Questions tagged [intuition]

Mathematical intuition is the instinctive impression regarding mathematical ideas which originate naturally without regard to formal mathematical proofs. It may or may not stem from a cognitive rational process.

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Intuition behind vector field along a local parametrization of a manifold.

In our course on several variable calculus,the following notion was defined: Vector Field along a parametrization: Definition: Suppose $M$ is a $k$-manifold in $\mathbb R^n$.Let $q$ be a point on the ...
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Condition for $\int_{1}^{+\infty} x^a(x-1)^b dx$ be convergent?

I did some full tests $\int_{1}^{+\infty} x^a(x-1)^b dx$ and apparently it is always convergent if $a<-1$, $b>-1$ and $a+b<-1$. Intuitively I think that b must be greater than $-1$ because ...
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Why pyramid with sharp corners and edges is still a manifold intuitively

I'm new to manifolds and in my Computer Graphics class, we briefly explored the topic superficially through visual examples. From my intuitive understanding, we should be able to place a Euclidean ...
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intuitively Understanding meaning of a Combinatorics problem to reach solution

I have been recently taking course on Combinatorics and landed on following problem, Here is the formal statement of problem: A room contains a single bulb and $(2^{2^{10}}+2^{2^9})$ identical ...
Dheeraj Gujrathi's user avatar
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Is there a way of describing why multiplying complex numbers adds their angles intuitively?

Everywhere I'd looked for an explanation of this angle-adding phenomenon, it seemed to have been in one of two forms: Either something roughly like this: $$\left(\cos\left(a\right)+i\sin\left(a\right)\...
NaiDoeShacks's user avatar
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For a non-decreasing function $F$ why should we have the equality $\int_{a}^b f(x)dx\leq F(b)-F(a)$.

I am reading Patrick Billingsley for measure theoretic probability.I encountered the following theorem(Th $32.2$,page-$404$,Billingsley($1995$)).which states the following: Statement: A non-...
Kishalay Sarkar's user avatar
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Intuition behind $\lim _{x \to\infty} \frac{x^n}{a^x}=0$ [closed]

Could someone intuitively explain why $$ \lim _{x \to\infty} \frac{x^n}{a^x}=0 \quad(n \in \mathbb{N}, a>1) $$ Don't we get $\frac{\infty}{\infty}$ in this case, which is indeterminant?
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confusion over statement: If $x' = f(x)$, $f$ is continuously differentiable, then all solutions are monotone or constant.

Here's the statement: If $x' = f(x)$, an autonomous scalar dynamical system for $x\in\mathbb{R}$, where $f$ is continuously differentiable, then all solutions are either strictly monotone or constant....
eddie's user avatar
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How to express the root of an equation of degree $n$, in the product or the sum of two roots of degree $\alpha$ and $\beta$ with $\alpha\beta=n$ [duplicate]

Is there any way to transform the root of an equation of degree $n$ into the product of two roots of degree $a$ and $b$ with $a+b=n$? Practical example #1 $\phi$ is one of the solution of $x^2=x+1$ $\...
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Geometric properties of Snub Icosidodecadodecahedron and Medial Hexagonal Hexecontahedron ($U46$ and $U46'$)

I would like to calculate the closed form of some values ​​relating to $U46$ and $U46'$ (especially angles and volumes). I found this site where the values are given in term of root of equations ...
Math Attack's user avatar
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How to express a fixed root of an equation in terms of a fixed root of another equation.

I'll ask this question by giving a concrete example, which in my opinion is easier to understand: Premise: it is assumed that it is possible to do this because the two equations are connected related ...
Math Attack's user avatar
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Definition of tetration [duplicate]

We all know that many years ago we invented powers. e.g. $3^4$ meant how many times we multiply 3. i.e. $3^4=3\cdot 3\cdot 3\cdot 3$. But then, people started asking questions like what is the ...
Chess player's user avatar
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What does it mean to take the Jacobian of a system of Differential Equations?

When solving nonlinear differential equations, we often use the "Jacobian of the system" to determine if fixed points are stable. As an example, suppose I have a nonlinear system $$x_{t} = f(...
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Category of LCAG's "measures" the difference between the integers and the reals: what does this mean?

The Wikipedia article on Locally compact abelian Groups (https://en.m.wikipedia.org/wiki/Locally_compact_abelian_group) has the following excerpt in the Categorical properties section: Clausen (2017) ...
Pedro Lourenço's user avatar
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The special property of the principal part at a pole

I'm going through Ahlfors, and in the section about the Residue theorem, and he says that about the poles, a function $f$ can be expanded to $$f(z)=B_h(z-a)^{-h}+\cdots+B_1(z-a)^{-1}+\varphi(z)$$ he ...
Redcrazyguy's user avatar
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Intuition behind the the definition of homomorphisms and isomorphisms [duplicate]

I'm familiar with the definition (inverses and bijections, preserving operations) in the context of groups and vector spaces and do know the formal definitions of morphisms as a whole. However, what I'...
Adam Boussif's user avatar
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Why or when can we ignore specific requirements in probability problems due to symmetry?

