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4) to a volume integral and show that the integral is indeed identical to zero. 3) is indeed satis ed and that in the application of Stokes' law you can choose any surface as long as it is CHAPTER 7. THE THEOREM OF STOKES 62 bounded by the contour over which the line integration is carried out. This is a very useful result because often the surface integration can be simpli ed by choosing the surface carefully. 2 Stokes' theorem from the theorem of Gauss Stokes' law is concerned with surface integrations.

We now assume that the distance to the sun is perturbed from its original distance r to a new distance r + r, the perturbation in the position is therefore r = r ^r. 3). 24) Hint: consider the case where the radius is increased ( r > 0) and decreased ( r < 0) separately. 5. IS LIFE POSSIBLE IN A 5-DIMENSIONAL WORLD? 3: De nition of variables for the perturbed orbit of the earth. 24). e. mrv = m(r + r)(v + v). In what follows we will consider small perturbations and will retain only terms of rst order in the perturbation.

In this section we will determine this current using the theorem of Gauss. j j j j Problem b: In the following we need the time-derivative of (r t), where the asterisk denotes the complex conjugate. 13). 14) rst. @t V j j 2 r r 2 ; ; r r The left hand side of this expression gives the time-derivative of the probability that the particle is within the volume V . The only way the particle can enter or leave the volume is through the enclosing surface S . The right hand side therefore describes the \ ow" of probability through the surface S .

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A guided tour of mathematical physics by Snieder R.


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