El Jacobiano es
J(r,θ)=∂(x,y)∂(r,θ)=∣∂x∂r∂x∂θ∂y∂r∂y∂θ∣=∣cosθ−senθsenθrcosθ∣=rcos2θ+rsen2θ=r(cos2θ+sen2θ=r\begin{aligned} J(r,\theta) &= \frac{\partial(x,y)}{\partial(r,\theta)} = \begin{vmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta}\\ \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \end{vmatrix} = \begin{vmatrix} cos \theta & -sen \theta\\ sen \theta & rcos \theta \end{vmatrix}\\ &= r cos^2\theta + r sen^2\theta = r\big(cos^2\theta + sen^2\theta = r \end{aligned}J(r,θ)=∂(r,θ)∂(x,y)=∣∣∂r∂x∂r∂y∂θ∂x∂θ∂y∣∣=∣∣cosθsenθ−senθrcosθ∣∣=rcos2θ+rsen2θ=r(cos2θ+sen2θ=r