A guided tour of mathematical physics by Snieder R.

By Snieder R.

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21) and check this result for three spatial dimensions. At this point the constant A is not determined, but this is not important for the coming arguments. The minus sign is added for convenience, the gravitational eld points towards the sun hence A > 0. Associated with the gravitational eld is a gravitational force that attracts the earth towards the sun. 22) and is directed towards the sun. For simplicity we assume that the earth is in a circular orbit. 23) In equilibrium these forces balance: Fgrav + Fcent = 0.

1) the z -axis points out o the paper, this implies that the vector dS also points out of the paper. This means that in Stokes' law the sense of the line integration and the direction of the surface vector dS are related through the rule for a right-handed screw. There is something strange about Stokes' law. If we de ne a curve C over which we carry out the line integral, we can de ne many di erent surfaces S that are bounded by the same curve C . Apparently, the surface integral in the right hand side of Stokes' law does not depend on the speci c choice of the surface S as long as it is bounded by the curve C .

6) we have to de ne the relation between the vectors u and v. 4). 4). Since a rotation is an orthonormal transformation the inner product of two vectors is invariant for a rotation over 90 degrees so that (u n^) = (v ^t). 7). 6). What you have shown here is that Stokes' law for the special case considered in this section is identical to the theorem of Gauss for two spatial dimensions. 5). 5). When the disc is larger than the thickness R of the wire the surface integral of J gives the electric current I through the wire: I = J dS.

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