Questions about studying mathematics without formal instruction.

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6
votes
1answer
229 views

How to most efficiently remedy mathematical deficiencies

My spouse and I are currently both pursuing our undergraduate degrees. I'm double majoring in Computer Science & Mathematical Sciences and my spouse is double majoring in Economics & Finance. ...
4
votes
1answer
106 views

Construct a conformal mapping from $\Bbb C$ Onto $R$ if such a map exists. And explain why if does not exist.

Let $R$ be the domain obtained by removing the non negative real numbers from $\Bbb C$. Construct a conformal mapping from $\Bbb C$ Onto $R$ if such a map exists. And explain why if does not exist. ...
2
votes
1answer
57 views

Scheme-Theoretic Nakayama's Lemma

Let $X$ be a noetherian scheme and $\mathscr{F}$ a coherent $\mathscr{O}_{X}$-module. For a point $x \in X$, let $k(x)=\mathscr{O}_{X,x}/\mathfrak{m}_{x}$ be the residue field at $x$. (a) Suppose $x ...
1
vote
0answers
37 views

Classify all invertible meromorphic functions

I am studying complex analysis. And I see a topic, namely meremorphic functions But I cannot find any enough information about this function. So I have insufficient knowledge about this topic. ...
1
vote
0answers
36 views

What areas of mathematics are taught in a Computer Engineering course?

I'm planning on taking a Computer Engineering course next year, I study hard when it comes to math so I wanna know what area of mathematics I'm going to tackle during my course so I can study it ...
4
votes
0answers
211 views

Find a conformal map from the disc to the first quadrant.

Find a conformal mapping of the disk $x^2+(y-1)^2\lt 1$ onto the first quadrant $x, y \gt 0$ I did something, which may be false or not, I cannot exactly say anything. I used the composition of ...
1
vote
1answer
212 views

Showing how to find the vertices of the circle.

Find that the circle has four vertices. $$\gamma (t)=\langle R\cos (t/R), R \sin (t/R)\rangle$$ for $t\in [0,2\pi]$ I know the theorem: Every simple closed convex curve has atleast four ...
9
votes
2answers
599 views

Learning Abstract Algebra for a graduate degree

I would like to do a graduate degree in mathematics, and I have a full year before I will be able to do so (for personal reasons). I mainly have my weekends available to study. I am interested in ...
2
votes
1answer
380 views

How to find a conformal mapping of the first quadrant.

Find a conformal mapping of the first quadrant onto the unit disc mapping the points $1+i$ and $0$ onto the points $0$ and $i$ respectively. I think that i need to use "the change of variables ...
2
votes
2answers
200 views

Conformal map example $ f(z)=e^z$

I an studying the example-1. I understand $f(z)=e^z$ has a nonzero derivative at all points, hence it is everywhere conformal and locally $1-1$. But I dont understand th part I underlined with ...
3
votes
2answers
119 views

Why not $f(z)=z^2$ conformal at $z=0$?

$$f(z)=z^2$$ is not conformal at $z=0$ Why? Conformal definition: $f$ is conformal at z if f preserves angles there.
1
vote
2answers
199 views

Study regimen for discrete mathematics? - Lack high-school maths…

I have just gotten into college, and will be studying mathematics from next semester. (this course) Unfortunately I did not study mathematics for the last 2-3 years of high-school mathematics. What ...
1
vote
3answers
144 views

Solutions to $x+y+z=31$ and $x+2y+3z=41$

For the equations $$x+y+z=31$$ $$x+2y+3z=41$$ is there a elegant way or method to find all the positive solutions in integers? Thus far, I have been using trial and error (which is time consuming). ...
1
vote
1answer
57 views

How to construct a term of a particular type

I am reading the article "Introduction to Type Theory" by Herman Geuvers, the chapter explaining the Fitch style of natural inference. I stuck at the exercise 1.3 (first two are simple): ...
8
votes
2answers
159 views

Given $N$, count $\{(m,n) \mid 0\leq m<N, 0\leq n<N, m\text{ and } n \text{ relatively prime}\}$

I'm confused at exercise 4.49 on page 149 from the book "Concrete Mathematics: A Foundation for Computer Science": Let $R(N)$ be the number of pairs of integers $(m,n)$ such that $0\leq m < N$, ...
8
votes
7answers
4k views

Calculus book recommendations (for complete beginner)

Well I have not started calculus yet but I am really keen to. I would love if you suggest some books. Points to be noted: I really don't like the way textbooks are written so please no "textbooks" ...
1
vote
1answer
50 views

Open immersion from a proper scheme to a separated, irreducible scheme.

