是我,高斯男孩
e**** d*** integral

∫(-inf,inf)e(-x2)dx ← 这事高斯积分

原式^2
=∫(-inf,inf)e(-x2)dx*∫(-inf,inf)e(-y2)dy

=∬e(-x2-y^2)dσ D={(x,y)|x∈R,y∈R}

= ∫dθ∫re(-r2sin2θ-r2cos^2)dr D={(r,θ)|r∈[0,inf),θ∈[0,2π]}

=∫dθ∫re(-r2)dr

=2π/2

原式=√(π)

为甚么这里会多一个r?? → ∫dθ∫re(-r2sin2θ-r2*cos^2)dr

dxdy=|det(∂x/∂r,∂x/∂θ,∂y/∂r,∂y/∂θ)|ρdρdθ

=|det(∂rsinθ/∂r,∂rsinθ/∂θ,∂rcosθ/∂r,∂rcosθ/∂θ)|ρdρdθ

=|det(sinθ,rcosnθ,cosθ,-rsinθ)|ρdρdθ

=|-rsin^2θ-rcos^2θ|ρdρdθ

=rρdρdθ