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.

learn more… | top users | synonyms (1)

2
votes
0answers
23 views

Geometric Interpretation of Antiderivative?

Could someone please give me a geometric interpretation of the above theorem?
0
votes
0answers
36 views

Geometric interpretation of Cauchy-Goursat Theorem?

This theorem seems almost magical. The algebraic derivation doesn't really provide any insight into why it works. So could someone give me a geometric interpretation of it? This: Geometrical ...
1
vote
0answers
20 views

Geometric interpretation of analyticity?

Suppose the real valued functions $u(x,y)$ and $v(x,y)$ are continuous and have continuous first order partial derivatives in a domain $D$. If $u$ and $v$ satisfy the Cauchy Riemann equations at ...
3
votes
3answers
99 views

Self-studying Russell's Paradox

I'm self-studying and having trouble wrapping my head around Russell's paradox, even after looking here. I'd really appreciate a more intuitive explanation of the concept before I move on to ...
5
votes
3answers
55 views

Intuitive understanding of path integral formula

I have learned a general formula for a path/line integral $$ \int_a^b f\left(\mathbf{r}(t)\right) \|\mathbf{r}'(t)\|\ dt \tag{1} $$ and I'm trying to better understand it. Specifically, I'm ...
0
votes
1answer
42 views

Proof and interpretation of $\mathbb{E}[\mathbb{E}[X \mid Y, Z] \mid Z] = \mathbb{E}[X \mid Z]$

First, I understand that $\mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X \mid Y]]$, but how to prove that $$\mathbb{E}[\mathbb{E}[X \mid Y, Z] \mid Z] = \mathbb{E}[X \mid Z]?$$ Second, for ...
2
votes
1answer
30 views

Why is $x$ restricted this way? (limits of functions)

Here is a corollary from Ross' Elementary Analysis: Why is $x$ restricted this way?
2
votes
4answers
356 views

What does continuity *in general* mean?

I am looking from : http://en.wikipedia.org/wiki/Lipschitz_continuity Continuously differentiable $\subseteq$ Lipschitz continuous $\subseteq$ α-Hölder continuous $\subseteq$ uniformly continuous ...
1
vote
1answer
72 views

Why is conic isomorphic to $\mathbb{P(C^2)}$?

Given a quadratic form $C(x)=x_1 ^2+x_2 ^2 + x_3^2$ in $\mathbb{C}[x_1,x_2,x_3],$ we have a conic $$C=\{C(x)=0\} = \{[x_1:x_2:x_3]: x_1 ^2+x_2 ^2 + x_3^2 = 0\}$$ in $\mathbb{P(C^3)}$, given in ...
1
vote
0answers
17 views

Why is an open interval needed in this definition? (definition of a limit of a function)

Here's a part of the definition Ross' Elementary Analysis states for limits of a function: In both parts of the definition, why are open intervals needed? Would it fail if it were a closed interval ...
0
votes
4answers
135 views

Two plus two equals four when earth has one moon?

As is well known, we have the least intuitive of basic operations, the 'implication' or '=>'. Consider 'A => B'. Most beginners get stumped on the vacuous truth, that implication could be true even ...
1
vote
0answers
25 views

Quick Question on a Proof of Artin-Wedderburn Theorem

Artin-Wedderburn Theorem: Let $M$ be a semisimple $R$-module of finite type. Then $$\text{End}_R(M)\cong\bigoplus_{i=1}^k M_{n_i\times n_i}(D_i)$$ for some division rings $D_i$. Proof: ...
5
votes
2answers
57 views

Intuition behind sum of multiplication arithmetic sequence

Maybe this is a stupid question but please guide and enlighten me patiently. I have just known something fact that quite shocking me. Let start from this simple fact $$\sum_{k=1}^n ...
1
vote
2answers
25 views

Smallest open, dense, G-invariant subset of a metric space

Let $X$ be a metric space and $G$ be a topological group acting continuously on $X$. Let $ \mathcal S $ be the set of open, dense and $G$-invariant subsets of $X$. I need to take inverse limit (of ...
0
votes
1answer
49 views

How to visualize $ \mathcal P \ ( \ \mathcal P \ ( \ \mathbb R ^2 \ ) ) $?

So,$ \ \mathcal P \ ( \ \mathbb R ^2 \ ) $ , the power set of the set of all ordered pairs of real numbers, contains every imaginable (2D) function, black and white image and text as per its ...
3
votes
1answer
61 views

Role of determinant of the matrix of any Homology group.

I was thinking about the proof of the Lefschetz's Fixed point theorem and the ingeniuty of the Hopf's Trace formula, i.e. associating the trace of the matrix for deciding about the fixed points. Now ...
0
votes
0answers
9 views

Developing intuition for the modulus of continuity $\omega$?

