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The Jacobian determinant of a two-variable transformation is (∂u/∂x)(∂v/∂y) − (∂u/∂y)(∂v/∂x). It measures how much the map stretches area locally, and it is the factor that appears when changing variables in a double integral.
Jacobian
J = ∂u/∂x · ∂v/∂y − ∂u/∂y · ∂v/∂x
Change of variables
dA = |J| du dv
J = 8 − 3 = 5, so areas are stretched fivefold and orientation is preserved.
r, which is why dA = r dr dθ in polar coordinates.