高斯(dick)积分

∫(-inf,inf)e^(-x^2)dx ← 这是高斯积分

原式^2
=∫(-inf,inf)e^(-x^2)dx*∫(-inf,inf)e^(-y^2)dy

=∬e^(-x^2-y^2)dσ D={(x,y)|x∈R,y∈R}

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

=∫dθ∫re^(-r^2)dr

=2π/2

=π

原式=√(π)

*为什么这里会多一个r?? → ∫dθ∫re^(-r^2*sin^2θ-r^2*cos^2)dr

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

=|det(∂r*sinθ/∂r,∂r*sinθ/∂θ,∂r*cosθ/∂r,∂r*cosθ/∂θ)|ρdρdθ

=|det(sinθ,r*cosnθ,cosθ,-r*sinθ)|ρdρdθ

=|-r*sin^2θ-r*cos^2θ|ρdρdθ

=rρdρdθ