The Red Sox play the Yankees in a best-of-seven series that ends as soon as one team wins four games. Suppose that the probability that the Red Sox win Game $n$ is $\frac{n-1}{6}$. What is the ...
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Generalization of complete Bell polynomial $B_n(x_1,...,x_n)\mapsto B_{\nu}(f(\nu))$

I want to propose a question that may not have a solution Here I had asked a question: if it was possible to define the integral representation of the polygamma function $\psi^{(\nu)}(x)$ for $\nu>...
Math Attack's user avatar
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intuition behind deriving the equation of a double-napped cone

I know the equation of a double-napped cone is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z^2}{c^2}$ but don't fully understand how this is derived. For a right circular cone centered at the origin, ...
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Intuition behind duality in linear programming

I'm looking for an intuitive explanation of the duality principle in Linear Programming. About having a solution or not: Farkas' Lemma: $A x=b ; x \geq 0$ has a solution <=> $A^T y \geq 0 ; b^T ...
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How to solve $f(ix)=i f^{-1}(x)$

Is there some method to solve this equation? $$f(ix)=i\cdot f^{-1}(x)$$ I found these solutions: $x$ $c_1\arctan\left(\tanh\left(\frac{x}{c_1}\right)\right)$ $c_2\arcsin\left(\sinh\left(\frac{x}{c_2}\...
Math Attack's user avatar
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5 votes
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Visualising trigonometry Identity

Can someone help me visualize this trigonometry Identity Prove that : $$1+\tan A\times\tan \frac{A}{2} = \sec A$$ I got the answer by manipulating the lhs substitution lots of mundane stuff but ...
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Is flux the accumulation of divergence, and circulation the accumulation of curl?

What is the simple relationship expressed verbally between flux, circulation, div, and curl, as captured by Green's, Stokes', and Gauss' Theorems? Below is what I've been able to assemble: Can you ...
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What exactly is a ringed space?

The Question: What is a ringed space? Specifically, how does one think about them? Context: Ringed spaces are important for many fields of mathematics, but for me, I use them in the context of ...
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How do topologies capture the notion of space? A concrete example?

When trying to make sense of locales and comparing to usual topologies, I realized I have no idea how topologies relate to my everyday intuition of space. To make the question simpler, I'll restrict ...
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Intuitive explanation for the formula for the integral of a polynomial

I've only recently learned calculus and I am struggling to understand why integrals give the result that they do. I have watched 3blue1brown's videos which I have found extremely helping in ...
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Intuitive utility of the Jeffreys prior, eg. in Bernoulli trials

I understand the computation the Jeffreys prior, and also its historical motivation. I (somewhat) understand the theoretical desirability of a "prior-construction principle/method" that is ...
cambridgecircus's user avatar
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1 answer
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Intuition behind curvature 2-forms

I recently asked about the intuition behind connection 1-forms on pricipal bundles (Intuition behind connection 1-forms and Ehresmann connections). Thanks to the phenomenal answer I received, I now ...
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Intuition behind Inner product of matrices

I have learnt that the intuitive idea behind inner product space is finding angles between vectors. But what does inner product actually mean physically or intuitively when it comes to matrices.Can I ...
Shreya Jaganathan's user avatar
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Conceptually Understanding the Mathematical Definition for an Envelope of Family of Curves

Assuming we have a one-parameter, two-dimensional, family of curves, given by $f(x, y, p) = 0$, there are two requirements for the envelope (see https://en.wikipedia.org/wiki/Envelope_(mathematics)#) ...
Jacob Ivanov's user avatar
7 votes
2 answers
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Why is it valid to treat units as variables?

I've always taken for granted the fact that units can be treated as variables in mathematical expressions. If you have an object that travels $10m$ in $2s$, you can simply divide the length by the ...
Moyen Medium's user avatar
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How Are Voronoi Polygons Drawn?

Lately, I have come across an interesting concept in Geometry called Voronoi Polygons (https://en.wikipedia.org/wiki/Voronoi_diagram). Below is an example (using R programming with code from https://r-...
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A natural factor map between subshifts

My dynamical systems understanding is somewhat basic, so I wanted to ask the possible following question. Consider a two sided shift over a finite alphabet $\mathcal{A}$, denoted $\mathcal{A}^{\mathbb{...
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Intuition behind quadratic form of SPD matrices $S\Sigma S$

Is there an intuition behind the quadratic form of SPD (symmetric and positive definite) matrices: $$(S, \Sigma) \in \mathcal{S}_{++}\times\mathcal{S}_{++} \mapsto S\Sigma S$$ where $S$ and $\Sigma$ ...
Philipp123's user avatar
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Finding a Kähler manifold from a given exponential family. Intuition?

