Potential function,

u  = - ∂Φ / ∂x             v  = - ∂Φ / ∂y

Stream function

u  = - ∂ψ / ∂y             v  = - ∂ψ / ∂x  

Then          u  = - ∂Φ / ∂x  = -  ∂ψ / ∂y


v  = - ∂Φ / ∂y  = ∂ψ / ∂x

∂Φ / ∂x  =  ∂ψ / ∂y

∂Φ / ∂y  = - ∂ψ / ∂x

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