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Some things to think about: symmetrical wings, with the same curve on bottom and top, can fly perfectly fine. Flat wings, with no curve at all, don’t fly quite as well but with the proper angle of attack can also generate lift. Additionally, planes fly perfectly well upside down.
If this curve explanation were complete, how could those things work?
Not sure what you mean by this. But planes still generate lift when flying upside down. Wings with a symmetrical curve can also generate lift. Flat wings with no curve at all can also generate lift.
Pressure differences are definitely involved. That’s the only way air pushes against things, after all, so the fact that there is a lift force implies a pressure difference. However the cause of the pressure difference is rather complicated.