请帮忙解这个数分的题

来源：百度知道 编辑：UC知道 时间：2024/06/22 20:38:02
Find an analytic (symbolic) description of the mapping that assigns to a non-zero vector x∈ Rn the vector of length 2 that points in the direction opposite to x. In case n = 3 ,that is if x = (x, y, z) , determine the component functions of this mapping. Is it possible to define the image of the mapping at (0,0,0) such that it is continuous on R3 ? Justify your
answer.

题目的意思是找一个向量映射使得|f(x)|=2,且方向与x相反
以n=3为例
a=(x,y,z)≠0,|a|=(x^2+y^2+z^2)^(1/2)
f(a)=(-2x/[3^(1/2)|a|],-2y/[3^(1/2)|a|],-2z/[3^(1/2)|a|])
要让f在(0,0,0)连续,只需定义f((0,0,0))=(0,0,0)即可
验证很简单