>>558
一個目
x＝r cosθ, y＝r sinθ → dx＝cosθdr−r sinθdθ, dy＝sinθdr＋r cosθdθ
→ dr＝cosθdx＋sinθdy, dθ＝(−sinθdx＋cosθdy)/r
→ ∂r/∂x＝cosθ, ∂r/∂y＝sinθ, ∂θ/∂x＝−sinθ/r, ∂θ/∂y＝cosθ/r
∂^2f/∂x∂y＝(∂/∂x)(∂f/∂y)＝(∂/∂x)((∂g/∂r)(∂r/∂y)＋(∂g/∂θ)(∂θ/∂y))
＝((∂r/∂x)(∂/∂r)＋(∂θ/∂x)(∂/∂θ))((∂g/∂r)sinθ＋(∂g/∂θ)cosθ/r)
＝(cosθ(∂/∂r)−(sinθ/r)(∂/∂θ))((∂g/∂r)sinθ＋(∂g/∂θ)cosθ/r)
＝cosθ(∂/∂r)((∂g/∂r)sinθ＋(∂g/∂θ)cosθ/r)−(sinθ/r)(∂/∂θ)((∂g/∂r)sinθ＋(∂g/∂θ)cosθ/r)
＝cosθ((∂^2g/∂r^2)sinθ＋(∂^2g/∂r∂θ)cosθ/r−(∂g/∂θ)cosθ/r^2)−(sinθ/r)((∂^2g/∂r∂θ)sinθ＋(∂g/∂r)cosθ＋(∂^2g/∂θ^2)cosθ/r−(∂g/∂θ)sinθ/r)
＝(∂^2g/∂r∂θ)((cosθ)^2−(sinθ)^2)/r−(∂g/∂θ)((cosθ)^2−(sinθ)^2)/r^2
＋(∂^2g/∂r^2−(∂^2g/∂θ^2)/r^2−(∂g/∂r)/r)sinθcosθ
＝((∂^2g/∂r∂θ)/r−(∂g/∂θ)/r^2)cos2θ＋(1/2)(∂^2g/∂r^2−(∂^2g/∂θ^2)/r^2−(∂g/∂r)/r)sin2θ