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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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. ...
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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 ...
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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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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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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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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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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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99 views

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

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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36 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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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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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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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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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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142 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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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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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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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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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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34 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 ...
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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 ...
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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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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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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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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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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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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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63 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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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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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 ...
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Intuition. Equivalence of Characterization of Limits and Continuity (Abbott p106 t4.2.3, p110 t4.3.2)

What are the intuitions of these equivalences? Not questioning about proofs or any rigour. I question both equivalences jointly because they look similar. And Are there any figures? ...
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What is the intuition behind the name “Flat modules”?

I am studying Atiyah and MacDonald's book "Introduction to Commutative Algebra" and I have just read the definition of a flat module. It seems to me that if they have called that kind of modules ...
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if $g$ is continuous at $c$ and $g(c)\neq 0$, there exists an open interval containing $c$ on which $f(x)/g(x)$ is defined (Abbott p 113 q4.3.5)

Theorem 4.3.4.(iv) says that $f(x)/g(x)$ is continuous at $c$ if both $f$ and $g$ are, provided that the quotient is defined. Show that if $g$ is continuous at $c$ and $g(c)\neq 0$, then there exists ...
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Why study integrality?

Here are a few of the basic definitions related to integrality. (1) A polynomial in $R[x]$ is monic if its leading coefficient is $1$. (2) An element is integral over a ring $R$ if it ...
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What is the motivation behind a product solution?

Let's consider the simple differential equation: $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$$ And let's assume we have some regular homogeneous boundary conditions ...
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Intuition behind normal subgroups

I've studied quite a bit of group theory recently, but I'm still not able to grok why normal subgroups are so important, to the extent that theorems like $(G/H)/(K/H)\approx G/K$ don't hold unless $K$ ...
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Intuition or figure for Reverse Triangle Inequality $||\mathbf{a}| − |\mathbf{b}|| ≤ |\mathbf{a} − \mathbf{b}|$ (Abbott p 11 q1.2.5)

I acquiesce to Wikipedia's picture for Triangle Inequality. But without referring to Triangle Inequality at all, is there intuition or figure please for Reverse Triangle Inequality for all ...
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51 views

Intuition - Normal Subgroup Test - Fraleigh p. 141 Theorem 14.13

(1.) Not querying proofs or formality. I do this in my other question. Normal Subgroup Test says H is normal in G $\iff gH{g}^{-1}\subseteq H$ for all $g \in G$. What's the intuition of this ...
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41 views

Directional derivatives in any direction do not imply continuity?

I found an example where a function from R^2toR has directional derivatives at a point p at any direction however the function isn't continuous at p. I found this very weird because I thought that ...
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Equivalence Relation definitions of Coset - looks like 1-step Subgroup Test? [Fraleigh p. 97 theorem 10.1]

p. 4 We are especially interested in the case where the set is a group, and the equivalence relation has something to do with a given subgroup. That is, we want to partition a group G into subsets, ...
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How do we arrive at the definite integral to find area approximated by a sum of rectangles?

The area enclosed by a one variable function from a to b can be approximated by $n$ rectangles$$A \approx \sum_{i=1}^{n} f(x_i)(x_i-x_{i-1})$$ and if we let $n \rightarrow \infty$ we get $$A = ...
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Kolmogorov's $0-1$ law and constant RV

Kolmogorov's $0-1$ Law: For any terminal event $A$ we have that either $\mathbb{P}(A)=1$ or $\mathbb{P}(A)=0$. Alternatively any $F_{\infty}$ measurable random variable (so basically a terminal ...
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Prove That the Second Moment is Minimized with a Circle Packing

Graham and Sloane studied the problem of minimzing the second moment of disks on the plane, i.e. minimize $$ U = \frac{1}{d^2} \sum_{i=1}^{n} || \mathbf{p}_i - \bar{\mathbf{p}} ||^2 $$ s.t. ...
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Any group of prime order is cyclic - Proof blueprint [Fraleigh p. 100 Cory 10.11] [closed]

Not querying the proof or formality. I include only part of the proof. The order of the group is a prime number. Call it p. Hence by means of the definition of prime number, $p > 1$. Since the ...
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Intersection of Groups is a Group? Is a Union of Groups? - Fraleigh p. 66 Exercise 6.32h

This is a true or false question, hence are the answers supposed to follow quickly? Because the empty set has no identity element, hence $\emptyset$ is not a group. Hence I'm inquiring for ...
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Meaning of Normal Vector in Surface Integration

Is there a good interpretation of what the normal vector (and its magnitude) $$\mathbf{N}=\frac{\partial \mathbf{X}}{\partial s}\times\frac{\partial\mathbf{X}}{\partial t}$$ to the parametric surface ...
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Picture - Equivalence Relation & Classes, Partitions, Quotient Set, & other related ideas

To get intuition for them and to remember them, I'd be grateful for a picture that combines and embodies the key definitions regarding Equivalence Relations & Classes, Quotient Sets, and ...
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Show that $f(x,y)= \|x-y\|_2^2$ is differentiable

Problem: Show that $f: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ with $f(x,y)=\|x-y\|_2^2$ is differentiable and compute its differential at every point in the domain of $f$Note: $\| \cdot ...
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Physical significance of knot vector in B-spline.

A B-spline blending curve formulation is: $P(u)=\sum_{k=0}^np_k B_{k,d}(u)$ Given $n+1$ control points, B-spline blending functions are polynomials of degree $d-1$, $(1<d<=n+1)$. ...
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What is the relationship between the second isomorphism theorem and the third one in group theory?

The second isomorphism theorem [wiki] in group theory is as follows: Let $G$ be a group. $H \triangleleft G, K \le G$. Then: $HK \le G$, $(H \cap K) \triangleleft K$, and $K/(H \cap K) ...