(from X.L. Zhen) 计算二重积分 $$ex iint_{bR^2}e^{-(x^2+xy+y^2)} d x d y. eex$$
解答: $$eex ea iint_{bR^2}e^{-(x^2+xy+y^2)} d x d y &=iint_{bR^2} e^{-sex{sex{x+frac{y}{2}}^2+sex{frac{sqrt{3}}{2}y}^2}} d x d y\ &=iint_{bR^2} e^{-(u^2+v^2)}sev{frac{p(x,y)}{p (u,v)}} d u d vquadsex{u=x+frac{y}{2}, v=frac{sqrt{3}}{2}y}\ &=iint_{bR^2} e^{-(u^2+v^2)}frac{2}{sqrt{3}} d u d v\ &=frac{2}{sqrt{3}}int_0^infty e^{-r^2}cdot 2pi r d r\ &=frac{2pi}{sqrt{3}}. eea eeex$$