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2

My mentor, a delightfully eccentric mathematician, uses sounds quite often. I remember, when he introduced difference sets to a bunch of people, he gave one of the defining equations of algebraic design theory by making a little popping noise with his tongue after he was finished writing it on the blackboard. In the next lecture, when he wanted his students ...


0

The geometric idea is what you stated about the tangent line. The analytic idea is: $$f(x+\Delta x)=f(x)+f'(x) \Delta x + r(x,\Delta x)$$ where $r(x,\Delta x)$ is small if $\Delta x$ is small. That is, $g(\Delta x) = f(x+\Delta x)$ is nearly a linear function if $\Delta x$ is small enough. "Small" and "small enough" are quantified by the definition. ...


-1

$$\left(1-\frac{3}{2}\right)^2 =\left(2-\frac{3}{2}\right)^2 \implies $$ $$ \implies \left|1-\frac{3}{2}\right| =\left|2-\frac{3}{2}\right|$$ because $$\sqrt{x^2} = |x| $$


4

The problem is here: $a^2=b^2 \Rightarrow a=b \underbrace{\vee}_{\text{or}} a=-b$. In this case, only the second one counts: $(1-\frac32)^2=(2-\frac32)^2 \Rightarrow \require{cancel} \cancel{1-\frac32=2-\frac32} \vee 1-\frac32=-2+\frac32$, but the first one gives a contradiction, so it doesn't count.


1

If you wish to learn how to make 3d graphics to use in an existing game engine, you won't need more than very basic maths. However, if you wish to develop game engines, you will need linear algebra, vector calculus and graph theory skills, and you will need to learn how to program! Calculus' applications in game development are extensive. Physics is a ...


0

Given two real valued functions f and g: $$f(n)=O(g(n))$$ If $c>0$ and $n_0>0$: $f(n) \geq cg(n) \geq 0, \forall n\geq n_0$ From the definition it follows that for $n \geq 0$ $$a_0n^d + a_1n^{d-1} + a_2n^{d-2} + ... + a_d \leq |a_0|n^d + |a_1|n^{d-1} + |a_2|n^{d-2} + ... + |a_d|$$ $$\leq |a_0|n^d + |a_1|n^{d} + |a_2|n^{d} + ... + |a_d|n^{d}$$ ...


0

there is only one way to study maths...LEARN THEM! this is the bit that is always missing in maths. having acquired a certain amount of knowledge and mental dexterity is is necessary to 'learn' how to do a new problem. this ,learning is done by get an example go through it and through it ( with pencil in hand) until you understand it and you have ...


1

In mathematics a variable is a "temporary name". A name is a constant. When in mathematics we write an equation like : $x = x$ we are saying that, for every name $c$, if we put it in place of the variable $x$, what we get is always true. Why? Because, using the name "John" instead of $c$, the person named by the name "John" is always the same as the ...


0

I agree with RR and would just add that students in Europe who have been identified early-on for a math/science education might also take 2 or even 3 math subjects per year in high school and subsequently would not be bothered with requirements covering much history, social studies, health, etc. So they may learn more math in one year than the average ...


0

I'm in a same situation as you, in the sense that I'm teaching myself math. When I was in school I never paid much attention to math and I simply lost track, didn't build my foundation that subsequent math study depended on and hence really struggled. I made it through but just barely. Stuff such as calculus, integral and differential equations didn't mean ...


0

A lot of us share the same feeling towards math. Maybe our high school teachers were terrible. Or maybe we also wasn't that interested in math at that point in time. Time passed by and now you found yourself in a position that you really need to master the fundamentals of math in order to advance in your academic/professional life. I'll share a secret ...


2

I'm a bit buzzed after an evening of jazz, so I swat this with a cannonball. This follows from the divergence theorem. Let $\vec{u}$ be any constant vector. Let $V$ be that polyhedron, $\partial V$ its boundary, and $\vec{S}_i, i=1,\ldots, N,$ the vectors that you listed. By the divergence theorem the sum $$ \vec{u}\cdot\sum_i\vec{S}_i=\oint_{\partial ...


1

Prestige matters very little. Going to a smaller university where you can get more attention from your professors can be especially beneficial if they get to know you very well and write you good recommendation letters. Sometimes in better schools the professors are too busy doing their own research, and although the department is quite renowned in its ...


0

I had run into the exact same situation when I took a course in Abstract Algebra 1: called Group Theory. I also had a "Real" math course, it was a very rigorous proof oriented linear algebra class. But I ended up dropping the group theory course. The problem was the prof taught graduate level courses for 10 years, then after 10 years was asked to teach an ...


1

Much has been said about inappropriate pattern matching, and I agree with much of that opinion. But I really think that what is going on with many of these students is that instead of seeing mathematical patterns (albeit incorrectly), they are actually seeing equations and expressions as lexical patterns. $a(x+y)=ax+ay$ , ...


0

Advanced Engineering Mathematics, Zill Advanced Engineering Mathematics, Kreyszig Very well written text books and fun to read.


