求高数题设函数g(r)有二阶导数,f(x,y)=g(r),r=根号下x平方加y平方,求证:f对x的二阶偏导数加上f对y的二阶偏导数=g"(r)+g'(r)/r

Question

求高数题
设函数g(r)有二阶导数,f(x,y)=g(r),r=根号下x平方加y平方,求证:f对x的二阶偏导数加上f对y的二阶偏导数=g"(r)+g'(r)/r

Answer 1

由轮换对称性,只需要计算αf/αx,α^2f/αx^2 即可,把x换作y就是另一个偏导数
r对x的偏导数是x/r,所以
αf/αx＝g'(r)×x/r
α^2f/αx^2＝g''(r)×(x/r)^2＋g'(r)×[r－x×(x/r)]/r^2＝g''(r)×x^2/r^2＋g'(r)×[r^2－x^2]/r^3
所以,α^2f/αx^2＝g''(r)×y^2/r^2＋g'(r)×[r^2－y^2]/r^3
所以,α^2f/αx^2＋α^2f/αx^2＝g''(r)＋g'(r)/r