The Jacobian measures how a coordinate map scales small areas.
WHAT HAPPENS IF…For r=2, Δr=0.5, Δθ=0.2, how large is the approximation error?
Interactive exploration
Loading the mathematical model…
Connect the picture to the ideadet∂(r,θ)∂(x,y)=r
For (x,y)=(r cosθ,r sinθ), derivative columns are (cosθ,sinθ) and (−r sinθ,r cosθ), with determinant r. The anchored sector has exact area (rΔr+Δr²/2)Δθ, while its tangent parallelogram has area rΔrΔθ. At r=0 the map is singular.
A detail that matters
Area scaling uses the absolute determinant; polar radius is nonnegative here.