Partial Differentiation

設球坐標向量F=<u,v,w>表示r方向變化量為u, θ方向變化量v, φ方向變化量為w
本題函數w=f(r),由定義知:
(1) grad(w)=<f'(r),0,0>
(2) div<f'(r),0,0>= [r^2*f'(r)]'/r^2= [r^2*f"(r)+ 2r*f'(r)]/r^2= f"(r)+2f'(r)/r
故∂^2 w/∂x^2+...+∂^2 w/∂z^2=div<grad(w)>=div<f'(r), 0, 0>
=f"(r)+ 2f'(r)/r= ∂^2 w/∂r^2+ (2/r)*(∂w/∂r)

另法: 公式
∆w=(∂^2/∂x^2+∂^2/∂y^2+∂^2/∂z^2)w
=(1/r^2)*(∂/∂r)(r^2*∂w/∂r)+1/(r^2 sinθ)*(∂/∂θ)(sinθ*∂w/∂θ)+(∂^2w/∂φ^2)/(rsinθ)^2
=(1/r^2)*(∂/∂r)(r^2* f'(r))+0+0
=(1/r^2)*[2r*f'(r)+r^2*f"(r)]= f"(r)/r^2+ 2f'(r)/r
=∂^2 w/∂r^2+(2/r)*(∂w/∂r)