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The mathematical proof doesn't take into account that the contour is moving with the fluid. See for example the treatment in . I would fix this myself but I dont feel sure enough yet.
Indeed; I was just coming here to post this myself. Also, Γ = Γ(t) do the derivative on it should be straight 'd' only. And what are the ω and Φ that appear suddenly? (It's also a bit worrying that baratropicity isn't used explicitly anywhere in the proof, and it would be nice to start the proof from a governing equation!)
Unfortunately, the cleanest way of proving it involves knowing how line elements translate in the flow, which is kind of assumed in your reference. Also the treatment there isn't quite as general as it could be, since ρ is assumed to be constant. I'll try and have a go at fixing things when I get a moment...