For questions about visualizing mathematical concepts. This includes questions about visualization of mathematical theorems and proofs without words.

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2
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2answers
57 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
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1answer
136 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
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2answers
58 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
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1answer
41 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
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1answer
56 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
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0answers
55 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 ...
4
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1answer
85 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 ...
2
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2answers
77 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
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3answers
121 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 ...
0
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1answer
71 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
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1answer
78 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 ...
0
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3answers
120 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
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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 ...
0
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2answers
136 views

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 ...
1
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1answer
211 views

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 ...
2
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1answer
257 views

Cauchy-Schwarz Inequality - Proof using Projections [Lay P379 Thm 6.7.16]

t If $u=0$, then the inequality becomes $ 0 \le 0 $, which is true. See Practice Problem 6.7.1 on P382. If $u\neq 0$, let $W$ be the subspace spanned by $u$. $1.$ How would one determine to ...
0
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5answers
187 views

Proof strategy for $(\Leftarrow)$: If $g \circ f = id_A$, then $f$ onto $\Leftrightarrow$ $g$ 1-1. [Chartrand 3Ed P239 9.72]

For nonempty sets $A$ and $B$ and functions $f \colon A \to B$ and $g \colon B \to A$, suppose that $g \circ f =$ the identity function on $A$. $(♦)$ (e) $(\Leftarrow)$ Assume that $g$ is ...
2
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1answer
123 views

Intuition - Countable iff Surjection iff Injection [Velleman P310 Thm 7.1.5]

Define $I_n = \{1, 2, ..., n \} $. Let $A$ be a nonempty set. TFAE : (i) $A$ is finite (ie: a bijection $h:A\rightarrow I_{N}$ exists) or A is countably infinite (ie: a bijection ...
1
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2answers
103 views

Proof strategy for $(=>)$: If $g \circ f = id_A$, then f onto $\iff$ g 1-1. [Chartrand 3Ed P239 9.72]

For nonempty sets A and B and functions f : A → B and g : B → A, suppose that $g \circ f =$ the identity function on A. $(♦)$ (d) $(=>)$ Assume that $f$ is onto. This means there exist ...
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20answers
17k views

Visually deceptive “proofs” which are mathematically wrong

Related: Visually stunning math concepts which are easy to explain Beside the wonderful examples above, there should also be counterexamples, where visually intuitive demonstrations are actually ...
2
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3answers
335 views

Polar coordinates for $xz$-plane: $z = r\sin\theta$ ? [Stewart P1091 16.7.25]

$1.$ The unit disk is projected onto the xz-plane, so shouldn’t $x = 1\cos \theta$ and $\color{red}{z = 1 \sin \theta} $? User Semsem below kindly identified the problem: The normal to the ...
2
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2answers
223 views

Problem using Stokes's Theorem - Boundary Curve, Unit Normal Vector [Stewart P1097 16.8.5]

$\Large{1.}$ How does one determine the boundary curve, denoted as C, to be the plane $z = -1$? I’m flummoxed because $S$ here is given as bottomless. I'm not enquiring about formal or rigorous ...
4
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3answers
336 views

Visualizing mathematics and geometry

Im writing a paper on the role of visualization in mathematics and specifically geometry. I was wondering if it is possible to represent any arbitrary system of relations and manipulable objects ...
17
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6answers
2k views

Visual explanation of the following statement:

Can somebody fill me in on a visual explanation for the following: If there exist integers $x, y$ such that $x^2 + y^2 = c$, then there also exist integers $w, z$ such that $w^2 + z^2 = 2c$ I know ...
0
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2answers
121 views

Is this a counterexample to “continuous function…can be drawn without lifting” ? (Abbott P111 exm4.3.6)

I'm au courant with http://math.stackexchange.com/a/288133 and http://math.stackexchange.com/a/422001. They're both Abbott P111 exm 4.3.6 which proves "a continuous function is sometimes described, ...
0
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1answer
70 views

How to visualize four dimensional tic-tac-toe?