Fix a scheme $S$ and let $X$ and $Y$ be $S$-schemes. Assume that $X$ is proper over $S$ and $Y$ is separated over $S$. Let $f: X \rightarrow Y$ be an open immersion of $S$-schemes. If $Y$ is ...
1
vote
1answer
82 views

Trying to understand an exercise using factorials with induction

Exercise: Prove that (n + 1)! - n! = n(n!) for any n $\ge$ 1 Given Answer: I will skip the basic step since I understand that part. (n + 2)! - (n + 1)! = (n + 1)!(n + 2) - n!(n + 1) I understand ...
0
votes
1answer
106 views

Resources for exploring math without a teacher

The ability to understand the beauty of math requires rigorous study. However, most people do not have access to the kind of training pure math requires. Many of my friends easily get interested in ...
0
votes
0answers
20 views

Question regarding differentation with respect to functions

I am reading some papers which include differentiation wrt functions rather than real numbers. I follow the proofs, and am able to verify that they hold, but still do not feel comfortable that I would ...
3
votes
1answer
108 views

Definition of $ 1 + \cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{\ddots}}}}$

Is there a definition of $ 1 + \cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{\ddots}}}}$? I am somewhat familiar with continued fractions; that is, I am aware that their convergence depends on whether ...
4
votes
1answer
79 views

The curve has constant torsion.

Question: Show that when the curve $c_1=c_1(t)$ has constant torsion $\tau$, the curve $$c_2=c_2(t)=-\frac{1}{\tau}N+\int_{t_0}^{t}B(u)du$$ has constant curvature $-\tau$ or $+\tau$. What I ...
0
votes
1answer
75 views

$(\frac{1}{\kappa})^2+(\frac{\dot{\kappa}}{\kappa^2\tau})^2=r^2$

Show that for a curve lying on a sphere of radius r with nowhere vanishing torsion, the following equation is satisfied: $$(\frac{1}{\kappa})^2+(\frac{\dot{\kappa}}{\kappa^2\tau})^2=r^2$$ Please ...
3
votes
1answer
85 views

Showing the parametrically representation of hyperbolic paraboloid. And how to find the curves $u$ and $v$ be constant.

Show that the hyperbolic paraboloid can be represented parametrically as $$r(u,v)=\langle a(u+v), b(u-v), uv\rangle$$ Find the curves $u$ is constant and $v$ is constant. I guess I need to use the ...
2
votes
1answer
96 views

21, Not Touched Maths Since GCSE. Want to start learning again. Where to Start?

I am 21 and have got into computer programming. Doing very well in my degree. Would love to get into computer science but feel I am being held back by my basic knowledge of maths. I got an A at GCSE, ...
3
votes
1answer
45 views

Show that $\dot{n_s}=-\kappa_s t$

I found the question in a differential geometry textbook while studying. This question seems so intesting to me. So please help me solving it. I know that $$\dot t =\kappa_s n_s$$ and $$\kappa ...
3
votes
1answer
72 views

If $\gamma$ is spherical, the equation $\frac{\tau}{\kappa}=\frac{d}{ds}(\frac{\dot{\kappa}}{\tau \kappa^2})$ holds.

Question: Please help me doing this question. In fact, there is its solution as I posted below. But I don't understand the answer. Please explain this more clearly. Thank you for helping. ...
1
vote
1answer
49 views

Showing a set of functions $F$ is bounded

I have a set of functions given by; $$F = \{f:[0,1]\rightarrow\mathbb{R}|\int_0^1 f(x)dx = 0, |f(x)-f(y)|\leq|x-y|, x,y\in[0,1]\}.$$ I have a solution for the question so my questions are about the ...
4
votes
4answers
131 views

Calculate $\sum\limits_{n=1}^\infty (n-1)/10^n$ using pen and paper

How can you calculate $\sum\limits_{n=1}^\infty (n-1)/10^n$ using nothing more than a pen and pencil? Simply typing this in any symbolic calculator will give us $1/81$. I could also possibly find this ...
8
votes
2answers
118 views

How to obtain $y$

The question was written with dark-blue pen. And I tried to solve this question. I obtained $x$ as it is below. But I cannot obtain $y$ Please show me how to do this. By the way, $\gamma (t)$ ...
2
votes
1answer
67 views

Verify that an ellipse has four vertices.

Verify that an ellipse has four vertices. The ellipse is given by $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ And I took $$x=a\cos t$$ and $$y=b \sin t$$ for $t\in [0,2\pi]$ Please can someone help ...
0
votes
1answer
61 views

From $ \sum^\infty_{\lfloor \log n \rfloor + 1}n/{2^r} $ to $ \sum^\infty_{r=0}1/2^r $?

$$ E[h] = E[\sum^\infty_{r=1}I_r] = \sum^\infty_{r=1}E[I_r] $$ $$ = \sum^{ \lfloor \log n \rfloor}_{r=1}E[I_r] + \sum^\infty_{\lfloor \log n \rfloor + 1}E[I_r] $$ $$ \leq \sum^{ \lfloor \log n ...
0
votes
1answer
90 views

Joint distribution of two marginal normal random variables

Question: Suppose we have: \begin{align*} \begin{bmatrix} X_1 \\ X_2 \end{bmatrix} \sim N\left(\begin{bmatrix} 6 \\ 3 \end{bmatrix}, \begin{bmatrix} 12 & 3 \\ 3 & 2 \end{bmatrix} \right) ...
1
vote
1answer
54 views