3

At a point where a function is not $C_2$ (i.e. not twice differentiable), there will be a discontinuity in the curvature of its graph. In other words, the curvature will jump abruptly from one value to another. Designers can see these jumps, if they are large enough, but many people can't. You can try some experiments with circular arcs that join ...


0

Let $\vec{a}, \vec{b}, \vec{c}$ be the position vectors of the vertices of the triangle $\Delta A B C$ and $M_1$ is opposite $A$, $M_2$ opposite B, $M_3$ opposite $C$. Now, the sides $AC=\vec{c} - \vec{a}, BC=\vec{c}-\vec{b}, AB=\vec{b}-\vec{a}$. Therefore, the medians are $AM_1=\vec{b}-\vec{a}+ \frac{1}{2}(\vec{c}-\vec{b}) = ...


4

If you're really planning on having a career in applied math, I'd say that learning about algorithms and data structures and programming languages typically used for numerical computation (plus getting a grip on computer algebra systems like Mathematica) will be far more useful than learning the ins and outs of web development.


1

I recently graduated with an undergraduate degree in Actuarial Mathematics. I took courses in both the pure and actuarial math branches. Actuarial Specific Subjects you may be interested in: Probability + Stats - Basic probability theory is necessary. Any additional knowledge of stats is highly recommended. Theory of Interest - Learn about time value of ...


1

What would you consider an underrating of the golden ratio to be? Ron Knott maintains a very detailed webpage on the phi ratio and was one of the discussants in Melvyn Bragg’s radio program on the Fibonacci series (along with Marcus du Sautoy and Jackie Stedall), which I highly recommend. http://www.bbc.co.uk/programmes/b008ct2j During the discussion on ...


0

My alma mater, the University of Texas at Arlington, offers math degrees with various emphases, somewhat equivalent to minors. Computer Science Actuarial Science/ Statistics Accounting Education Pure math- considerered essential for grad. school prep. by some advisers a few other programs. See specifics and career options here


9

My advice has two parts: 1) Don't worry about other people. Chances are, the successful people you refer to got that way from thinking about their work and not by focusing on the performance or others. I often found myself falling into this trap as an undergraduate. I'd say about half of what should have been math time was spent worrying about my relative ...


12

I didn't always ace my exams in school. Being a mathematician has nothing to do with test scores. My advice is to make mathematics personal. It's not about anybody else but you. In my opinion, it's best to compete with yourself. Try and have your own relationship with mathematics, understand it in a way that is your own, and, as always, engage mathematics ...


0

Visualisation in ancient times: Sum of squares Let's go back in time for about 2500 years and let's have a look at visually stunning concepts of Pythagorean arithmetic. Here's a visual proof of \begin{align*} \left(1^2+2^2+3^2\dots+n^2\right)=\frac{1}{3}(1+2n)(1+2+3\dots+n) \end{align*} The Pythagoreans used pebbles arranged in a rectangle and linked ...


2

This may be an unsatisfying answer, but: A function $f(x)$ is twice differentiable if neither the graph of $f(x)$ nor the graph of $f'(x)$ has sharp bends in the region of interest. If I were given $$f(x) = \left\{ \begin{array}{l l} x^2 & \quad \text{if $x \geq 0$}\\ -x^2 & \quad \text{if $x < 0$} \end{array} \right.,$$ I ...


0

One way to do this is to arrange that the unit price $p_N$ falls by a certain fraction $f$ when the quantity $N$ increases tenfold. This is a property of the function $$ p_N = p_1 N^{-k} $$ where $p_1$ is the price of one unit sold individually and $k$ is a constant. $k$ is found by setting $$ 10^{-k} = 1 - f $$ which gives $$ k = -\log_{10}{(1-f)} $$ ...


0

In France at one time (until reforms in the mid-90's, I think), students learned about normal subgroups and ideals of rings before they ever saw a matrix. The reason this happened in France is that engineering school entrance exams are very competitive there, and mathematics weighs more heavily than other subjects in the selection. (Most mathematicians, ...


6

If you're concerned about time, I don't think reading Calculus by Spivak is the best thing to do. Either Zorich or Apostol is a great choice. I would say that they're "intermediate" in difficulty. Zorich contains more in-depth discussions of topics, and more examples than does Apostol. If your goal is only to move on to Royden, you'll probably cover the ...


2

Have you seen the cool "MSE-driven" graphs of maths (well ... of MSE question tags) posted on meta? They only include the 64 most popular tags, so a lot of more advanced and/or specialised areas are left out, but I still think they provide some insight.


1

Circle division by chords is also a nice one IMHO. It is easy to explain, it shows that you shouldn't make assumptions too early, and the proof of the correct formula can be understood with high school algebra. [Picture from Wolfram MathWorld.]


2

A useful introduction to proof by induction is to take a $2 \times 2$ square with one corner removed as in the diagram below. The problem is to prove that using these shapes, one can cover any $2^n \times 2^n$ square with one corner removed as in the next diagram.


4

Something easily grasped by pretty much anyone are problems related to covering chessboards with dominoes. (I'm especially fond of these because learning about them was what initially got me interested in mathematics.) A domino exactly covers two squares of chessboard. Can you always cover an $8\times 8$ chessboard with dominoes? Of course this is easy by ...


