If $a^2+b^2+c^2+ab+bc+ca \le 0\quad \forall a, b, c\in\mathbb{R}$, then find the value of the determinant $$ \begin{vmatrix} (a+b+2)^2 & a^2+b^2 &...

If I have a right triangle, of the form: $ABC$, with right angle at B: Then, $BM = CM = AM$. Then, as I show that: $BM = AM$ The simplest demonstration is...

I was pondering the topic of complex numbers and how we just look at the them and extract the real and imaginary parts because it's easy given that w...

What is an orientation of a surface? What difference does it make if you pick one versus the other?

In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotan...

I am trying to prove that the derivative of 1/x exists. Is my proof correct? Please help me! Proof. According to the definition of the derivative. It is n...

Given the IVP $$ y'' + y = f(t) , \qquad\quad y(0) = 0 , \quad y'(0) = 0 , \tag{1}$$ where $$ f_{k} (t) = u_{0} + 2 \sum_{k=1}^{n} (-1)^{k}...

This is my first proof related to linear functions. It refers to the linear-algebra-$\textit{linear}$ (not the calculus-$\textit{linear}$). Please comment...

An element $S$ is said to be totally ordered set if for all $a, b\in S\implies$ either $a\leq b$ or $b\leq a.$ An algebraic structure $(S, +, \cdot)$ is s...

My question is very simple yet I've gotten nowhere with it. Is there any way one can, without directly or indirectly referencing any differential equ...