Prove that $a_n \times b_n \to 0$ for $n \to \infty$

I want to prove this example: If $a_n \to 0$ for $n \to \infty$ and $(b_n)_n$ is bounded. Prove that $a_n \times b_n \to 0$ for $n \to \infty$. My first guess is that I should use the definition ...
-1
votes
1answer
42 views

how to find the signed normal

$$\gamma (t)= (R\cos (t/R), R\sin (t/R))$$ $$\dot {\gamma (t)}=(-\sin (t/R), \cos (t/R))$$ $$n_s= (-\cos (t/R), -\sin (t/R))$$ where $n_s$ is the signed normal. the instructor has found the $n_s$. ...
1
vote
2answers
404 views

Proof of the limit laws (Analysis)

Hi everyone I'd like to know if my arguments for the next proof are sound or needs some changes to be correct. I hope they are not a little flaws. Proposition (limit laws): Let $(a_n)_{n=m}^\infty$ ...
0
votes
0answers
30 views

Convergence of $xe^x - R$

Basing my question on one of the previous questions I have passed before Root of the function $f(x)=xe^x-R$, I was wondering why does $xe^x - R$ always converge? I was told that the function will ...
57
votes
6answers
3k views

Strategies for Effective Self-Study

I have a long-term goal of acquiring graduate-level knowledge in Analysis, Algebra and Geometry/Topology. Once that is achieved, I am interested in applying this knowledge to both pure and applied ...
0
votes
0answers
114 views

Proof verification of some properties of the exponent.

I'd like to know if the proof of the following laws of exponent is correct. First of all thanks in advance and I apologize for the extension of the following proof but it's one exercise. I hope there ...
2
votes
2answers
44 views

how to calculate frenet serre eqautions

how to calculate frenet serre eqations of the helix $$\gamma : \Bbb R \to \ \Bbb R^3$$ $$\gamma (s) =(\cos (\frac{s}{\sqrt 2}), \sin (\frac{s}{\sqrt 2}), (\frac{s}{\sqrt 2}))$$ i know the ...
6
votes
4answers
262 views

Beyond the Exercises? [closed]

I've entertained and become bored with quite a few interests, though mathematics has more or less been my central passion throughout my high school/middle school life. I've only recently started into ...
0
votes
3answers
132 views

Natural Deduction (FeedBack)

I am looking for feedback to three proofs (alternatively derivations) that I have constructed. The first is: Theorem. Injectivity does not imply surjectivity. Proof: Suppose $\{\phi\} \vdash ...
0
votes
1answer
471 views

how to calculate the curvature of an ellipse

how can I compute the curvative of an ellipse given by $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ do i need to take $x=acos(t)$ and $y=bsin(t)$? please show me a way how to solve this? thank you for ...
3
votes
2answers
90 views

Proof of the infinite descent principle

Hi everyone I wonder to myself if the next proof is correct. I would appreciate any suggestion. Proposition: There is not a sequence of natural numbers which is infinite descent. Proof: Suppose for ...
0
votes
1answer
51 views

calculation of vector product $\ddot{\gamma} (t) \times \dot{\gamma} (t)$

$$\ddot{\gamma} (t) \times \dot{\gamma} (t)=(-a\cos t, -a\sin t, 0)\times (-a\sin t, a\cos t, b)$$ The writer will get the following result $$(-ab\sin t, ab\cos t, -a^2)$$ but I don't know how to ...
2
votes
1answer
64 views

solve this equation $z(z+y)dx+z(z+x)dy=0$

I need to solve this following equation $$z(z+y)dx+z(z+x)dy=0$$ I get this from above equation $$\frac{dx}{z(z+x)}+\frac{dy}{z(z+y)}=0$$ After there, I dont know what I need to do.
0
votes
1answer
96 views

$\liminf$ and $\limsup$ question

I am trying to learn about $\liminf$ and $\limsup$, as I have struggled with the definition of these, and mostly just avoided questions about this in the past. I have had a go at answering a question ...
0
votes
1answer
83 views

Abelian Group elements and inverses

Let G be a finite abelian group, say, $G={e,a_1,a_2...a_n}$ Prove the following: a)$(a_1a_2...a_n)^2=e$ b)If there is no element x $\neq$ e, x=x^(-1), then $a_1a_2...a_n=e$ c)If there is exactly ...
0
votes
1answer
55 views

Find solution(primitive) of the equation

I want to find its solution of the following equation $$ydx+xdy+2zdz=0$$ answer: Keeping $z$ constant; I obtain that $$ydx+xdy=0$$ or $$\frac{dx}{x}+\frac{dy}{y}=0$$ Then I get $$U(x,y,z)=xy$$ ...
0
votes
1answer
398 views

Prove the least upper bound property using $\mathbb{Q}$-Cauchy sequences.

Hi everyone I'd like to know if the next proof is correct. I'd appreciate any suggestion mainly in the points marks with (1) and (2). Theorem: Let $E$ be a nonempty subset of real numbers which has ...