欧拉齐次函数定理
2024-9-28
如果\(f(x_1,x_2,\cdots,x_n)\)是\(m\)次齐次函数, 则
$$f(ax_1,ax_2,\cdots,ax_n)=a^mf(x_1,x_2,\cdots,x_n)$$如果\(f(x_1,x_2,\cdots,x_n)\)是\(m\)次齐次函数, 那么有
$$\sum_{i=1}^n{\frac{\partial{f}}{\partial{x_i}}}x_i=mf$$证明：两边对\(a\)求导，可得：
$$ma^{m-1}f(x_1,x_2,\cdots,x_n,)=\sum_{i=1}^n{\frac{\partial{f(ax_1,ax_2,\cdots,ax_n)}}{\partial{ax_i}}}\frac{\partial{ax_i}}{\partial{a}}=\sum_{i=1}^n\frac{\partial{f(ax_1,ax_2,\cdots,ax_n)}}{\partial{ax_i}}x_i$$令\(a=1\), 得到：\(mf=\sum_{i=1}^n\frac{\partial{f}}{\partial{x_i}}x_i\)