news
Finding the error in this proof that 1=2
I have a "proof" that has an error in it and my goal is to figure out what this error is. The proof: If $x = y$, then $$ \begin{eqnarray} x^2 &=& ...
Apr 13, 2026
Archive: News
news
Do we have a "short five lemma" for any two of the isomorphisms?
$\require{AMScd}$ The "short" Five Lemma concerns the famous form of exact commutative diagram: $$\begin{CD}0@>>>A@>>>B@>>>C@&g...
Apr 13, 2026
news
Generic topology on a vector space?
For a (possibly infinite-dimensional) vector space $V$, I thought about the following topology $\tau$: Let $O \in \tau$ if every $x \in O$ has the propert...
Apr 13, 2026
news
Weird indefinite integral homework questions
I'm solving a couple of integration problems using the method of changing variables, and would like assistance with two particular problems that I ca...
Apr 13, 2026
news
ring without zero-divisors
Suppose we have a ring (could be infinite) without zero-divisors. I have to prove that if $xy=1$ then also $yx=1$ for some $x$ and $y$ in the ring. I real...
Apr 13, 2026
news
Finding out the volume of an inscribed and circumscribed sphere in a prism
A prism with an equilateral triangle in its base, whose height is two times bigger than base-side, is given. Find out the ratio of volumes of inscribed an...
Apr 13, 2026
news
Polar angle of a given point
How to find out polar angle of a given point $A(x_1,y_1)$ relative to another point $B(x_2,y_2)$ in a 2D space?
Apr 13, 2026
news
Finding the differential equation for a simple pendulum.
The motion of the simple pendulum is described by the following differential equation $$\frac{d^2 \theta}{dt^2}+\frac{g \sin \theta }{l}=0$$ Multiply thro...
Apr 13, 2026
news
Linearization with Jacobian Matrix
Assuming I have $\frac{dx}{dt}=5x^2+2xy+x$ $\frac{dy}{dt}=xy-y$ which leads to a jacobian matrix $$\begin{pmatrix} 10x+2y & 2y \\ y & x-1 \end{pma...
Apr 13, 2026
news
Normal vector to a surface and binormal vector to a curve that lies on the surface
Is the normal vector to a surface on the same direction as the binormal vector to a curve that lies on the surface ? To be more precise, consider a curve ...
Apr 13, 2026