I have played three dimensional tic-tac-toe with three players before, and we had no problem visualizing it. We drew three layers on a sheet of paper and just remembered all the different ways you ...
1
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1answer
64 views

Which line is the antiderivative and why?

The graph of a function $f$ is shown. Which graph is an antiderivative of f and why? This should be easy but I keep second guessing myself so I thought I'd check with you magnificent people. I'm ...
1
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1answer
65 views

Tools or Resources for pictures and visualizations

The popularity of books like Visual Group Theory and Visual Complex Analysis validates the importance of pictures and visualization for complex subjects. Unfortunately, I'm not aware of similar books ...
0
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0answers
145 views

Intuition for $\inf(AB) = \inf(A)\sup(B)$. Difference for sets and functions? (Abbott pp 199 q7.4.5)

1. What's the intuition for $\inf(AB) = \inf(A)\sup(B)$? Figure please? I know I must posit $A,B \subseteq R$ as bounded sets. If they're unbounded, $\sup$ doesn't exist. I believe $\inf(AB) = ...
0
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3answers
110 views

If $\int^{b}_a f > 0$ then there is some interval and $\delta > 0$ on which $f(x) \ge \delta$ (Abbott pp 199 q7.4.4d)

True or False. If $\int^{b}_a f > 0$, then $\exists \; [c,d] \subseteq [a,b]$ and $\delta > 0$ such that $f(x) \ge \delta$ for all $x \in [c,d]$. 1. We need to determine if true or false. ...
0
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1answer
191 views

Cauchy's Generalized Mean Value Theorem. Required function. (S.A. pp 140 t5.3.5)

Cohen, Henle. Calculus pp 827, (http://www.vias.org/calculus/09_infinite_series_10_06.html) I revised the footnote in pp 14 http://math.uga.edu/~pete/2400calc2.pdf. This theorem can be illustrated ...
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1answer
286 views

If f' exists and f'(c) > 0 then f'(x) > 0 for all |x - c| < d for some d. (S.A. pp 137 question 5.2.8b)

If $f'$ exists on an open interval, and there is some point $c$ where $f'(c) > 0$, then there exists a d-neighborhood $\{x \in \mathbb{R} : |x - c| < d\} = V_d(c)$ around c in which $f'(x) > ...
0
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1answer
60 views

logic - how to model or diagram conditional probabilities of multiple related scenarios.

I am interested in modeling questions and specific outcomes so that i can evaluate conditional probabilities and mathematical expectation. I am looking for a way to diagram or otherwise describe the ...
2
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3answers
75 views

Intuition - $fr = r^{-1}f$ for Dihedral Groups - Carter p. 75

Name $r$ = clockwise 90 deg. rotation and $f$ = flip across the square's vertical axis = the brown $\color{brown}{f}$ in my picture underneath. Zev Chonoles's $f$ is different. Carter fleshes out why ...
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1answer
184 views

Intuition - Theorem - A group homomorphism preserves normal subgroups - Fraleigh p. 149. Theorem 15.16

p. 128, 129. Theorem 13.12. Let $h$ be a homomorphism of groups $G \to G'$. III. If $S \le G$, then $h[S] \le \color{red}{G'}$. IV. If $S' \le G'$, then $h^{-1}[S'] \le G$. p. 149. ...
0
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1answer
61 views

Eigenvalues and Corresponding Eigenspace Bases

Could someone describe the eigenvalues of $ \left( \begin{array}{cc} 2 & 1 \\ -1 & 2 \end{array} \right) $, as well as the bases of the corresponding eigenspaces? I received eigenvalues ...
4
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3answers
270 views

Visual explanation of $\pi$ series definition

Can you visually explain why the following is true: $$ \frac{\pi}{4} = \sum\limits_{k=0}^\infty \frac{(-1)^k}{2k + 1} = \frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}\ldots\approx 78.5\% ...
0
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0answers
128 views

Intuition of Picture - Collapse, Factor Group, Homomorphism, Normal Subgroup - Fraleigh p. 144 Figure 15.1

Let $N \unlhd G$. In the factor group $G/N$, the subgroup $N$ acts as identity element. Regard N as being collapsed to a single element, to the identity element. This collapsing of N together ...
1
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0answers
114 views

