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0

The easiest software you can used for that is Graphing Calculator 3D. It also generats nicer graphs than even Mathematica. You can modify and rotate your graphs live so it's great for inspecting the visual properties of various graphs. But before you can use it you need to convert your 3D equations from implicity to explicit. That is, all equations should ...


2

Multiplication is repeated addition. So $\sqrt{2}\cdot \sqrt{8}=4$.Similarly division is repeated subtraction. Solve the differential equation $$\dfrac{dy}{dx}=y$$ Easy! Multiply both sides by $dx$, so $\dfrac{dy}{y}=dx$ now integrating both sides $\ln{y}=x+C$. I don't know how many people understand this multiplication by $dx$(Frankly I don't). Even ...


0

I've can think of two: 1) For teaching elementary calculus, treating $\dfrac{dy}{dx}$ as a fraction is a great way of working with things like the chain rule, separable differential equations etc. 2) "To multiply by $10$ add a zero to the end" works perfectly for integers, and fails spectacularly for decimals.


3

A vector is just an arrow in space or an object which components transform in some particular way under rotations (see the question I asked in the Physics site). In a similar fashion, a tensor is just a matrix. The fundamental theorem of calculus is the definition of a definite integral; it works, but it's wrong: $$\int_a^b f(x)dx=F(b)-F(a)$$ Given a ...


7

We tell students in an introductory algebra class, say, that mathematicians invent axioms and then study the properties of the resulting object or theory. The story goes that some bright mathematician had the idea to write down the four group axioms before anyone knew anything about group theory, and then he and other mathematicians worked out all the known ...


0

Does the use of "wrong" definitions/formalizations count for your purposes? For example, in high school it is probably reasonable (although not essential) to use the term "vector space" to mean "finite-dimensional vector space". This avoids the need to spend more time on the Axiom of Choice than is really called for by the curriculum, to justify the ...


4

I'm surprised nobody mentioned square roots before: I am 17 now and have always been told the square root of a negative number does not exist and can not be taken. I am interested in mathematics, so I know this to be false (though I don't really know what imaginary numbers are). To not teach this at first makes sense, because when one first learns square ...


6

You've identified one of the biggest issues already: we considered $\Bbb R$ (and $\Bbb R^n$) to be given and just kind of "the number line of every number you possibly can think of" As a result, there are some very important theorems that I'll bet you didn't prove formally (although you probably did draw pictures corresponding to them), such as the ...


-3

Calculus is rigorous but it may not be exact. Infinitesimals and limit theory are saying something similar - that if an increment of an argument is small enough then any terms containing it after a manipulation can be discarded because they will not be measurable (in the common sense of the term). Manipulation almost invariably involves one cancellation (due ...


-1

Its more of a Physics, but Physics are also Mathematics. So Maybe Black Body? http://en.wikipedia.org/wiki/Black_body


3

Usually you are tought that topological spaces provide a nice framework for doing topology. But when you really sit down and try to prove some of the "obvious homeomorphisms" (for example $|X \times Y| \cong |X| \times |Y|$ for simplicial sets $X,Y$ with geometric realizations $|X|,|Y|$, or the exponential law $C(X \times Y,Z) \cong C(X,C(Y,Z))$), you will ...


1

In physics there is the point charge, and to a broader extent, any point particle, which is strictly an idealization. The point charge is an idealized model. It is dimensionless and has infinite charge at the point.


3

The velocity addition: The first semester of my study I've been taught that if a guy walks with veolicty $v_g$ in a train that has velocity $v_t$ then his velocity with comparison to someone out of the train is $$v_r=v_g+v_t.$$ During the second semester I have been taught (by the same teacher) that $$v_r= \frac{v_g+v_t}{1+\frac{v_gv_t}{c^2}},$$ using the ...


1

The existence of Gabriel's Horn. That is, there exists an object with a finite volume but with an infinite surface area. Mathematically, we can show this to be true by taking a 360-degree revolution of the graph $y=\frac{1}{x}$ in the domain $[1,\infty)$ about the $x$-axis, and calculating its volume ($\pi$) and its surface area (infinite). This is ...


7

Euclid's 5th postulate: two lines at right angles to the same line will never meet--and only if they are both at right angles. Well, in a perfect plane universe that might be true, but in the real world, non-Euclidean geometry is more "true": large triangles on the surface of the planet don't add up to 180 degrees, etc.


1

Commonly I use the following letters as indices: $$\text{Discrete}: i,j,k,l,m,n,p,q,r,s\\ \text{Continuous}: \alpha, \beta, t, \epsilon$$ And these as sizes of discrete sets: $$K,l,L,m,M,n,N,P,Q,r,R,s,S$$ That's of course opinion based, but these are my favorites and $m,n$ only when not in use as an index. $r,s$ occur as sizes mostly in numerical context ...


