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Method of Characteristics / fluid simulation

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Hello, I am trying to implement the fluid Solver by Jos Stam (paper Stable Fluids). However, I don't really understand the maths he is doing in the appendic where he explains the method of characteristics. Can anybody give me a mathematical derivation why I can take simply trace back the velocity in time for the term (- u \dot \nabla) u. I really would like to understand the maths behind this method, but his explanation (and the ones I have found on the net) are simply not detailed enough for me to follow every step. Thanks, Icebraker

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