Intution, Figure. Negation of Continuity and Uniform Continuity (S.A. pp 117 T4.4.6)

Every time I need negation, I have to write out all the logical symbols to negate manually. I know how to determine these negations myself. But I want to compehend intuition or figure like ...
1
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0answers
104 views

Intuition on Axiom of Completeness

♪ (J. Stewart. Calculus 6th ed. pp 682) Axiom of Completeness = AoC = A nonempty set of real numbers that has an upper bound has a least upper bound. AoC is an expression of the fact that there ...
4
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3answers
1k views

Direct proof. Square root function uniformly continuous on $[0, \infty)$ (S.A. pp 119 4.4.8)

(http://math.stanford.edu/~ksound/Math171S10/Hw8Sol_171.pdf) Prove for all $e > 0,$ there exists $d > 0$ : for all $x, y \ge 0$, $|x - y| < d \implies |\sqrt{x} - \sqrt{y}| < e$. (a) ...
1
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1answer
295 views

Characterize normal subgroups - Find all subgroups of $S_3$ conjugate to $\{id, (1,3) \}$ - Fraleigh p. 143 14.29

(27.) A subgroup H is conjugate to a subgroup K of a group G (viz. p. 141 $K \le G$ is a conjugate subgroup of $H$), if $i_g[H] = gHg^{-1} =K$ for some $g \in G$. Show that conjugacy is an ...
3
votes
2answers
257 views

Why is this a good picture of a covector?

I'm reading a book about applied differential geometry and the author says: "suppose $V$ is a finite dimensional vector space. For a given covector $\omega \in V^\ast$, the set $\hat{\omega}$, of ...
1
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1answer
40 views

Draw following set in $\Bbb{R}^3$

How can I draw following set in $\Bbb{R}^3$? $$ M = \left\{(x,y,z) : \sqrt{x^2 + y^2} \le z \le 1 \right\} $$ I have the answer in the book but I don't want to check it before i try to solve it ...
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1answer
50 views

If $\phi[H] \subseteq H'$, homomorphism from G to G' induces homomorphism from G/H to G'/H' - Fraleigh p. 143 14.39

Let $H \trianglelefteq \text{ group } G$ and let $H' \trianglelefteq \text{ group } G'$. Let $\phi$ be a homomorphism of G into G'. Show that if $\phi[H] \subseteq H'$, then $\phi$ induces a natural ...
3
votes
2answers
109 views

Intuition - Fundamental Homomorphism Theorem - Fraleigh p. 139, 136

Let $\phi: G \to H$ be a group homomorphism with $K = \ker\phi$. Then $G/K \simeq \phi[G]. $ The hinge to the proof is to define $\Phi: G/K \to \phi[G]$ given by $\Phi(gK) = \phi(g)$. Then we must ...
1
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1answer
95 views

Visualize meaning of quotient in quotient map, group - etc?

What are the reasons for the name "Quotient" in Quotient map, group - etc? Overhead picture shows each of the three cosets in $A_4$ is mapped to a single - gray - node. But this isn't division? ...
2
votes
0answers
160 views

Visualize normal subgroup, normalizer, cosets.

A few important aspects of the relationship $H \lhd N_G(H) \le G$ are highlighted in Figure 7.31. First, the size of $N_G(H)$ is some multiple of |H|, and the size of G is some multiple of $N_G(H)$, ...
4
votes
2answers
177 views

Visualize Fundamental Homomorphism Theorem for $\phi: A_4 \rightarrow C_3$

Question 1. How do you see $\ker\phi = V_4 $ = Klein 4 group ? Book doesn't give formula for $\phi$? Question 2. What's $H$ in $i(aH) = \phi(a)$? I think $H = \ker\phi$ ? Question 3. Why is $i: ...
1
vote
0answers
47 views

Visualizing a complex curve

This may be as much a question about computers as about math. Let $C=\{f(r,s,t)=0\}$ be a curve in $\mathbb{CP}^2.$ By forgetting about the points at infinity we can view $C$ as a surface in ...