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2
votes
1answer
39 views

How would you draw $(A\setminus B)\times (A\setminus B) = (A\times A)\setminus (B\times B)$?

I know it's useful to prove set equalities to make a quick sketch of the sets described. How can I draw this one? $$(A\setminus B)\times (A\setminus B) = (A\times A)\setminus (B\times B)$$
8
votes
2answers
130 views

0 to the power of 0, what does the essential discontinuity actually look like?

So having watch this clip by Numberphile which explains why $0^0$ is undefined https://www.youtube.com/watch?v=BRRolKTlF6Q And also this http://mathforum.org/dr.math/faq/faq.0.to.0.power.html And ...
33
votes
5answers
6k views

The Stupid Computer Problem : can every polynomial be written with only one $x$?

When I was a child, I wanted to be a mathematician so I asked my parents to buy me a computer to make super complex calculations. Of course, they were not crazy enough to buy an expensive super ...
1
vote
2answers
53 views

What is the fourth dimension of a Tesseract?

Is the fourth dimension of the Tesseract time? That is why it is represented as a moving 3D structure on Wikipedia? I am asking because I have trouble understanding what it is.
-1
votes
1answer
35 views

Books on the visual/graphical aspects of geometry

Are there any books providing a general overview of the visual/graphical aspects of geometry? For example, Tilings (e.g. hyperbolic) and tessellations Plane/space filling shapes/objects (e.g. ...
1
vote
0answers
36 views

Explaining Spin(3)

I’m going to discuss the action of Spin(3) on Euclidean vectors. This thing has several alternative names: “versors”/“rotation quaternions”, “quaternionic adjoint representation”, “quaternion action ...
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 ...
0
votes
0answers
8 views

Are there tools for presentation and vizualization of deduction?

I read that Kalish and Montague introduced a natural deduction method (http://en.wikipedia.org/wiki/Donald_Kalish), which can be easily implemented in software. Any other tools who can help a logician ...
0
votes
1answer
22 views

How to visualize implicit functions

I have a task of visualizing few implicit functions. Firstly lets say I have the following function of $N$: $$\epsilon = \sqrt{\frac{8}{N}\ln \left( \frac{4(2N)^{50}}{0.05} \right)}$$ Now this is ...
4
votes
0answers
58 views

How to visualise Bollobas' 1965 theorem?

Theorem $[n]=\{1,\ldots,n\}$. Let $\lbrace (R_i, S_i), i \in I \rbrace, R_i, S_i \subset [n]$ be such that $R_i \cap S_i = \emptyset, R_i \cap S_j \ne \emptyset (i \ne j)$. Then $$\sum_{i \in I} ...
0
votes
0answers
9 views

Draw an ellipse corresponding to a bivariate normal distribution

Let $$\mu=\left(\begin{array}{c}\mu_1 \\ \mu_2\end{array}\right)$$ and $$\Sigma=\left(\begin{array}{cc}\Sigma_{1,1} & \Sigma_{1,2} \\ \Sigma_{2,1} & \Sigma_{2,2}\end{array}\right)$$ be ...
2
votes
1answer
74 views

Why is it not possible to visualise a 4th dimension object? [duplicate]

By drawing a cube on a paper or by seeing it on a screen (a 2D surface - see Figure below), we can sort of visualise how a 3D cube would look like. I was wondering whether we will be able to ...
1
vote
1answer
25 views

How to visualize probability distributions in terms of sets - joint and marginal?

Let there be two sets, $\mathcal{X},\mathcal{Y}$, both finite, and they represent the set of values that the discrete random variables, $X,Y$ can take. $\mathcal{P}_{Y|X}$ be all possible ...
0
votes
1answer
17 views

visualizing a function from the plane into the reals

If $$(x_1,y_1), (x_2,y_2),(x_3,y_3)$$ are points in the plane and if $a,b$ are fixed real numbers, how can I visualize $$f=(ax_1+b-y_1)^2+(ax_2+b-y_2)^2+(ax_3+b-y_3)^2$$ as a function from the plane ...
4
votes
5answers
277 views

How to graph/visualize complicated inequalities

I'm having trouble visualizing areas defined by for example, $$ x^2 + y^2 \leq 2y $$ Or $$ (x^2 +y^2)^2 \leq 2a^2(x^2 - y^2) $$ What is the thought process in picturing these regions?
0
votes
0answers
21 views

Affine Transformation and Continuous Deformation

How do these two concepts relate? Thus far I have a (what I think is a) good intuitive idea of a continuous deformation- the visual basically looks like the boundary being stretched so that it never ...
2
votes
2answers
66 views

How can I visually imagine the area of a circle divided by $\pi$?

