Recent content by THSMathWhiz

Any comments or feedback on this theorem and proof are appreciated, including improvements and past publications. Specifically, I'd like to know if someone else has come up with a formula for counting the number of invertible squares modulo N. Definition. If x\in(\mathbb{Z}/N)^\times...

Suppose we had ##u,v\in\mathbb{R}^2## such that ##u=B-A## and ##v=C-A##, then ##u-v=B-C##. Then ##a=||u-v||##, ##b=||v||##, and ##c=||u||##. So $$||u-v||^2=||u||^2+||v||^2-2uv\cos A,$$ or $$(u-v)\cdot(u-v)=u\cdot u+v\cdot v-2u\cdot v.$$ This is true and can easily be checked..

Here's what HomogenousCow is trying to say. Consider the expansion of ##\exp(t)## where ##t=-x^2##. The Taylor expansion of ##\exp(t)## is $$\sum_{n=0}^\infty \frac{t^n}{n!},$$ so substituting gives you $$e^{-x^2}=\sum_{n=0}^\infty \frac{(-x^2)^n}{n!}=\sum_{n=0}^\infty \frac{(-1)^n...

Consider the function f(x)=\begin{cases}(x+2)^3,&x<-2,\\0,&-2\leq x\leq 2,\\(x-2)^3,&x>2.\end{cases} You can see that f(x) is continuous on \mathbb{R} and that f'(x)\geq0\forall x\in\mathbb{R}. However, this function fails the horizontal line test on [-2,2] and therefore does not have a...

A formal cross product is defined for \mathbb{R}^3 and \mathbb{R}^7. There is a projective cross product for \mathbb{R}^2, which returns a directed scalar. Given \mathbf{u}=\langle u_1,u_2\rangle and \mathbf{v}=\langle v_1,v_2\rangle, the "cross product" is...

The first sum can be rewritten as \displaystyle \sum_{k=1}^\infty (2k)\frac{x^{2k-1}}{(2k)!}=\sum_{k=1}^\infty \frac{x^{2k-1}}{(2k-1)!} since (2k)!=(2k)(2k-1)!. To change the base index, replace k by k+1. You then get the sum \displaystyle \sum_{k=0}^\infty \frac{x^{2k+1}}{(2k+1)!}.

A global maximum is the absolute greatest value that a function reaches on its domain. For example, the function f(x)=x^3+x^2-17 x+15 has no global maximum, but g(x)=\sin(x) has global maxima at (\frac{\pi}{2}+2\pi n,1),\,\,n\in\mathbb{Z}. A local maximum is the greatest value that a function...

The natural logarithm function \ln{x} has two equivalent definitions. 1) \ln{x} is the function such that \exp(\ln{x})=\ln(e^x)=x, and 2) \ln{x}=\int_1^x \frac{dt}{t}. Proof. From definition 1 to 2: The derivative of \exp(\ln{x}) is \exp(\ln{x})\frac{d}{dx}\ln{x}. But \exp(\ln{x})=x, so...

But the only way to find the angular acceleration is to use the moment of inertia of the rod-spheres-bug system, which requires me to... Wait a minute... The two spheres make the density of the system non-uniform... Which means I need to calculate the second moment of inertia manually...

The moment of inertia for a thin, uniform rod of length L and mass m with its axis at its center is m*L^2/12. Does m = 2M, the mass of just the spheres, or m = 5M, the mass of the whole rod-spheres-bug system?