu$ be a Borel measure on $\mathbb R^n$ with compact support, and $0 < \mu(\mathbb R^n) < \infty$. We define the lower derivative and derivative of $\mu$ at $x\in \Bbb R^n$ by
$$\underline{...
- 10.3k
1
vote
0
answers
36
views
Motivation behind the definition of paracompact.
I am self-studying topology.I encountered the definition of a paracompact space.A collection $\mathcal A$ of subsets of $X$ is said to be locally finite if each point in $X$ has an open neighborhood ...
- 2,005
0
votes
0
answers
22
views
Motivation behind definition of product measure.
Consider two $\sigma$-finite measure spaces $(X,\mathcal F,\mu)$ and $(Y,\mathcal G,\nu)$.In standard measure theory books they define product measure as follows:
$(\mu\times \nu)(E)=\int_X\nu(E_x)d\...
- 2,005
2
votes
3
answers
95
views
If $\pi$ is a representation of a lie algebra $\mathfrak{g}$, why is $\pi(x)$ not required to be invertible?
The definition of a representation of a group $G$ is a homomorphism $\pi: G\to GL(V)$. So here $\pi(x)$ is an invertible linear map $V \to V$.
The definition of a representation of a lie algebra $\...
- 2,789
6
votes
4
answers
150
views
Significance of Cauchy's theorem.
I am studying complex analysis. In the beginning of the chapter on integration theory there is a theorem known as Cauchy's theorem. It states that:
If $\Omega\subset \mathbb C$ be open and $f:\Omega\...
- 2,005
1
vote
1
answer
51
views
Motivation for the definition of orbifold euler characteristic
Let $S$ be a normal projective surface with quotient singularities only. For each singular point $p\in S$, $(S,p)$ is locally analytically isomorphic to $(\Bbb C^2/G_p,0)$ for some finite subgroup $...
- 2,034
1
vote
2
answers
102
views
What is the motivation behind the definition of Lebesgue measurable set?
We study the definition of Lebesgue measurable set to be the following:
Let $A\subset \mathbb R$ be called Lebesgue measurable if $\exists$ a Borel set $B\subset A$ such that $|A-B|=0$,where $|.|$ ...
- 2,005
4
votes
1
answer
52
views
Intuition modular equivalence of elements in general groups with element in subgroup (Herstein)
Let $G$ be a group, $H$ is a subgroup of $G$; for $a,b \in G$ we say $a$ is congurent to $b \mod H$, written as $a \equiv b \mod H$ if $ab^{-1} \in H$
Herstein, Topics in abstract algebra, page 34.
...
- 6,560
7
votes
0
answers
121
views
Is there a connection between free–forgetful adjunctions and tensor-hom adjunctions?
In the Wiki article on adjunction https://en.wikipedia.org/wiki/Adjoint_functors, there is a motivation section that talks about how adjunctions can be viewed as "Solutions to optimization ...
- 5,694
-1
votes
1
answer
29
views
Condition for measurable function.
We define measurable functions as follows:
Definition
Let $(X,\mathcal S)$ be a measurable space.Then $f:X\to \mathbb R$ is called measurable function if $f^{-1}(B)\in \mathcal S$ for any Borel set $B\...
- 2,005
18
votes
3
answers
598
views
How can construction of the gamma function be motivated?
The gamma function extends the factorial function. This can be proved inductively using integration-by-parts.
But if you didn't already know that the gamma function had this property, and you wanted ...
- 1,373
1
vote
1
answer
62
views
Help With the Similarity Solution Method
I am having trouble understanding the similarity solution method for solving partial differential equations. I have been able to replicate the highly spoon-fed example, but all of the other, more ...
- 1,666
2
votes
1
answer
65
views
Morphisms set of sub-$\infty$-category closed under equivalences
In Markus Land's Introduction to Infinity-Categories, he defines sub-$\infty$-categories the following way (p. 56):
Definition. A sub-$\infty$-category $\mathscr{C}'$ of an $\infty$-category $\mathscr{...
- 6,802
3
votes
0
answers
75
views
What does linear equivalence geometrically mean for varieties?
Suppose we have a sufficiently nice scheme or say we are working with an abstract nonsingular variety $X$ over an algebraically close field. In this setting, one can study (Weil) divisors. It is then ...
2
votes
1
answer
52
views
Motivation for the norm of the quotient space
I'm studying Functional Analysis, more specifically, the quotient space. Since more often we see the $\sup$ as some norm definitions, I was wondering, what is the motivation for the quotient norm to ...
- 35
3
votes
1
answer
129
views
Motivation behind this proof of the Basel problem?
