Questions tagged [motivation]

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

353 questions
37
votes
5 answers

Fractional Calculus: Motivation and Foundations.

If this is too broad, I apologise; let's keep it focused on the basics if necessary. What's the motivation and the rigorous foundations behind fractional calculus? It seems very weird & beautiful to me. Did it arise from some set of applications?…
36
votes
7 answers

Why is it important to study combinatorics?

I was having a discussion with my friend Sayan Mukherjee about why we need to study combinatorics which admittedly, is not our favorite subject because we see very less motivation for it (I am not saying that there does not exist motivation for…
user75930
36
votes
4 answers

Motivating (iso)morphism of varieties

I am reading course notes on algebraic geometry, where a morphism of varieties is defined as follows ($k$ is an algebraically closed field): Let $X$ be a quasi-affine or quasi-projective $k$-variety, and let $Y$ be a quasi-affine or…
Daan Michiels
  • 3,196
  • 15
  • 39
34
votes
2 answers

Motivation of stable homotopy theory

A stable homotopy category can be obtained by modifying the category of pointed CW-complexes: objects are pointed CW-complexes, and for two CW-complexes $X$ and $Y$, we take $$\lbrace X,Y \rbrace = \mathrm{colim}[\Sigma^kX,\Sigma^kY]$$ as a hom-set…
H. Shindoh
  • 1,970
  • 15
  • 24
32
votes
3 answers

Motivation for Eisenstein Criterion

I have been thinking about this for quite sometime. Eisentein Criterion for Irreducibility: Let $f$ be a primitive polynomial over a unique factorization domain $R$, say $$f(x)=a_0 + a_1x + a_2x^2 + \cdots + a_nx^n \;.$$ If $R$ has an irreducible…
31
votes
1 answer

Motivation for abstract harmonic analysis

I am reading Folland's A Course in Abstract Harmonic Analysis and find this book extremely exciting. However, it seems Folland does not give many examples to illustrate the motivation behind much of the theory. Thus, I wonder whether there is…
Hui Yu
  • 14,131
  • 4
  • 33
  • 97
29
votes
3 answers

Motivation of Vieta's transformation

The depressed cubic equation $y^3 +py + q = 0$ can be solved with Vieta's transformation (or Vieta's substitution) $y = z - \frac{p}{3 \cdot z}.$ This reduces the cubic equation to a quadratic equation (in $z^3$). Is there any geometric or algebraic…
28
votes
4 answers

Motivation for/history of Jacobi's triple product identity

I'm taking a short number theory course this summer. The first topic we covered was Jacobi's triple product identity. I still have no sense of why this is important, how it arises, how it might have been discovered, etc. The proof we studied is a…
dfeuer
  • 8,689
  • 3
  • 31
  • 59
28
votes
6 answers

Could *I* have come up with the definition of Compactness (and Connectedness)?

Ok, buckle up for a rather long question. I've spent a large portion of today learning about compactness, stemming mainly from this wikipedia article about point-set topology. The article mentions three main things: continuity, connectedness, and…
D.R.
  • 5,864
  • 3
  • 17
  • 44
26
votes
2 answers

Why do we want probabilities to be *countably* additive?

In probability theory, it is (as far as I am aware) universal to equate "probability" with a probabilistic measure in the sense of measure theory (possibly a particularly well behaved measure, but never mind). In particular, we do assume…
26
votes
3 answers

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 the right object of study because you "can't do…
hunter
  • 25,480
  • 3
  • 35
  • 61
25
votes
5 answers

An equation that generates a beautiful or unique shape for motivating students in mathematics

Could anyone here provide us an equation that generates a beautiful or unique shape when we plot? For example, this is old but gold, I found this equation on internet: $$ \large\color{blue}{ x^2+\left(\frac{5y}{4}-\sqrt{|x|}\right)^2=1}. $$ …
Tunk-Fey
  • 24,769
  • 9
  • 78
  • 106
24
votes
4 answers

Motivation behind the definition of GCD and LCM

According to me, I can find the GCD of two integers (say $a$ and $b$) by finding all the common factors of them, and then finding the maximum of all these common factors. This also justifies the terminology greatest common divisor. However, the…
24
votes
4 answers

Why does the generalised derivative have to be a linear transformation?

I am starting to learn Real Analysis and I have come across the generalised definition of the derivative for higher dimensions. I realise that the derivative being a linear transformation nicely accommodates the one dimensional case where the…
24
votes
0 answers

Why is Fourier Analysis effective for studying uniform distributions

On his great expository article about the naturality of the Zeta function in number theory, Tim Gowers makes the following claim: When it comes to the primes, we find that we do not have a good feeling for which numbers are primes, but we do know…
1
2
3
23 24