2

This is more like a long comment. I am having an argument with someone who thinks that it's never justified to teach something that is not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is to iteratively learn and unlearn along the way. In math, or in science? If we're talking about science then ...


0

The letters $\alpha$ and $\beta$ are common for indices.


5

Here's a technical one I ran across at the beginning of grad school which always stuck with me. During questions after a colloquium talk on condensed matter physics, the speaker mentioned the fact that defects in these systems could be classified by topology. In the process, he equated the notion of homotopy methods with the concept of winding numbers. This ...


15

How about this notorious one I remember from high school? "$f(x)$ is just a fancy name for $y$."


8

Often in physics you pretend like higher order differentials vanish. For example, you have $(x+dx)(x+dx)$, you multiply through and say that the $(dx)^2$ term is $0$. You could try to appeal to nonstandard analysis for manipulating the differentials, but of course the amount of work involved to develop such a theory is hidden.


0

How about e=mc^2? This is taught as fact in almost every physics class, and is commonly 'known' to the public, but is actually a simplified version of the real formula. You see, this says that the energy of an object is a product of it's mass and the speed of light squared. This completely disregards the energy that an object can have due to its velocity ...


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Here are two false concepts taught in lower grades (explicitly or implicitly): 1) Every plane figure has an area. Elementary kids are not told that some figures have no area (or might not have an area, if you leave out the Axiom of Choice). 2) A set is a collection of objects satisfying any particular relation. (Some middle-school books avoid this by ...


22

Thinking of Dirac delta function as a function works reasonably well up to a certain point. For example, every physicist knows that $$\int_{-\infty}^{\infty}e^{i\omega x}dx=2\pi\cdot\delta(\omega),$$ but only a small part of them really studied the theory of distributions.


-1

This is a new interesting 4x4 Magic Sqaure, which I believe will be interesting to School Children. Here each element of the square is a square number. This was provided by Dr. Geoffrey Campbell 509020 is the sum of rows and columns 29^2 | 191^2 | 673^2 | 137^2 || 509020 -------+--------+--------+--------++-------- 71^2 | 647^2 | 139^2 | 257^2 ...


0

A story which I heard when I was in Primary School motivated me to understand the Power of Exponentials. The story goes like this--..........A Brahmin Priest presented the King with the Chess Board and explained to him how to pay War Games on this board . The King was pleased and asked the priest , what he wanted as a reward ..........The priest asked the ...


17

That recursion should be $S_n = 10 S_{n-1} + n$. Solving the recursion, we get $$S_n = \dfrac{10^{n+1}}{81} - \dfrac{9n + 10}{81}$$ If $n$ is odd, say $n=2k-1$, write this as $$ S_n = \left(\dfrac{10^{k}}{9}\right)^2 \left(1 - \dfrac{9n+10}{10^{2k}}\right)$$ so that $$\sqrt{S_n} = \dfrac{10^k}{9} \left(1 - \dfrac{9n+10}{2 \times 10^{2k}} - ...


0

There is a negative application for databases. Often record numbers in a database are assigned sequentially. Another common thing is to want to partition records into multiple databases serving as one big database. If you don't know what the maximum value would be, you might be tempted to divide the records between databases based upon string comparison of ...


3

Many objects in Computer Graphics are given by polygonal meshes. For processing these, techniques from Discrete Differential Geometry apply. See also: The Discrete Differential Geometry Forum Polygon Mesh Processing, especially Differential Geometry slides (Discrete) Differential Geometry slides


0

Hard! Consider the following simplification. You are in 3-space. You have a 2-space membrane that you can push 3-D objects through. You see 2-D cross sections of the 3-D object as you push the object through the 2-D membrane. OK, now add 1 to the above narrative: You are in 4-space. You have a 3-space membrane that you can push 4-D objects through. You ...


4

Well, I can't say how much more time a mathematician put in his work more than other scientist but I can tell you something. The father of one of my colleagues is a mathematician. He is now 91 years old and still working on math every day except Sunday for 5 hours. Maybe is just a passion involved. When you are determined to do something and it doesn't ...


4

I would not call this "part of mathematics" beautiful nor aesthetic, although sometimes it can be a pleasant waste of time. Most of the computable integrals, sums and products can be found in the books like Gradshteyn-Ryzhik or Prudnikov-Brychkov-Marychev. Programs like Mathematica or Maple, as well as theoretical physicists, solve this kind of problems ...


0

What mathematicians mean by 'infinitesimal' and how the word is commonly used are sometimes different. The common definition is that it is a quantity too small to be measured so the simple answer to your question is No. However, we could still ascribe values to sub-measureable quantities, see Could we assign a numerical value to an infinitesimal? And of ...