If I have a circle with an area of 100 units^2, and I divide it by $\pi$, how can I imagine that visually in my mind? Since 100 / $\pi$ =~ 31.83, and the square of that is =~ 5.64, I currently ...
4
votes
2answers
70 views

Visualizing Lie algebra of SO(3)

Let $SO(3)$ be the Lie group of 3D rotations. Rotation about z-axis by an angle $\phi$ is represented in standard basis by this matrix: $$ \begin{pmatrix} \cos \phi & -\sin\phi & 0 \\ ...
1
vote
2answers
40 views

Parallel Lines Intersecting in the Projective Plane

My question is about visualizing projective space, in particular the real projective plane $\mathbb{P}^2(\mathbb{R})$. I know there are different ways to define this space, but in each we can say that ...
3
votes
3answers
71 views

Viewing an abelian group using cayley diagram

I cannot understand this way of viewing whether a group is abelian using cayley's diagram: (from Visual group theory book) What I can't understand is that while checking being abelian we check ...
1
vote
1answer
35 views

Order of an element in direct product using cayley's diagram

How can I find the order of element (1,1) of the group $C_4\times C_3$ visually in the diagram below :
0
votes
1answer
35 views

Analysing/Visualising shape of multi-variate function.

I have an unknown function $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ for which I'm determining a first order Taylor approximation through a non-linear optimization process in six variables (the ...
2
votes
1answer
38 views

Criteria of regularity that a Cayley's Diagram should meet .

As referred in the Visual group theory Book by Nathan Carter- The unofficial definition of a group says that : A group is a collection of actions satisfying the rules: 1. there is a predefined list ...
1
vote
2answers
262 views

Visualization of Eratosthenes’ sieve

In otherwise great paper on prime numbers, I found following visualization of Eratosthenes’ sieve: I found it somewhat scary and confusing. Is there any better visualization of Eratosthenes’ sieve ...
0
votes
1answer
63 views

Line integrals of vector fields-positive, negative, or zero

I have a question about line integrals of vector fields being positive, negative, or zero. If you are measuring the work it takes to "push" a point on the curve through the vector field, does this ...
1
vote
0answers
76 views

Geometrical application of generation function for permutation

It is quite well known that the generation function for permutations is represented as $$(1+x)(1+x+x^2)\dots(1+x+x^2+x^3...+x^{n−1})$$ (See, e.g., question The generating function for permutations ...
1
vote
1answer
54 views

Plotting the intersection of multiple surfaces with WolframAlpha

I want to plot the intersection of two surfaces like in this post. But if I enter the much simplified expression ContourPlot3D[{x^2 + y^2 + z^2 - 4=0, xy=1}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}] ...
86
votes
5answers
10k views

How were 'old-school' mathematics graphics created?

I really enjoy the style of technical diagrams in many mathematics books published in the mid-to-late 20th century. For example, and as a starting point, here is a picture that I just saw today: ...
2
votes
1answer
41 views

Usefulness of alternative constructions of the complex numbers

Complex numbers $\mathbb{C}$ are usually constructed as $\mathbb{R}^2$ together with a suitable multiplication. But this is not the only possible way, one can get to the complex numbers. One ...
1
vote
2answers
43 views

$2\times$ distance to a line${} = {}$distance to a point, in geogebra

While preparing for my math exams, I got this question: give the locus (if thats the right word) where $2$ times the distance to the line l $x=8$ equals $1$ time the distance to point F $2,0$. I was ...
0
votes
0answers
24 views

Nyquist limit explanation

Kindly explain Nyquist in easy words. The actual question is as follows. We can attempt to display sampled data by simply plotting the points and letting the human visual system merge the points into ...
1
vote
0answers
20 views

Different ways of visualizing a certain classes of single variable complex functions

