Most Popular
1500 questions
64
votes
23 answers
Interesting results easily achieved using complex numbers
I was just looking at a calculus textbook preparing my class for next week on complex numbers. I found it interesting to see as an exercise a way to calculate the usual freshman calculus integrals $\int e^{ax}\cos{bx}\ dx$ and $\int e^{ax}\sin{bx}\…
Adrián Barquero
- 15,654
64
votes
5 answers
Gradient of a dot product
The wikipedia formula for the gradient of a dot product is given as $$\nabla(a\cdot b) = (a\cdot\nabla)b +(b\cdot \nabla)a + a\times(\nabla\times b)+ b\times (\nabla \times a)$$ However, I also found the formula $$\nabla(a\cdot b) = (\nabla a)\cdot…
Euler....IS_ALIVE
- 4,867
64
votes
3 answers
Geometrical interpretation of Ricci curvature
I see the scalar curvature $R$ as an indicator of how a manifold curves locally (the easiest example is for a $2$-dimensional manifold $M$, where the $R=0$ in a point means that it is flat there, $R>0$ that it makes like a hill and $R<0$ that it is…
Daniel Robert-Nicoud
- 30,936
64
votes
5 answers
How do you rotate a vector by a unit quaternion?
Given a 3-variable right-handed vector v that is a translation measured in local space and a unit quaternion representing an orientation from local to world space, how do you use the quaternion to rotate the vector from local space to world space?…
Narf the Mouse
- 823
64
votes
11 answers
Striking applications of linearity of expectation
Linearity of expectation is a very simple and "obvious" statement, but has many non-trivial applications, e.g., to analyze randomized algorithms (for instance, the coupon collector's problem ), or in some proofs where dealing with non-independent…
Clement C.
- 68,745
64
votes
8 answers
Why do the French count so strangely?
Today I've heard a talk about division rules. The lecturer stated that base 12 has a lot of division rules and was therefore commonly used in trade. English and German name their numbers like they count (with 11 and 12 as exception), but not French:…
Martin Thoma
- 10,267
64
votes
2 answers
Why are (representations of ) quivers such a big deal?
Quivers are directed graphs where loops and multi-arrows are allowed. And we can talk about representations of quivers by assigning each vertex a vector space and each arrow a homomorphism. Moreover, Gabriel gives a complete classification of…
Hui Yu
- 15,629
64
votes
9 answers
On the functional square root of $x^2+1$
There are some math quizzes like: find a function $\phi:\mathbb{R}\rightarrow\mathbb{R}$ such that $\phi(\phi(x)) = f(x) \equiv x^2 + 1.$ If such $\phi$ exists (it does in this example), $\phi$ can be viewed as a "square root" of $f$ in the sense of…
user1551
- 152,549
64
votes
4 answers
A "new" general formula for the quadratic equation?
Maybe the question is very trivial in a sense. So, it doesn't work for anyone. A few years ago, when I was a seventh-grade student, I had found a quadratic formula for myself. Unfortunately, I didn't have the chance to show it to my teacher at that…
64
votes
1 answer
Show that the countable product of metric spaces is metrizable
Given a countable collection of metric spaces $\{(X_n,\rho_n)\}_{n=1}^{\infty}$. Form the Cartesian Product of these sets $X=\displaystyle\prod_{n=1}^{\infty}X_n$, and define $\rho:X\times X\rightarrow\mathbb{R}$ by…
Set
- 8,465
64
votes
7 answers
Does $\zeta(3)$ have a connection with $\pi$?
The problem Can be $\zeta(3)$ written as $\alpha\pi^\beta$ , where ( $\alpha,\beta \in \mathbb{C}$ ), $\beta \ne 0$ and $\alpha$ doesn't depend of $\pi$ (like $\sqrt2$ , for example)? Details Several $\zeta$ values are connected with $\pi$ , like:…
GarouDan
- 3,488
64
votes
2 answers
Is there a step by step checklist to check if a multivariable limit exists and find its value?
Do we rely on certain intuition or is there an unofficial general crude checklist I should follow? I had a friend telling me that if the sum of the powers on the numerator is smaller then the denominator, there is a higher chance that it may not…
Yellow Skies
- 1,770
64
votes
1 answer
Which Algebraic Properties Distinguish Lie Groups from Abstract Groups?
This question is motivated by a previous one: Conditions for a smooth manifold to admit the structure of a Lie group , and wants to be a sort of "converse". Here I am taking an abstract group $G$ and looking for necessary conditions for it to admit…
Lor
- 5,868
64
votes
8 answers
Lemma/Proposition/Theorem, which one should we pick?
This is something that confuses me. I have read a few mathematical texts and they often seem to use Lemma/Proposition/Theorem if they have a particular statement. Now, which one to use? A lemma can be something you need to prove a more important…
user1120
64
votes
11 answers
Why does the discriminant in the Quadratic Formula reveal the number of real solutions?
Why does the discriminant in the quadratic formula reveal the number of real solutions to a quadratic equation? That is, we have one real solution if $$b^2 -4ac = 0,$$ we have two real solutions if $$b^2 -4ac > 0,$$ and we have no real solutions if…
user487950