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