On a past paper from our examination committee there seems to be a proof to the Basel problem that is plucked out of thin air. It doesn't match any of the proofs on Wikipedia (at least by first glance)...
- 1,073
4
votes
2
answers
134
views
Why do you calculate the dot product that way?
I am learning dot product these days. I understand the geometric meaning of one vector's interpretation in the same direction of the other to calculate the work in terms of force and distance in the ...
- 41
3
votes
3
answers
140
views
Why is it important that the $p$-adic absolute value satisfy multiplicativity?
I suppose my question is really "why do we require norms in general to satisfy multiplicativity?". I ask this because for the usual absolute value on $\mathbb R$, I never feel like ...
- 5,694
6
votes
1
answer
122
views
Doubts on roster notation of sets
Simple examples of sets are often described via roster notation (aka enumeration notation) like $\{0,1,4,9\}$ - simply write down the elements of the set between the two set delimiters (curly ...
- 1,511
1
vote
0
answers
123
views
What is a $\sigma$-algebra, really?
I know that a $\sigma$-algebra is a suitable generalization of the notion of sample space, in the following sense:
Consider a sample space $\Omega$ and a collection $\mathscr{F}$ of subsets of $\...
- 10.2k
1
vote
1
answer
79
views
What are the important things to study in infinity categories?
I am reading about infinity categories.
My source is http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf
My aim is to think categorically so that all the constructions I deal with are natural (...
- 1,804
1
vote
0
answers
16
views
Definition of a pseudo-free circle action
Let us consider $S^1$ as a Lie group, and suppose we are given a smooth $S^1$-action on a closed manifold.
According to this paper (https://www.jstor.org/stable/60608), the action is said to be pseudo-...
- 1,710
14
votes
3
answers
843
views
Why choose sets to be the primitive objects in mathematics rather than, say, tuples?
Sets are defined in such a way that $\{a,a\}$ is the same as $\{a\}$, and $\{a,b\}$ is the same as $\{b,a\}$. By contrast, the ordered pair $(a,a)$ is distinct from $(a)$, and $(a,b)$ is distinct from ...
- 14.1k
1
vote
2
answers
63
views
Reference request for Separation axioms in Topology.
I am an undergraduate student of Mathematics and I want to study the topic "Separation Axioms" of general topology.I have already studied Basis,Subbasis,Product topology,Countability axioms,...
- 2,005
3
votes
1
answer
113
views
What is the motivation behind defining continuity at isolated points?
The naive conception of a continuous function is that it is a function whose graph can be drawn without lifting your pencil off the page. In introductory analysis, we often define continuity at a ...
- 14.1k
3
votes
2
answers
228
views
What is the intuition behind product topology?
Let $X$ denote the cartesian product of the sets $X_i,i\in \mathbb N$ where $(X_i,\tau_i)$ are topological spaces. Define product topology by defining $\mathcal S=\bigcup\limits_{i\in \mathbb N}\...
- 2,005
5
votes
2
answers
185
views
Motivation for the Nijenhuis tensor
I'm learning about complex and almost complex structures on smooth manifolds, in particular the Newlander-Nirenberg theorem. Recall that for an almost complex structure $J$ on a smooth manifold, the ...
- 707
26
votes
3
answers
582
views
is it possible to motivate higher differential forms without integration?
You have been teaching Dennis and Inez math using the Moore method for their entire lives, and you're currently deep into topology class.
You've convinced them that topological manifolds aren't quite ...
- 25.5k
2
votes
0
answers
109
views
Simplicial schemes vs simplicial presheaves
My question is not very precise, I apologize in advance.
Essentially I am wondering in the context of "homotopy theory for schemes, in the broad sense, be it in motivic homotopy theory or $\...
- 121
16
votes
4
answers
3k
views
Why are abelian groups of interest? What is their usefulness?
I am reading about Abelian groups
So apparently it is a set, with an associative binary operation, and identity element, an inverse operation and the binary operation must also be symmetric.
But it is ...
- 1,507
6
votes
1
answer
178
views
Intuition of Algebraic Partition of Unity
In Vakil's Rising Sea (Nov. 18, 2017 Draft) on p. 130 - 131, he gives a proof that his definition of the structure sheaf on affine schemes indeed defines a sheaf. He coins this proof an argument by &...
- 6,802
0
votes
2
answers
128
views
Why are Fuchsian groups interesting?
I am recently reading the book "Fuchsian groups" by Katok and now on Chapter $2$. I am curious about why Fuchsian groups are interesting. I look it up online and find answers here. Those are ...
