Maxwell's theorem (after James Clerk Maxwell) says that if a function $f(x_1,\ldots,x_n)$ of $n$ real variables is a product $f_1(x_1)\cdots f_n(x_n)$ and is rotation-invariant in the sense that the value of the function depends on $x_1,\ldots,x_n$ only through $x_1^2+\cdots+x_n^2$, then it's an exponential function of that sum of squares (so $f$ is a "Gaussian function").
Say a class of undergraduates knows only enough mathematics to understand what the theorem says (so maybe they've never heard of inner products or orthogonal matrices), but they're bright and can understand mathematical arguments. What proof of Maxwell's theorem do you show them? Can you keep it short and simple?