1

And yet although we may think of $10^{10^{56}}$ as a large number (positive integer), there are a lot more positive integers that are larger than there are that are smaller. P.S. $10^{10^{56}}$ is quite puny compared to the renowned googleplex = $10^{10^{100}}$.


1

Let's call $10^{56}$ a grxy. Then, $10^{10^{56}}$ is one with a grxy zeroes after it. That's a very large number, depending on whether you're comparing it to $1$ or $10^{10^{56}}-1$. I'm sure you know what a cell is. The number of cells in the human body is $10^{43}$. Obviously, $43\ll10^{56}$. So it's no wonder that compared to $1$ or $0$, $10^{10^{56}}$ ...


2

If you were to write out the number $10^{10^{56}}$, it would be a $1$ followed by $10^{56}$ zeroes. That is, $$10^{10^{56}} = 1\underbrace{000\dots000}_{\large{10^{56}\ \text{zeroes}}}$$ To give you some sense of scale, there are only about $10^{80}$ (i.e. a $1$ followed by $80$ zeroes) atoms in the observable universe. $$10^{80} = ...


0

There's also an interesting three-volume book called "Mathematics - Its Content, Methods, and Meaning" written by Aleksandrov, Kolmogorov, and others. Originally written in the 50s and translated into English (MIT Press) in the 60s.


0

Well, as an applied math student, I also love Russian books so much. I found most Russian mathematicians are also interested in writing books, so it may be convenient to search by the authors. Here are some that I know: Geometry: S.P.Novikov (as you mentioned), Fomenko (he has many books, including a book on "geometric intuition" and a nice textbook, with ...


1

Math is always fun to learn. Here are some of the images that explain some things beautifully visually


0

The question demonstrates the difference between expressions and functions. And the question of finding the maximal domain for a function determined of an expression depends on what one mean with the expressions in the formula. Either is $\sqrt x$ a function defined for $x\geq 0$ or it is an other function, defined for other domains. Anything goes but has to ...


27

Let us define the functions: $$ f(a,b)=\frac{\sqrt{a}}{\sqrt{b}}\quad\,\,\text{and}\,\,\quad g(a,b)=\sqrt{\frac{a}{b}}. $$ Then $f$ and $g$ AGREE on the intersection of their domains. However, they have different domains: $$ \mathrm{Dom}(f)=\{(a,b): a\ge 0,\,\,b>0\},\\ \mathrm{Dom}(g)=\{(a,b): a\ge 0,\,\,b>0\}\cup\{(a,b): a\le 0,\,\,b<0\}. $$ ...


3

Actually, you are right- as functions of two real variables, they have different domains. For example take $a=-1,b=-1$


1

If the curve $x = \cos t + \cos 2t$, $y=\sin t + \sin 2t$ passes through the point $(-1,1)$, then there must be some value of $t$ such that $-1 = \cos t + \cos 2t$ and $1 = \sin t + \sin 2t$. Your first task is to find that value of $t$. Once you have that, you can evaluate the derivative (which you have correctly calculated) at that value of $t$, which ...


2

I like the pictures from chapter 0 of Hatcher for that purpose. And also Louis Kauffman made some nice pictures for the Fox calculus. And Thurston's 3-manifolds book has good pictures as well. "Normal maths works with squares, triangles, perfect circles; it counts beans. In topology we're freer to work with more interesting shapes." (cue picture from ...


1

This may be useful for a somewhat different perspective. My personal feeling is that it is (I would say) of foremost importance that you become comfortable and nurture a proficiency with proofs. The topics you mention, when studied in the depth I would think you want to explore all have a "proof component." While this point could be argued, typically in ...


2

Yes, there are all sorts of problems that mathematicians can and do solve for a living, either in an academic setting or in industry. For example, you might look at What Kind of Problems Might I Work On?.


0

You should take a look at the field called Operations Research. It combines many interesting elements of both mathematics and computer science. The problem like the one you linked to reminds me of some problems I solved working as a Research Scientist in industry. If the corporate environment does not suit you, then there are research labs and professorships ...


2

Two books; I guess yesterday, somebody asked again about the question of Regiomantus, Finding the widest angle to shoot a soccer ball from the sideline using optimization!! I knew of this from some book i had 40 years ago, but it is in two books that can be purchased (or borrowed) , Heinrich Dorrie (translated) 100 Great Problems of Elementary Mathematics ...


1

This is another rectangular problem, but I like it because the student can take one of at least 3 approaches. A spider in on the floor in the north-west corner of a room. He would like to crawl to the south-east corner on the ceiling. What is the path that minimizes the distance that he has to crawl? Possible approaches When the spider reaches the edge ...


1

The examples you gave my themselves are elementary but good examples already. However, if you want to expand even more, you have many options. I'll name the ones I can think of at the moment. Economics has a lot of great maximization problems at various levels, especially microeconomics. Physics, chemistry, and biology use optimization problems a lot. An ...



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