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.
4,436 questions
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What is the correct definition of a limit point in real analysis?
This question relates to two (seemingly) conflicting definitions of Limit Points in real analysis.
The definition of limit points and closed sets from my notes is written as:
A much more general ...
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Is there an intuitive way to understand this formula for the magnitude of the projection of one vector onto another? [duplicate]
Let's say it's 200 B.C. and you're tasked with building all of modern math from the ground up. Let's say also that we already intuitively understand the concepts of a "vector", the "...
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Why are we satisfied to explain the power of Lebesgue integration just by saying it is ''horizontal, rather than vertical''?
A standard intuition found in textbooks for the power of the Lebesgue integral compared to its Riemann counterpart is that "We integrate by taking horizontal slices, rather than vertical ones.&...
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Connections between the surface $x^{3}+y^{3}+z^{3}-3\cdot x\cdot y\cdot z=1$ and the curve $\left(x,y,z\right)=\left(h_{3,n}\left(t\right)\right)$?
What are other relationships between the surface $x^{3}+y^{3}+z^{3}-3\cdot x\cdot y\cdot z=1$ and (the curve defined by) functions $x=h_{3,0}\left(t\right)$, $y=h_{3,1}\left(t\right)$, $z=h_{3,2}\left(...
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A technical question regarding forward passes [closed]
I am a mathematician writing an article on rugby forward passes and am looking for a little help with a definition.
Issue is this:
If I am standing on the 25 metre line and pass the ball laterally ...
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Isomorphism between geometric tangent vectors and derivations
I am trying to understand the proof of the following Theorem:
Let $a \in \mathbb{R}^n$.
a) For each geometric tangent vector $v_a \in \mathbb{R}_a^n$, the map $D_{v|a}: C^{\infty}(\mathbb{R}^n) \...
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Is it a must for a singular solution to have the dependent variable in it in case of PDE?
Exercise Question given in text book :
For the question (ii) here, from the equation $(2)$, we could eliminate $a$ and $b$ and say that $xy=1$ is the singular solution. But that was not done here. As ...
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Large numbers estimation
For commonly discussed enormous yet finite numbers, such as Graham's Number or TREE(3), is there any computation of their order of magnitude that can be expressed like $\log(N)$ or $\log(\log(\log(...(...
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How to show that if $(X_i)_{1 \leq i \leq n}$ are an ordered n-tuple of sets, then the Cartesian product, is a set
I am trying to do the following exercise from Tao's Analysis I but I think I may not have the correct intuition on how to approach the proof:
Show that if $(X_i)_{1 \leq i \leq n}$ are an ordered n-...
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How to combine the difference of two integrals with different upper limits?
The following link describes the Integral Comparison Test and its proof, along with this diagram which I understand.
This next theorem (and its proof) is what I am trying to understand:
There is ...
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Trying to understand Slope Fields in ODE
I am trying to understand Slope Fields.
I only found explainations but no straight up definitions. This is what I got thus far:
Consider the general first order ODE $$y'(x)=f(x,y)$$ Then at each
...
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Why is the product between a quaternion and its conjugate a scalar?
I am reading "Naive Lie Theory" book by Stillwell. When the author introduces quaternions, these are viewed as matrices:
$$q = a\mathbf{1}+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$$
where
$$\...
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Derivation of the integrating factor
Is this a valid derivation of the integrating factor?
\begin{align}&\frac{d\mu }{dx}=\mu P(x) \longrightarrow \frac{1}{\mu }d\mu =P(x)dx\longrightarrow \int \frac{1}{\mu }d\mu =\int P(x)dx\\\...
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What a homomorphism means [closed]
"I'm studying universal algebra and am currently working on the topic of algebra homomorphisms.
I'm currently trying to capture the essence of a homomorphism using Feynman's method. As I ...
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Conditional probabilities in a sudoku
While making sudoku puzzles I came up with the following question:
Suppose there is a square $a$ in which there can be a $3$ or a $4$.
Obviously, the probability of there being a $3$ is equal to that ...
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Equivalence class of Devil's Staircase function
In trying to clarify the picture of $L^p$ spaces in my head, I've been trying to get to grips with what equivalence classes of functions really look like. That they differ on say $\mathbb{Q}$ is not a ...
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Understanding the order on $\mathbb{N\times N}$ defined by $(m,n)\leq (p,q)\iff \frac{2m+1}{2^n}\leq\frac{2p+1}{2^q}$. [closed]
I encountered an order relation $\leq$ on the set $\mathbb{N\times N}$ defined by $$(m,n)\leq (p,q)\iff \frac{2m+1}{2^n}\leq\frac{2p+1}{2^q}\text{, for every }(m,n),(p,q)\in \mathbb N\times \mathbb N$$...
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Intuition on the Tangent Space of Orthogonal Matrices
I am studying Lie Groups with a shaky background on manifolds and I am having problems with a proof that my professor showed about the dimension of the Orthogonal group $O_n(\mathbb{R})$, which ...
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Intuition behind the representation theory of semi-direct products (Mackey's theorem)
I've been attempting to understand the irreducible representations of semi-direct products involving abelian groups; this is primarily motivated by a desire to understand Wigner's classification of ...
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A property of spiral closed curves
I have a question related to a simple property of a spiral closed curve, by which I mean figures of the following kind:
I want to somehow prove that this kind of a closed curve satisfies both the ...
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Why this particular change of variables in this PDE?
I am trying to study some methods of resolution of PDEs, for my exam of mathematical methods for physics. Currently I am reading “A guide to mathematical methods for physicists” (volume 2) by Petrini, ...
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$\infty$-Categories as Framework with "Better Homotopical Behaviour"
In a comment below this question Zhen Lin indicated that one of the major advantages to work with $\infty$-categories instead of ordinary ($1$-)categories is that these exhibit better homotopical ...
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Need help understanding the value of this infinite continued fraction intuitively
If I have the fraction,
\begin{equation} \cfrac{1}{1-\cfrac{1}{1-1}} \end{equation}
the value of it is undefined because it involves a division by zero.
The same holds for a finite number of ...
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Can we use L'Hôpital's Rule in treating the ODE.
Suppose we have the ODE ($N \ge 2$):
$$
-\frac{1}{(\psi(r))^{N-1}}\left[(\psi(r))^{N-1} u^{\prime}(r)\right]^{\prime}=f(u(r))
\tag{0}
$$
Then we will get
$$
(\psi(r))^{N-1} u^{\prime}(r)=-\int_{r_0}^...
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Intuition for group 2-cocycle
Is this intuition about cocycles appropriate?
If I have an abelian group $A$ and a group $G$ and I want to build a new group where the multiplication has a twist like $(a, g)\cdot (b, h) = (a+b+\omega(...
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