Next, we addressed a question which a student asked me at the end of last lecture, involving the application of Stokes' theorem to the divergence theorem [Transparancy 2]. This leads to an apparent contradiction. The contradiction is resolved by realizing that the divergence theorem applies to a closed surface, wheras Stokes' theorem applies to a surface bounded by a curve.
The rest of the lecture was spent on an introduction to the Transport Theorems. First, the Flux Transport Theorem was presented [Transparancy 3], followed by the Reynold's Transport Theorem, [Transparancy 4].
A second form of the Reynold's Transport Theorem was also given, making use of an expression for the convective derivative. (The convective derivative is the time rate of change of a scalar field due to two processes: an intrinsic time-dependence of the scalar field, and the velocity of the time-dependent region which is under consideration.)
A "proof" of the flux transport theorem was discussed [Transparancies 5, 6, and 7]. It involves applying the divergence theorem to the volume swept out by the time-dependent surface under consideration. (A more rigorous proof is given in Chapter 5 of the textbook, for the interested student. We won't be covering this in class.)
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