I'm wondering how to go from: $$ \mathrm{Exponential~ families} \implies \mathrm{Kähler~manifolds} $$ I read that the tangent bundle of an exponential family naturally forms a Kähler manifold. I also ...
John Zimmerman's user avatar
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One of the Specific Intuitions behind Central Limit Theorem

This is a relatively vague question since it occurred during the class couple of semesters back when the Professor tried to explain the "intuition" of Gaussian distribution. The information ...
Partial T's user avatar
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differential dynamic programming : Intuition, key idea and difference from dynamic programming

I cam across 'Differential Dynamic Programming' in a course on Optimal Control. In this course , we were introduced to Dynamic Programming prior to DDP. I went through the Wikipedia Post on ...
Amor Rei's user avatar
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1 answer
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Nature of the set of deltas in limit definition

I was proving the fact that if f is differentiable at a, then (cf)’(a)=cf(a)’ After showing it, I started to play with some ideas and wondered from intuition how to see that depending on the value of ...
Alejandro's user avatar
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Why is optimal transport theory so relevant?

I see plenty of papers published with "optimal transport" in their title and I know that at least 2 Fields medal in the last 10 years were assigned for something related to optimal transport ...
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How to understand change of variables intuitively?

I've been trying to prove or have an intuitive understanding of the change of variables, and I tried it for the function $f(x)=x^2$ using $u(x)=x^2$, the transformed function then becomes $g(u)=u$. ...
Shady Abdulmunim's user avatar
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Algebraically why must a single square root be done on all terms rather than individually?

Let's assume we know that $x+9=10$. I understand this is illegal: $$\sqrt[]{x} + \sqrt[]{9} = \sqrt[]{10}.$$ And this is correct: $$\sqrt[]{x + 9} = \sqrt[]{10}.$$ Is there an intuitive way to ...
Mark Thomas's user avatar
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1 answer
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Intuition behind $Q_t=\sum \langle M^{\alpha},M^{\alpha}\rangle_t+\sum |A^{\alpha}|^3_t+|A^{\alpha}|_t+t$

Consider the semimartingale $Z$, which by Doob decomposition can be written as $Z=M+A$, where M is a martingale and A is the process of total bounded variation. I am trying to make sense of the ...
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Definition of nested partition of the circle

Below is an excerpt from the paper Boundary torsion and convex caps of locally convex surfaces, in which the author defines a so-called nested partition of the circle. I am having a hard time ...
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Explanation of $\{A_i\bigcup B_j\}_{i,j\in I\times J}$

It is in my understanding that a family, $\{A_i\}_{i\in I}$ is a collection of sets $A_i$ with $i$ taking on all possible values within $I$. Of course this is the less formal definition; the most ...
Camelot823's user avatar
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1 vote
3 answers
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Why is it "obvious" that the expected value of a continuous uniform distribution is (a+b)/2?

For a continuous uniform random variable X with support on an interval [a,b], where a<b, one can always calculate the expected value, by integrating, to arrive at the value of (a+b)/2. However, ...
S_M's user avatar
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Interpretation of an example for the Hamilton-Jacobi-Equation

I have the following example of the Hamilton-Jacobi Equation (and first some notations): $$ \partial_t u+H(x, \nabla u(t, x))=0 $$ The associated Hamiltonian equations are $$ \dot{\gamma}=H_p(\gamma, ...
Taleofwoe's user avatar
2 votes
4 answers
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Why does the function $e^{ix}$ have a real part, without using the Euler's formula

I would like to intuitively understand why $e^{ix}$ has a real part, if the the function $e^{ix}$ has an imaginary argument. I know that $$e^{ix}=\cos x + i\sin x$$ and I don't need convincing that it ...
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What is the intuition behind the method of characteristics for a second order PDE?

I understand the idea behind the method of characteristics as applied to first-order PDEs: watching how $u(x,y)$ changes along special curves $(x(s),y(s))$ simplifies the problem to a set of coupled ...
Mohit Kumar's user avatar
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2 answers
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Why is the directional derivative a dot product of a vector and a gradient?

In this question I'm looking for an intuitive explanation, which could provide me some "a-ha!" moments. As we all know, the derivative is just a $\tan$ of an angle in the triangle, whose ...
i cant's user avatar
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Intuition behind Eisenstein's proof of quadratic reciprocity using a function of a complex variable

I have read about one of Eisenstein's proof of quadratic reciprocity using a function of a complex variable, presented here (chapter 2.2, pp.11-15), which I summarized below. It's quite ingenious, and ...
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