Mathematics

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1500 questions
86
votes
14 answers

How to effectively and efficiently learn mathematics

How do you effectively study mathematics? How does one read a maths book instead or just staring at it for hours? (Apologies in advance if the question is ill-posed or too subjective in its current form to meet the requirements of the FAQ; I'd…
user7273
86
votes
8 answers

When not to treat dy/dx as a fraction in single-variable calculus?

While I do know that $\frac{dy}{dx}$ isn't a fraction and shouldn't be treated as such, in many situations, doing things like multiplying both sides by $dx$ and integrating, cancelling terms, doing things like $\frac{dy}{dx} =…
xasthor
  • 1,426
86
votes
12 answers

What are the applications of functional analysis?

I recently had a course on functional analysis. I was thinking of studying the mathematical applications of functional analysis. I came to know it had some applications on calculus of variations. I am not specifically interested in applications of…
gamma
  • 954
86
votes
9 answers

What is the meaning of the third derivative of a function at a point

(Originally asked on MO by AJAY.) What is the geometric, physical, or other meaning of the third derivative of a function at a point? If you have interesting things to say about the meaning of the first and second derivatives, please do so.
Gil Kalai
  • 1,143
86
votes
6 answers

Why is a full turn of the circle 360°? Why not any other number?

I was just wondering why we have 90° degrees for a perpendicular angle. Why not 100° or any other number? What is the significance of 90° for the perpendicular or 360° for a circle? I didn't ever think about this during my school time. Can someone…
P K
  • 1,121
86
votes
3 answers

What is the "standard basis" for fields of complex numbers?

What is the "standard basis" for fields of complex numbers? For example, what is the standard basis for $\Bbb C^2$ (two-tuples of the form: $(a + bi, c + di)$)? I know the standard for $\Bbb R^2$ is $((1, 0), (0, 1))$. Is the standard basis exactly…
Casey Patton
  • 1,493
86
votes
11 answers

Fake induction proofs

Question: Can you provide an example of a claim where the base case holds but there is a subtle flaw in the inductive step that leads to a fake proof of a clearly erroneous result? [Note: Please do not answer with the very common all horses are the…
Daniel W. Farlow
  • 23,258
85
votes
14 answers

The Monty Hall problem

I was watching the movie $21$ yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the wrong answer) which revolves around…
Avicinnian
  • 961
85
votes
8 answers

Very *mathematical* general physics book

I am searching for a book to study physics. So far, I've been suggested Resnick, Halliday, Krane, Physics , but it doesn't seem to be very suited for a math major. Can you suggest some more mathematical books? By mathematical I mean: rigorous, with…
Dal
  • 8,632
85
votes
2 answers

Is $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ algebraic or transcendental?

I thought it was easy to show that $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ is irrational, but found a gap in my proof. Simple finite approximations show the denominator cannot be small, though, strongly suggesting irrationality. However, can it be…
user2566092
  • 26,500
85
votes
4 answers

Gradient of 2-norm squared

Could someone please provide a proof for why the gradient of the squared $2$ -norm of $x$ is equal to $2x$ ? $$\nabla\|x\|_2^2 = 2x$$
user167133
  • 973
85
votes
3 answers

Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)

I am trying to understand the differences between $$ \begin{array}{|l|l|l|} \textbf{vector space} & \textbf{general} & \textbf{+ completeness}\\\hline \text{metric}& \text{metric space} & \text{complete space}\\ \text{norm} & \text{normed} &…
vonjd
  • 9,110
85
votes
8 answers

Difference between Fourier transform and Wavelets

While understanding difference between wavelets and Fourier transform I came across this point in Wikipedia . The main difference is that wavelets are localized in both time and frequency whereas the standard Fourier transform is only localized in…
chatur
  • 1,262
85
votes
29 answers

General request for a book on mathematical history, for a VERY advanced reader.

I am aware that there are answered similar questions on here, however I am specifically after a text that would be engaging for a professor of mathematics, also Fellow of the Royal Society (FRS). He is unwell and in the hospital, and I would like to…
Kino
  • 999
85
votes
5 answers

Intuitive explanation of a definition of the Fisher information

I'm studying statistics. When I read the textbook about Fisher Information, I couldn't understand why the Fisher Information is defined like this: $$I(\theta)=E_\theta\left[-\frac{\partial^2 }{\partial \theta^2}\ln P(\theta;X)\right].$$ Could anyone…
maple
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