I am currently in a course which is going through rigorous definitions of continuity, convergence, integrals, etc. I am trying to develop an intuition in understanding the modulus of continuity, ...
6
votes
2answers
73 views

If f(x)dx is a rectangle with height f(x) and width dx, what is f(z)dz in complex analysis

I am trying to intuitively understand the multiplication $f(z)dz$ in complex analysis. For instance, $f(x)dx$, we are all aware, is a rectangle with height $f(x)$ and width $dx$ so its multiplication ...
3
votes
1answer
45 views

Find $T_\mathrm{id}\left(\mathrm{Diff}(S^1)\right)$

We established on last tutorial that $T_\mathrm{id}(\mathrm{Diff}(S^1))$ are vector fields on $S^1$. I'd be grateful for any explanation (formal or intuitive) standing behind this answer.
-1
votes
2answers
45 views

F(A ∩ B) ⊆ F(A) ∩ F(B) laymen translation?

I am suppose to prove the above statement but i have got diffculty understanding it in the first place. Could anyone help me translate it into laymen language?
0
votes
2answers
63 views

Looking for intuïtive explanation why contour integral of $\frac{dz}{z} $equals $2\pi i$ in complex analysis

$$\oint \frac{dz}z = 2\pi i$$ I've seen the derivation of it using the parametrisation. Since this result is used all the time in my complex analysis course, i'd like to understand this ...
2
votes
3answers
33 views

Extensions Fields and Zeros of Polynomials

I've just digested the proof for Kronecker's theorem that states every polynomial $f(x)$ over a field $F$ has a zero in an extension field of $F$, mainly, the factor ring of polynomials ...
1
vote
1answer
67 views

Building intuition in group theory

I'm finding it hard to translate abstract results of group theory into something that intuitively makes sense. Putting this into a concrete example: if $f:G\to H$, $Im(G)$, is a subgroup of $H$? Is ...
5
votes
2answers
130 views

Significance of homology groups of a topological space

I am studying homology groups of topological spaces. In books I have found that the $n$th homology group counts the number of "$n$-dimensional holes" which exist in that space. If I consider homology ...
0
votes
2answers
29 views

Intuitive understanding of relationship between unit vectors and position vector

On Wolfram Mathworld they give a unit vector in the $\textbf{x}_n $ direction as: $$ \hat{\textbf{x}_n} \equiv \dfrac{\frac{\partial \textbf{r}}{\partial x_n}}{\lvert \frac{\partial ...
7
votes
3answers
228 views

What's a good motivating example for the concept of a slice category?

What nice example can one give a beginner to really motivate the idea of a slice category, before they've met the more general notion of a comma category? There's the toy example of a poset category ...
8
votes
1answer
53 views

Intuition behind a certain limit.

We want to find $\displaystyle\lim_{\theta\to\frac{\pi}{2}} b_1-a_1$, we are given $c=1$ and that $\cdot=90^{\circ}$ This is my solution; $$\begin{equation}\sin \theta=\frac{b_1}{a_1} \iff b_1=a_1 ...
0
votes
0answers
28 views

Pairwise Independent Events that are not Independent

Let a ball be drawn from an urn containing 4 balls, numbered 1,2,3,4 . let E = {1,2}, F = {1.3}, G = {1,4} . If all outcomes are assumed equally likely, then P(EF) = P(E)*P(F) = 1/4 P(EG) = ...
1
vote
2answers
46 views

Is it possible to gain intuition into these trig compound angle formulas - and in general, final year high school math?

Does anyone have any insight into the trig sum and difference formulas? The formulas in themselves are very elegant, but I don't really like the proofs that have been given, even the geometric proofs. ...
0
votes
0answers
32 views

Perimeter of an ellipse intuition help

I am aware that you can take the circumference of an ellipse using an elliptic integral and haven't looked much into it, but that seems to be a bit extreme and i was taking a personal look at conic ...
1
vote
2answers
25 views

I don't understand the definition of a base of a metric space

Definition: A collection {$v_n$} of subsets is said to be a base for X if for every x $\in$ $X$ and every open set $G$ $\subset$ $X$, such that x $\in$ $G$ we have x $\in$ {$V_n$} $\subset$ $G$ for ...
0
votes
1answer
22 views

A Base of a metric space intuition

From what I have read online and from what I have read in Rudin, a collection of open sets $\lbrace$$V_{n}$$\rbrace$ is said to be a base for a metric space $X$ if every open set in $X$ can be ...
1
vote
2answers
68 views

Intuitive Understanding of the First Isomorphism Theorem

I've been reading some things about an intuitive understanding of the first isomorphism theorem, and there's just one more part that I do not understand. Specifically, the first isomorphism theorem ...
0
votes
0answers
34 views

Understanding what $P - P \log(P)$ means for an event of probability $P$

Let $(\Omega, \Sigma, \mathbb{P})$ be a probability space, $X$ be a random variable, and $E \in \Sigma$ be an event with $\mathbb{P}(E) = P$. Then $P - P \log(P) \in [0, 1]$, for all $P \in (0, 1]$, ...
2
votes
2answers
79 views

What is the motivation to build measure theory?