- 3
100
votes
8
answers
11k
views
Is Bayes' Theorem really that interesting?
I have trouble understanding the massive importance that is afforded to Bayes' theorem in undergraduate courses in probability and popular science.
From the purely mathematical point of view, I think ...
- 3,718
4
votes
2
answers
104
views
Why do fixed point theorems appear all over mathematics?
For example, the Banach fixed-point theorem is applied in the proof of the Picard–Lindelöf theorem about the uniqueness of solutions of ordinary differential equations and the Lefschetz fixed-point ...
- 41
1
vote
0
answers
40
views
The motivation for approaching problems using discretizing tools
Introduction of the question
I'm a student in mechanical engineering and we do have a lot of mathematic. We have algebra, calculus and numerical analysis classes. We are tending to learn how to ...
- 21
1
vote
0
answers
69
views
Motivating the multiplication of real numbers
What are some of the ways in which one can motivate the multiplication of real numbers?
The sum of two real numbers can be thought in terms of jumps along a horizontal line. For example, (3)+(-2) may ...
- 527
0
votes
2
answers
123
views
Motivation behind studying Alternating Groups
Going through "A Book of Abstract Algebra" by Charles Pinter now.
At the end of Chapter 8 Permutations of a finite set, he says that:
"The set of all even permutations in $S_n$ is a ...
- 340
0
votes
1
answer
212
views
Combinatorial Technique in Selberg's Symmetry Formula
Before I ask my question, I would like to inform, I am new to this topic from a non-math background, I am trying to understand the topic.
In Selberg, A. (1949). An Elementary Proof of the Prime-Number ...
5
votes
1
answer
387
views
What is the natural motivation for smooth/étale/unramified morphisms restricting from formally smooth/étale/unramified morphisms?
Also at MO.
Smooth (resp. étale) morphisms are just locally finitely presented + formally smooth (resp. étale) morphisms.
For unramified morphisms, it is originally defined in EGA as locally finitely ...
- 891
-1
votes
1
answer
96
views
Tricky questions on one-point compactifications
We understand the one-point compactification of a topological space $X$ is the special way to build a compact space from $X$ by adjoining just one additional point such that $X$ is densely embedded.
I ...
- 821
3
votes
1
answer
178
views
probability and generating functions...motivations and analogies
I have a question similar in spirit to this one.
In essence, what does a generating function (moment, probability, characteristic, other?) "do" to a random variable $X$, and how are the ...
- 2,370
1
vote
3
answers
103
views
Motivation to define $\limsup$ and $\liminf$ of sets
Let $\{A_n\}$ a collection of set. We define $$\limsup_{n\to \infty }A_n:=\bigcap_{n\in\mathbb N}\bigcup_{m\geq n}A_m\quad \text{and}\quad \liminf_{n\to \infty }A_n:=\bigcup_{n\in\mathbb N}\bigcap_{m\...
- 259
4
votes
0
answers
87
views
Geometric Motivation for Inner Product
I think some background will make the kind of answer I'm looking for clearer. I'm trying to think of an elementary proof of the Pythagorean Theorem. I don't like the geometric proofs because they all ...
- 4,403
2
votes
2
answers
171
views
What is the motivation for Liu's definition of an algebraic variety?
I'm currently trying to understand the motivation for Liu's definition of an algebraic variety and in particular, how it arises from and generalises Milne's definition. The question is actually ...
- 469
2
votes
1
answer
260
views
Application of the Bertrand integral
I'll have to explain to my students in an exercise class for what values of $\alpha,\beta\in \mathbb{R}$ the so-called "Bertrand integral" converges
$$ \int_e^\infty \frac{dt}{t^\alpha (\ln(...
1
vote
1
answer
150
views
Motivation for the Notion of Regular Function in Algebraic Geometry
I am studying algebraic geometry, and naturally came across the notion of regular functions as a way to make any algebraic variety a space with functions.
My problem with the notion of regular ...
2
votes
2
answers
436
views
Intuition about Euler's Theorem on homogeneous equations
I wonder, what would be the intuition or motivation to studying Euler Formula for homogeneous function
$f:\mathbb{R}^k \to \mathbb{R}$ such that $f(tx) = t^n f$, for all $t>0$ .
$\sum x_i \frac{\...
- 95
6
votes
3
answers
346
views
How do I go about self-studying Maths? Do I create a routine? Answer a bunch of Q's? [closed]
Im 19F & I'm starting my Computer Science course this year.
Before starting it, I wanted to make sure I had good GCSE & A level Maths knowledge.
The problem is I don't know how to go about ...
- 85