Volume of a revolution for an ellipse?
How do you even integrate an ellipse? The question is... Rotating the ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ about the $x$-axis generates an ellips...
Apr 19, 2026 1 VIEWS
Topic: General
5104 Articles 
Derivation of the density function of student t-distribution from this big integral.
My lecturer posed a question where we derive the density function of the student t-distribution from the Chi-square and Standard normal distribution. I wo...
Apr 19, 2026 1 VIEWS
What is essence of second order derivative?
Let $u(x)\in C^2(R)$ is a real function. so: $$ u'(x)=\lim_{\Delta x\rightarrow 0} \frac{u(x+\Delta x)-u(x)}{\Delta x} $$ And: $$ u''(x)=\l...
Apr 19, 2026 1 VIEWS
What is the intuition behind $\cos^2(x)$ being the same as $(\cos x)^2$?
Shouldn't $(\cos x)^2$ be $\cos^2 \cdot x^2$? as $(xy)^2$ is $x^2y^2$? What is the intuition behind it? Please use simple language as I am not a math...
Apr 19, 2026 1 VIEWS
Long Division Algorithm Proof
As I was doing my homework today, a sudden thought popped into my head. Why does our long-division algorithm work and how can I prove it? Why does it the ...
Apr 19, 2026 1 VIEWS
What is the magnitude of displacement?
I am studying arclength for a path $c(t)=(x(t),y(t),z(t))$. I understand that we integrate the derivative of speed etc. My question is, geometrically, wha...
Apr 19, 2026 1 VIEWS
Two fair dice are rolled. What is the conditional probability that at least one lands on 6 given that the dice land on different numbers?
$F =$ The probability that the dice land on different numbers $F = {(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,4),(3,5),(3...
Apr 19, 2026 1 VIEWS
Euler-Mascheroni constant bound for the Omega constant
The Omega constant $\Omega = \rm{W}_0(1) = 0.56714329\ldots$ is defined as the real root to the equation $x {\rm e}^x = 1$ and corresponds to the value of...
Apr 19, 2026 1 VIEWS
Why are viscosity solutions to Hamilton-Jacobi equations interesting?
This is a soft question. For my background, I am an undergraduate math major just finishing my first course in PDE. In my PDE class, we have been learning...
Apr 19, 2026 1 VIEWS
Solving 2D heat equation with separation of variables
Solve: $$ \begin{cases} u_t = Ku_{xx}\\ u(0,y,t) = u(\pi,y,t) = 0\\ u_y(x,0,t) = u_y(x,\pi,t) = 0\\ u(x,y,0) = 1 \end{cases}$$ for $0 < x < \pi$, $0...
Apr 19, 2026 1 VIEWS
How to find the positive and negative roots of a function?
Iam trying to solve the following question: Find all numbers $a$, such that the equation $x^2-ax-a = 0$ has one positive root and one negative root. I...
Apr 19, 2026 1 VIEWS
How come $32.5 = 31.5$? (The "Missing Square" puzzle.)
Below is a visual proof (!) that $32.5 = 31.5$. How could that be? (As noted in a comment and answer, this is known as the "Missing Square" puzzle.)
Apr 19, 2026 1 VIEWS