I started reading about measure theory on wikipedia, and downloaded some PDFs, but they all start defining things that I can understand, but can't imagine the motivation to define these things. ...
1
vote
0answers
34 views

Intuition for problem involving binomial random variable

Question: The below algebraic solution is simple enough. But is there a way to "see" the answer using a clever trick or intuition? Given the algebraic solution, I feel like there should be. I just ...
0
votes
0answers
16 views

How could one go about constructing this relatively simple contagious diffusion-reaction model?

How could one go about constructing a contagious diffusion-reaction model showing the relationship between disease (e.g. Ebola) and number of available healthcare workers in an unevenly distributed ...
3
votes
2answers
97 views

Find the time interval between oscillations of SHM.

Parts i) and ii) I can solve. But for part iii) I can't do, as I don't know which equation describes the SHM motion? Is it $y=0.5sin(1.2t)$ or $y=0.5cos(1.2t)$ or $x=0.5sin(1.2t)+2.5$? I thought ...
1
vote
0answers
27 views

Practical Intuition behind converge of random sequence $X_n \to X$

Can someone come up with a practical example as to what it means for a set of random variables to converge i.e. $X_n(\zeta) \to X(\zeta)$? Specifically, what is the meaning of $n$ and $\zeta$?
17
votes
6answers
2k views

Why do some mathematical ideas seem counter-intuitive?

Suppose you play the following game: There's a certain buy-in, and at every turn you flip a coin. If anytime you flip a tail, you lose the game and leave with your winnings. If you flip a head on the ...
2
votes
1answer
27 views

Motivation for Definition of Measurable Function

I'm having trouble understanding why a function is defined as "measurable" if the preimage of every measurable set is measurable. I see the parallel to the definition of continuity, and the latter ...
0
votes
1answer
53 views

Why does the order not matter? Partial D

When taking partial derivatives, why does the order not matter as long as the function is continuous? Any proof, intuitive or rigorous?
0
votes
1answer
62 views

Physical significance of the fact that the cardinality of the real number line is the same as a finite interval of the real number line

It is known that the cardinality of the real number line is the same as a finite interval of the real number line. Is there a physical meaning of this apparently conter-intuitive statement?
2
votes
2answers
70 views

Matrix exponentiation intuition.

What does $x^A$ intuitively mean if $x \in \mathbb{C}$ and $A$ is any matrix? Also, what if we had $x$ being a matrix too? Last but not least, what happens if we have a complex $x$ raised to a ...
1
vote
1answer
32 views

general mean value theorem

Can anyone give me the intuitive explanation of the general mean value theorem stated in my notes as under: Let $f:U\rightarrow \mathbb R$ and $U\subseteq \mathbb R^n$ and let $f$ is differentiable ...
2
votes
1answer
39 views

Intuition for Euler's Partition Theorem

Euler's Partition Theorem states the following: Every number has as many integer partitions into odd parts as into distinct parts. I played around with small examples (I wrote out the partitions ...
1
vote
0answers
31 views

Dirichlet Distribution - the underlying intuition.

I'm not a math expert, but I need dealing with some math tools for natural language processing research. One of the most common tools is the Dirichlet distribution. I know that with a multinomial ...
0
votes
1answer
31 views

Conflict between geometric intuition and computed answer

Evaluate the integral $\int_{C} z ds$ where C is the intersection of $x^{2}+y^{2}=4$ and $z=0$ (oriented clockwise as viewed from above). My interpretation of this problem yields the following ...
0
votes
0answers
22 views

Negative integrals spherical coordinates

Need help interpreting a result prompted by the following question. Solve the triple integral of $\sqrt{x^{2} + y^{2} +z^{2}}$ on the region Q where Q is bound by $z=-\sqrt{9-x^2-y^2}$ and the x-y ...
0
votes
0answers
28 views

Geometric intuition for mixed partial derivatives

I'm trying to better understand exactly what $f_{xy}(x,y)$ at a point is geometrically, and possibly understand why $f_{xy}$ and $f_{yx}$ should be equivalent, not just because the math happened to ...