**234 motivation questions.**

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 ...

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\...

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 &...

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 ...

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.

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 ...

This week, I come to the end of the first year analysis, and suffer from a "crisis of motivation." With this question, I want to chase away my thought, "Why is it important to study the general ...

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 ...

$\newcommand\CC{\mathsf{C}}$The notion of a category is well-known. There are multiple equivalent definitions; small categories can be seen as an internal category in $\mathsf{Set}$, that is
a couple ...

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 $a,b,x\in A$:
$a\circ a =a$;
$(x\circ a)\circ a=x$...

I am about to finish my first year of studying mathematics at university and have completed the basic linear algebra/calculus sequence. I have started to look at some real analysis and have really ...

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 ...

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 ...

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.

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 & ...

I feel like I have a pretty good way to talk about $\varepsilon$-$\delta$ to physics and engineering students (and possibly students in comp sci). But I am not very sure what I can do for chemistry ...

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 ...

When solving linear nonhomogeneous equations, we deal with two types of solutions:
particular
homogeneous
Why do we have these two types of solutions for differential equations? What does each of ...

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 $...

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 ...

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'...

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}...

What is the physical meaning of the fractional integral and fractional derivative?
And many researchers deal with the fractional boundary value problems, and what is the physical background?
What ...

Simply as the title says. I've done some research, but still haven't arrived at an answer I am satisfied with. I know the answer varies in different fields, but in general, why would someone study ...

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 ...

What is the most motivating way to introduce continued fractions? Are there any real life applications of continued fractions?

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 ...

I'm trying to learn linear algebra on my own but I am stuck on the definition of a linear subspace.
Let's assume I want to find out if $S$ is a subspace of $\mathbb{R}^2$, where $ S = [X_1 , X_2] $ ...

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 ...

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), ...

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 ...

Right now, I am reading Larry Washington's book "Introduction to Cyclotomic Fields." In Chapter 8 of this book, the unit group of the ring of integers in a cyclotomic field (or its totally real ...

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_{\...

I've already asked about the definition of tensor product here and now I understand the steps of the construction. I'm just in doubt about the motivation to construct it in that way. Well, if all that ...

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

(Reminder: A $T_0$ topological space, also known as a Kolmogorov space, is a space where the topological structure "recognizes" that different points are different: No two points have exactly the same ...

I know that the spectral theorem tells us that in the case of a real inner product space, an operator is self adjoint if and only if there is an orthonormal basis with only eigenvectors of that ...

Landau's inequality is a pretty result that bounds the Mahler measure (product of absolute value of leading coefficient and absolute values of roots which have absolute value at least 1) by the norm ...

The informal intuition for the limit of a function is this:
What is the value of the function $f$ as $x$ gets infinitely close to $c$?
How on earth does this monster
$$ \lim_{x \to c} f(x) = L \...

I'm seeking examples of concepts or theorems in school mathematics that are better understood when we generalize (when we deal with a more abstract concept where the former concept is a special case ...

I just got through reading section I.6 in Hartshorne, and I have no idea why anybody would ever have invented these concepts. I agree that what you can do with it is pretty neat, but how would anyone ...

Suppose for every $i\in\mathscr I,$ $X_i$ is a topological space.
The product space has as its underlying set the product set $X =\prod \limits_{i\,\in\,\mathscr I} X_i$ and as its open sets product ...

Why people in sixteenth century (or the people now ) interested in solving the cubic?
There were (I think) no number theoretic or relation to science that time, and the only impression I get from ...

I tried to understand the definitions of cofibration and fibration through wikipedia. But I didn't get any motivation why such things are important, and through which basic natural question they come? ...

I'm trying to understand adjoint functors better and I have to admit I'm a little bit confused by the idea of the unit/counit of an adjunction.
I have written out units, counits and their triangle ...

Disclaimer: My apologies for making such a long question. The question is possibly also rather specific, but I hope that (some parts of) it might be useful in general.
Background: I have recently ...

I'm currently following an introduction to Riemannian geometry i.e. connections, curvature and isometric immersions (the Gauss, Codazi and Ricci equations).
I find the introduction to Riemannian ...

The Hilbert Symbol it superficially similar to the Legendre Symbol: it measures whether or not solutions to some polynomial exist. In the case of the Legendre Symbol it was clear for me that it is a ...

I'm learning inequalities for the first time, and except a paragraph by Paul Zeitz in his book Art and Craft of problem solving, none actually give much motivation of why should I care about ...

I'm learning polynomials (for competitions) the first time, with having a little number theory and combinatorics experience.
The difficulty is that I can't understand
why polynomials are so ...

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