2

I heard here that "it takes roughly ten thousand hours of practice to achieve mastery in a field."


-3

For an infinite line segment, its length is infinite inches, infinite kilometers, infinite parsecs, or infinite angstroms - all are same. Angstroms and light-years does not make any difference as infinity + infinity = infinity * infinity = infinity As infinity is not a number but a concept, it cannot be compared with any real number. A crude definition of ...


1

Kreyszig's book Introductory Functional Analysis with Applications is a classic that might be good at your level. I like the way the fundamentals of functional analysis treated in this set of lecture notes. For more a more advanced book, consider Conway's A Course in Functional Analysis. It's also useful to really have a solid intuitive understanding of ...


0

Some thoughts on why many authors might prefer the approach via opens: See nets are nice if you want to grasp what being and getting close means in general topological spaces sure and this way in many cases give first hints at how to obtain a proof for some theorems. That is probably also the reason why you were able to reproduce a proof for Tychonoffs ...


0

There's a very recent book by Ulianov Montano called Explaining Beauty in Mathematics: An Aesthetic Theory of Mathematics. Here's the first paragraph of the blurb: This book develops a naturalistic aesthetic theory that accounts for aesthetic phenomena in mathematics in the same terms as it accounts for more traditional aesthetic phenomena. Building upon a ...


22

I often wonder the same thing, and for concerns and interests around these lines found myself reading "The Cambridge Handbook of Expertise and Expert Performance." (I would suggest reading this more if you are interested - it is a well-supported book and says lots of things like practicing well for a few decades along with a supportive home environment is ...


14

I recommend reading Allyn Jackson's short biography of Alexander Grothendieck. Here's a highlight to give you an idea of how much Grothendieck worked: Even among mathematicians, who tend to be single-minded and highly devoted to their work, Grothendieck was an extreme case. "Grothendieck was working on the foundations of algebraic geometry seven days a ...


14

I do not know how many hours did "great" mathematicians spend on doing research but I don't think it is of any importance. The most important think IMHO is to not worry to much about the result of your work, not compare yourself with others (from the past as well as from the present). Find what is really interesting for You in mathematics and have fun !


0

I think that the only answer I could give is that as long as you like what you're doing, there's no wrong way to study. If you try to change your method and by doing that you don't have any pleasure, stop ! Of course there are boring and tough periods, but if you enjoy it more than it sucks... You'll do it !


0

I think the best advice I can give to a prospective student is maximise your future entropy, that is take a wide selection of courses initially from all sections of math- pure,applied,statistics, computer science/programming, cryptography, that will leave your options open for future specialisation when your find your niche what you really want to do, also ...


3

A Wikipedia page that seems to be relevant: $\ldots$ blackboard bold in fact originated from the attempt to write bold letters on blackboards in a way that clearly differentiated them from non-bold letters, and then made its way back in print form as a separate style from ordinary bold, possibly starting with the original 1965 edition of Gunning and ...


3

I have no references here, just stories my professor told us. As far as I know, the double lines were not double lines in the beginning, but rather boldened lines, so people wrote R with a slightly thicker vertical line to denote the real numbers. The laziness of mathematitians and difficulty of writing a bolder line that students will recognise on a ...


2

It is a great mystery as to why there are so many excellent Russian born geometers/topologists (S. Novikov, M. Kontsevich, V. Voevodsky, G. Perelman, I. Shafarevich, D. Fuchs, M. Postnikov) and so few good textbooks in Russian on those subjects. So let me list the textbooks that are (in my opinion) still on par with the best modern textbooks: 1) M. M. ...


0

This is a book by kirillov ,titled what are numbers?,it could help you http://www.math.upenn.edu/~kirillov/MATH480-S08/WN1.pdf http://www.math.upenn.edu/~kirillov/MATH480-S08/WN2.pdf


0

I'll hazard an answer. For both plans and videos, notes etc... much can be found from MIT's OpenCourseWear. I would say your general method of studying is wise except you probably already realize it's a bit heavy on lectures. I also wouldn't say it's necessarily wise to just work all the problems. Whatever you do, it needs to stretch your mind. Try asking or ...


1

http://padic.mathstat.uottawa.ca/~mnevins/latex/sample.pdf, here is a good short and simple guide. other is the following: www2.kenyon.edu/Depts/Math/Aydin/.../Report.doc, which goes to the main aspects of the body of a math report. Hope to be useful. Greetings.


2

I'm going to prove two other vector calculus identities to check if I've understood physicists notations well :D I'm going to use Einstein's summation notation as well. ;-) 1. $$\nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B})$$ $$\nabla \cdot (\mathbf{A} \times ...



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