We may use Stokes' Theorem to deduce this by applying it to a tiny rectangle normal to the w₁ direction with sides obeying one of dw₂  = 0 and dw₃  = 0, and then cyclically permuting the indices.
The surface integral in Stokes' theorem then becomes

u₂u₃dw₂dw₃(v)₁

while the path integrals become

which gives us:

Exercises