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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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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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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