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For n functions f_i of n variables x_i, the Jacobian, written [(∂( f₁ , f₂ , ... , f_n ))/( ∂( x₁ , x₂ , ... , x _n ) )] is the following determinant:

           ∂f1 ∂x1 ∂f1 ∂x2 ... ∂f1 ∂x n ∂f2 ∂x1 ∂f2 ∂x2 ... ∂f2 ∂xn : : : ∂fn ∂x1 ∂fn ∂x2 ... ∂fn ∂xn           

It is useful when we need to perform a change of coordinates in an integral with several variables; it gives eg the ratio of dxdydz to df(x,y,z)dg(x,y,z)dh(x,y,z).