I've been interested in properly calculating a Jacobian Integral (that is, an integral in which a change of variables occurs). I'm sure that, by one's reading this, you've all probably already heard of how Jacobian Integration was used to calculate $$\int_{-\infty}^{\infty}e^{-x^2}dx = \sqrt{\pi}...

i have that $f:\mathbb{R}\rightarrow\mathbb{R}$ continuous such that $|f(t)|\leq c [\phi(|t|)|t|+\phi_*(|t|)|t|]$ $(u_n)\subset W^{1,\Phi}(\mathbb{R}^N)$ such that $u_n\rightarrow u$ How to prove that $$\int_{\mathbb{R}^N} |f(u_n)-f(u)| |v| dx\rightarrow 0~\text{for all}~ v\in W^{1,\Phi}(\math...

A related (and to me, when I first saw it, much more suprising) Fun Fact: Divide the n-dimensional cube in half in each of $n$ dimensions, to create $2^n$ smaller cubes of edge length 1/2. Inscribe a ball in each of these subcubes, and then construct the smallest ball tangent to each of those (...