>>160 下
まず
∂r/∂x = (∂/∂x)(x^2+y^2+z^2)^(1/2)
= (1/2)(x^2+y^2+z^2)^(-1/2)*2x
= (x^2+y^2+z^2)^(-1/2)*x
= x/r

(∂/∂x)(x/r) = (1/r) - (x/r^2)(∂r/∂x)
= (1/r) - (x/r^2)(x/r)
= (1/r) - (x^2/r^3)
で、y,z も同様だから

div(r↑/r) = (∂/∂x)(x/r) + (∂/∂y)(y/r) + (∂/∂z)(z/r)
= (3/r) - ((x^2+y^2+z^2)/r^3) = (3/r) - (r^2/r^3)
= 2/r