# Tagged Questions

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

20k 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 ...
427 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 disk ...
263 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 ...
441 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 ...
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 ...
217 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, ...
98 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 ...
82 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 ...
69 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 ...
124 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. ...
270 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 ...
358 views

336 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. Theorem ...
93 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 ...
315 views

71 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 ...
135 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 ...
148 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? ...
255 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 $N_G(H)$, ...
240 views

63 views

### Graphic of function similar to Sin(x) but with scaled size

I wont to build a graphical representation of numbers, as "waves": Similar as the graphic of sin x but for each number N it gets bigger and bigger crosses the <...
65 views

69 views

68 views

### 3-space viewer?

Is there a software package that would allow visulaizing/rendering some example structures in 3-space? Specifically, I'm thinking of something that would provide a 3-D rendering of, say, 3-vectors (...
(32.) Let $H \le$ group G and let $a, b \in G.$ Prove or disprove. If ${aH= bH},$ then $Ha^{-1} = Hb^{-1}.$ $\color{blue}{Ha^{−1}} = \{\color{magenta}ha^{−1} | h ∈ H\} = \{\color{magenta}{h^{−1}... 1answer 156 views ### What are all the boundary curves for this combined cone and cylinder? [2013 10C] Consider the bounded surface S that is the union of$x^2 + y^2 = 4$for$−2 \le z \le 2$and$(4 − z)^2 = x^2 + y^2 $for$2 \le z \le 4.$Sketch the surface. Use suitable parametrisations for the ... 1answer 103 views ### Can something like$\text{Hom}(V,K)$be visualised? I have no trouble visualising vector spaces like$\Bbb R^3$and (e.g.) a subspace of dimension$2$, which would just be a plane through the origin of a$3$-D space, but I'm having trouble visualising ... 1answer 56 views ### Collection of Intuitive / Visual Derivations of Mathematical Concepts and Formulas I find it difficult to simply memorize mathematical formulas in engineering without understanding what it means and what the result is like, but I realized that many mathematical relationships can be ... 0answers 177 views ### What group are the group of symmetries of these figures isomorphic to - Fraleigh p. 85 Theorem 8.23, 24, 26 In this section we discussed the group of symmetries of an equilateral triangle and of a square. In Exercises 23 through 26, give a group that we have discussed in the text that is isomorphic to the ... 2answers 546 views ### How to Visualize Diagonally Opposite Vertices Consider a cube that exactly fills a certain cubical box. As in Examples 8.7 and 8.10, the ways in which the cube can be placed into the box corresponds to a certain group of permutations of the ... 1answer 582 views ### Animations or Pictures of Group of Rigid Motions (or Rotations) of the Cube Consider a cube that exactly fills a certain cubical box. As in Examples 8.7 and 8.10, the ways in which the cube can be placed into the box corresponds to a certain group of permutations of the ... 2answers 88 views ### Isomorphism of Group with the Image of the Group - Fraleigh p. 82 Lemma 8.15 I found multifarious duplicates that I listed at http://math.stackexchange.com/a/631364/53934. I edged the purple part because my answer proves it more efficiently. I remember that any function$...
Theorem 5.1. Cayley's Theorem: Every group is isomorphic to a collection of permutations. Figure 5.31. A multiplication table for the group $V_4$, with nodes numbered 1 through 4 to facilitate ...