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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The relationship between the intercepts and the remainder in the remainder theorem

The polynomial remainder theorem states that when a polynomial $P(x)$ of degree $> 0$ is divided by $x-r$ ($r$ being some constant) the remainder is equal to $P(r)$, that is: $$\begin{array}l If &...
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1answer
702 views

What is the significance of integrating a function?

Now i understand how important these things can be in terms of very small changes or finding area under curves and otherwise. However, when we integrate a function such as y = x we get (x^2)/2, and ...
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191 views

General formula for dependent probability distributions

Recently I encountered the following problem: What is the mean distance between two random points on a unit square? I understand pen and paper methods for solving this exist however I'm ...
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206 views

What functions have the property that $\frac{d}{dx}f(x) = c \cdot f(x+1)$?

If we are allowed to pick any real-valued constant $c$ that helps, when does $$\frac{d}{dx}f(x) = c \cdot f(x+1)$$ In other words, when does the derivative of a function $f(x)$ equal some constant ...
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1answer
83 views

Can we possibly exchange summation and integration with negative values?

This is an attempt to go further than this answer. Essentially, we have either a summation of an integral: $$\sum_x{ \left( \int{ f(x)dx } \right) } \tag{1}$$ ...or an integral of a summation: $$\...
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1answer
79 views

When can we use substitution for both integrals and summations?

This question is partially inspired by Qiaochu Yuan's answer to "Will moving differentiation from inside, to outside an integral, change the result?". Essentially, I would like to know, if we have: ...
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3answers
696 views

Why is the expected number coin tosses to get $HTH$ is $10$?

Can someone please explain why is the expected number of coin tosses to get the sequence of $HTH$ is $10$? What is the intuition and formulas behind this?
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278 views

Developing intuition in algebraic geometry through differential geometry?

I'm interested in algebraic geometry (I'm working through Ravi Vakil's notes and also have worked with curves and general varieties in the past), and I have seen some basic definitions from ...
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157 views

Can't see the intuition behind the validity of this formula: $\exists x(\exists yP(x,y) → \forall z \exists wP(z,w))$

I know that $$\vdash_{\mathcal G}\exists x(\exists yP(x,y) → \forall z \exists wP(z,w))$$ (I have read and done a syntactic proof of this.) And therefore also $$\models \exists x(\exists yP(x,y) → ...
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1answer
59 views

When do Entries Remain, after and despite Matrix Multiplication? [Strang P92 2.5.41]

Suppose $E_1, E_2, E_3$ are 4 by 4 identity matrices, except $E_1$ has $a, b, c$ in column 1 and $E_2$ has $d, e$ in column $2$ and $E_3$ has $f$ in column 3 (below the $1$ s). Multiply $L = ...
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234 views

Is there a deeper meaning when a number is squared? [closed]

In my opinion, math is about more than just memorizing equations, it's about numbers that are built in a way that represents our understanding of something. So I ask this, what does it mean ...
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461 views

How to understand blowing up a submanifold

I am trying to understand the idea of blowing up a submanifold of a smooth real manifold. The definition I know is replacing the submanifold by its unit tangent bundle (however, in the place I read ...
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0answers
47 views

What is a complete intersection?

I was reading and I encountered something that goes: We have degree $d$ polynomials in $s$ variables $F_1, ..., F_n$ with coefficients in integers. Let $X$ be the complete intersection defined by the ...
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1answer
56 views

Scale-invariance of $\int_0^\infty \frac{f(x)}{x} \ dx$

Let $f$ be some non-negative, measurable function on $[0,\infty)$. The quantity $\int_0^\infty \frac{f(x)}{x} \ dx$ is scale-invariant in the sense that, if one puts $f_c(x) := f(cx)$ for $c > 0$, ...
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1answer
66 views

Definition of a continuous function

I am struggling to understand a basic definition of a continuous function from a textbook: A function f is continuous if for all x, and for all $\epsilon>0$, there exists $\delta>0$ such that ...
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1answer
39 views

What is an intuitive extension of extreme-values and critical points in one variable to multiple variables?

While it is simple to grasp limits in multiple variables, since the formal definition extends in the obvious way, I am having a harder time grasping the same concept with critical points and extreme ...
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106 views

Terminology in forcing

In the context of forcing one reads the relation $p \leq q$ in a poset $P$ as "$p$ extends $q$". A typical example is the poset $P$ of finite partial functions, where one defines $p \leq q$ when $q \...
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2answers
158 views

Compound interest coumpounded n time per year formula. $A=P\left(1+\frac{r}{n}\right)^{nt}$ intuition behind it.

I know that the compound interest formula for the interest compounded annually is given by $$A=P(1+r)^t$$ I know the intuition behind it. But why the compound interest formula for the interest ...
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1answer
68 views

What is $R$-algebra and do I need to understand $R$-modules for it?