Single variable complex functions of a real variable are ubiquitous in engineering contexts such as control engineering and signal processing, and visualizing them is of utmost importance in designing ...
1
vote
1answer
55 views

Visualizing second-order Markov chain

You can visualize a first-order Markov chain as a graph with nodes corresponding to states and edges corresponding to transitions. Are there any known strategies to visualize a second-order Markov ...
16
votes
10answers
2k views

Visualizing the square root of 2

A junior high school student I am tutoring asked me a question that stumped me - I was wondering if anyone could shed some light on it here. We were talking about how the square root of 2 is an ...
0
votes
0answers
84 views

Intuition/Picture - Matrix Multiplication - Product of [Row or Column Vector] and Matrix [Lay P95]

This question is not a duplicate of the original, in which user Shuchang proved the question. Presently I'm asking about further intuition or a picture, and no proofs please. $1.$ Intuitively, in ...
0
votes
1answer
24 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 ...
0
votes
0answers
37 views

What kind of graph was github's impact graph?

Github used to have a graph called an impact graph. It feels almost like a Sankey diagram and almost like a stacked area chart. What is the name of this kind of graph? I couldn't find a better ...
2
votes
2answers
48 views

How to Complete Sketch of a function of two variables $ f(x, y) = 3x - x^3 - 2y^2 + y^4$ ? [Stewart P930 Question 14.7.4]

For $ f(x, y) = 3x - x^3 - 2y^2 + y^4$ $\implies$ $\partial_x f = 3 - 3x^2, \partial_y f = -4y + 4y^3$. Set both equations to 0 $\implies x = \pm $1 and $y = 0, \pm 1$. $1.$ To determine the ...
1
vote
1answer
80 views

Sketch Saddle Point of a function of two variables $ f(x, y) = 4 + x^3 + y^3 - 3xy$ [Stewart P930 Question 14.7.3]

As regards $ f(x, y) = 4 + x^3 + y^3 - 3xy$, I computed that (0,0) is a saddle point, and (1,1) is a local minimum. So I'm not asking about this, and am asking only about sketching contours. $1.$ ...
0
votes
2answers
51 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} ...
1
vote
1answer
39 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 ...
2
votes
1answer
52 views

Pictures for Expectation

Is there a good way to visualize the formula: $$ E(x) = \int_{0}^{\infty} 1 - F(X) \,\mathrm{d}x $$ ? for positive continuous random variables? I understand the formula as far as basic calculus ...
0
votes
0answers
43 views

Computer-aided study of elementary geometry

As a beginning student of elementary (euclidean plane) geometry, so far, I have gotten the impression that there are two major approaches to geometries: naive vs axiomatic. Being a humanities student ...
3
votes
1answer
52 views

Penrose tilings as a cross section of a $5$-dimensional regular tiling

Could somebody explain to me how a penrose tiling , which is not periodic, can be a cross section of a regular tiling in $5$ dimensions, which is periodic? It does not make sense to me how a periodic ...
4
votes
2answers
67 views

Which Cross Product for the Desired Orientation of a Hyperboloid ? [Stewart P1103 16.9.8]

P1103 16.9.$8.$ Evaluate the surface integral $\iint_S \mathbf{F} \cdot d\mathbf{S}$. $\mathbf{F} = (x^3y,-x^2y^2,-x^2yz)$ and $S$ is the surface of the solid bounded by the hyperboloid $x^2 + ...
7
votes
3answers
97 views

What does the vector space $\mathbb{R}^{\mathbb{R}}$ look like?

I can imagine $\mathbb{R}^{\mathbb{N}}$. For instance, the set of real series is part of this space, as is any infinite (but discrete numbered) tuple of reals. But how can I imagine ...
2
votes
1answer
40 views

Determine Cross Product with Left Hand vs Right Hand

If I perceive http://en.wikipedia.org/wiki/Cross_product correctly, then to determine the cross product With a right hand, let: the 1st vector in the cross product = your index finger = in red ...
3
votes
1answer
66 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 ...
2
votes
3answers
96 views

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? ...
6
votes
4answers
2k views

Factorial of 1,e+80

Recently I started being very fascinated in logistics, and out of the blue came the question into my head, what is the factorial of the amount of atoms in the observeable universe, which is said to be ...