I was given the following definition of $R$-algebra: Let $R$ be a commutative ring. An $R$-algebra is a ring $A$ (with $1$) together with a ring homomorphism $f : R \to A$ such that $f(1_R)...
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3answers
976 views

Line Integrals and Surface Integrals

Can someone please explain what surface integrals and line integrals are measuring? Is a line integral the arc length along a surface, and a surface integral is the surface area? Also, why is a line ...
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2answers
87 views

What's the fastest way to determine Eigenvalues & Eigenvectors of any 2 by 2 Matrix?

My instructor claims that it's inefficient and superfluous to compute eigenvectors de novo for each $2$ by $2$ matrix. He suggested a trick instead which resembles the eigenvectors and cases here. He ...
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1answer
116 views

Intuition or wisdom for stability and instability properties of locally linear system. Boyce, p513, Table 9.3.1

Our instructor requires us to memorize this table for our differential equations exam. So I wonder if anyone has some deeper intuition or observation to help with this? For example, I noticed that ...
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10answers
463 views

Why is it that $\int_a^b \int_c^d f(x)g(y)\,dy\,dx=\int_a^b f(x)\,dx \int_c^d g(y)\,dy$?

The title sums it up. It's simple to prove, but I'm wondering if there is a geometric interpretation?
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1answer
41 views

If $f(2\alpha-\theta) = f(\theta)$, then $\theta=\alpha$ is a line of symmetry of $r=f(\theta)$. How do you derive $f(2\alpha-\theta) = f(\theta)$?

For Polar Coordinates I know that for x-axis symmetry $f(-\theta)=f(\theta)$, for y-axis symmetry $f(\theta)=f(\pi-\theta)$, and for symmetry about the origin $f(\theta)=f(\theta+\pi)$. The big ...
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69 views

Why $\dfrac{d}{dt} \dfrac{dy}{dx} = \dfrac{d}{dx} [ \dfrac{dy}{dx} ] \quad \dfrac{dx}{dt} $ ? [Stewart P206 3.4.95, BDP P165 3.3.34]

If $y=f(x)$, and $x = u(t)$ is a new independent variable, where $f$ and $u$ are twice differentiable functions, what's $\dfrac{d^{2}y}{dt^{2}} $? By the chain rule, $\dfrac{dy}{dt} = \dfrac{dy}{dx} ...
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1answer
46 views

If $z = f(x, y)$, then why are $\partial_x z$ and $\partial_y z$ functions of x and y also? [Stewart P905]

This is Figure 5 from P905 which appears to show this, but Stewart doesn't write this explicitly or explain. I'm interested in an informal, intuitive explanation please. I'm not interested in a ...
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3answers
121 views

What is a smooth curve in $\mathbb{R}^2$ intuitively?

While studying for my exam, I've run into some problems understanding what a smooth curve in $\mathbb{R}^2$ is. I first thought that, intuitively, I could think of a piece of string on a piece of ...
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3answers
190 views

Why is $\cos(x)$ the derivative of $\sin(x)$? [duplicate]

The derivative of $\sin(x)$ is $\cos(x)$, and the derivative of $\cos(x)$ is $-\sin(x)$. Is there a simple proof of this, preferably using pictures?
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rank($A$)=rank($A^T$) [duplicate]

Is there an elementary explanation of why the row-rank of a matrix equals its column-rank (without using adjoint maps, resp. lots of technical computations)? What is the geometric intuition behind ...
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1answer
57 views

Intuition and Motivation - Linear Operator $T - \lambda_k I$ ? [Lay P270 Thm 5.1.2]

Let $T$ be a linear operator on a vector space V, and let $\lambda_{1},\ \lambda_{2},\ \ldots,\ \lambda_{k}$ be distinct eigenvalues of T. If $v_{1},\ v_{2},\ \ldots,\ v_{k}$ are eigenvectors of $T$ ...
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192 views

A graph of all of mathematics

In mathematics, one often makes (proves) statements on the basis of: Previously proven statements Axioms I like to think of these dependencies as a directed graph, with edges from the accepted ...
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1answer
55 views

Find the area bounded by the hypercycloid

Parametrization : $x = acos^3(t), y = asin^3(t)$ $a>0$ If you can solve it for me that would be awesome.:D If not, can you give me some hints? Tell me how to set it up and stuff. It's solveable ...
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0answers
97 views

understanding Green's theorem Intuition

The idea of it is to find the area of a region, yet I keep seeing vector fields popping up all over the place. Take this example from my text book: Find the region enclosed by the two graphs: $y = ...
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2k views

What, fundamentally, is the reason for the shape of a sin curve?

Say we have a metal bar in space aligned horizontally and we start rotating it counter-clockwise about its left end. Then, the sin of the angle from between the horizontal and the bar is the y ...
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1answer
246 views

Stokes' Theorem Explanation

Can someone explain what Stokes' Theorem is measuring? What would taking the integral of a vector on a surface give you? When would you use it? This is the only definition I have and I don't really ...
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2answers
318 views

What's the intuition behind definition of chaotic function?

I read books A First Course in Discrete Dynamical Systems by Richard A. Holmgren and An Introduction to Chaotic Dynamical Systems by Robert L. Devaney. I want to understand which concepts of "chaos" ...
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207 views

Gaining Linear Algebra Intuition — Subspaces

So I aced linear algebra over the fall semester, though I'm deeply troubled in that I struggle to really describe what I did. I cannot say with confidence what it all meant, nor do I have any sort of ...
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1answer
110 views

Example for finite dimensional analog of integral transforms

I understand that integral transforms are generalisations of the dot product of functions that could be interpreted as infinite dimensional vectors. The most significant advantage then is that ...
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1answer
41 views

Finding Cos of an angle between matrices

I have two $2\times 2$ matrices and it's asking me to find the Cos of the angle between them. Firstly, how do yall visulize matrices and the angles between them? I think that's my first problem. ...
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1answer
64 views

Why must an inner function of a compound function be linear in order to integrate it using the power rule?

This is from my previous thread: $\int(1+x^2)^4\mathrm dx$ $\ne$ $\frac{(1+x^2)^{5}}{5(2x)}+C$? because differentiating back gives $\frac{d}{dx}(\frac{(1+x^2)^5}{10x}+C)=\frac{10x(5(2x)(1+x^2)^4)-10(...
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3answers
109 views

Intuition - Linear Congruence Theorem

Let a and b be integers (not both 0) with greatest common divisor d. Then an integer $c = ax + by$ for some $x, y \in Z$ $\iff d|c$. In particular, d is the least positive integer of the form ...
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325 views

Fermat's Little Theorem fails for composite instead of prime numbers.

I know Fermat's Little Theorem = Fermat-Euler's Totient Theorem when $n$ is prime. Elementary Number Theory, Jones, p83 writes if we simply replace p with a composite integer n, then the ...
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1answer
114 views

Ground plan of Forward direction - Let $p$ be an odd prime. Prove $x^{2} \equiv -1 \; (mod \, p)$ has a solution $\iff p\equiv 1 \; (mod 4)$

Prove by contradiction. Thence suppose NOT $p\equiv 1 \; (mod 4)$. Thence 3 possibilities remain: $4|p, 4|(p - 2), 4|(p - 3)$. But $p > 2$ is prime, thence $4 \not | p$. (1) How can you ...
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141 views

Natural example where $\ell_\infty$ distance appears.

The $\ell_2$ distance has a natural connotation: the straight line distance between two points "as the crow flies". Similarly, the $\ell_1$ distance has a natural connotation: the length of a path ...
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320 views

How can one visualize a homomorphic mapping.

It has been a year or so studying Group theory and Ring theory. Funnily enough, this is the part where i am able to solve most of the questions of the book quite easily, yet not fully understanding ...
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1answer
269 views

Ramanujan's personification of small positive integers

I dimly recall reading somewhere (perhaps in "The Man Who Knew Infinity"?) that Ramanujan associated personalities (perhaps it was mystical personalities, e.g. specific gods and goddesses?) with small ...
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6answers
88 views

Why does $ 1+2+3+\cdots+p = {(1⁄2)}\cdots(p+1) $ [duplicate]

I saw this from Project Euler, problem #1: If we now also note that $ 1+2+3+\cdots+p = {(1/2)} \cdot p\cdot(p+1) $ What is the intuitive explanation for this? How would I go about deriving the ...
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1answer
90 views

Geometric intuition behind subspaces in $\mathbb C^n$

While learning elementary linear algebra one develops a great deal of geometric intuition in $\mathbb R^n$. It helps to see the forest for the trees and leads through proofs. After meeting ...
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2answers
142 views

Gradient and Swiftest Ascent

I want to understand intuitively why it is that the gradient gives the direction of steepest ascent. (I will consider the case of $f:\mathbb{R}^2\to\mathbb{R}$) The standard proof is to note that the ...
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1answer
257 views

A sequence converges $\iff$ it's Cauchy. Proof of ($\Leftarrow$) (Abbott p 59 t2.6.4)

Lemma 2.6.3 $\implies (x_{n})$ is bounded. So use the Bolzano-Weierstrass Theorem to produce a convergent subsequence $(x_{n_{k}})$ . Set $x= \lim x_{n_{k}}.$ So $(x_{{n_{k}}}) \to x. \quad \Large{(♪)...