A guided tour of mathematical physics by Snieder R.

By Snieder R.

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1: Two di erent bodies with a di erent mass distribution that generate the same gravitational eld for distances larger than the radius of the body on the right. Problem c: Let us assume that the mass is located within a sphere with radius R, and that the mass density within that sphere is constant. 6) R3 ^r : Plot the gravitational eld as a function from r when the distance increases from zero to a distance larger than the radius R. Verify explicitly that the gravitational eld is continuous at the radius R of the sphere.

However, if we consider a vector eld that depends only on the coordinates x and y (v = v(x y)) and that has a vanishing component in the z -direction (vz = 0), then v points along the z -axis. 5) Problem a: Verify this. This result can be derived from the theorem of Gauss in two dimensions. 4)) and where ds denotes the integration over the arclength of the curve C . 4: De nition of the geometric variables for the derivation of Stokes' law from the theorem of Gauss. 3. 6) we have to de ne the relation between the vectors u and v.

2) for a sketch of this ow eld. 2: Sketch of an axi-symmetric source-free ow in the x,y-plane. 6) is expressed in cylinder coordinates. 6) an expression for the curl in cylinder coordinates will be derived. As an alternative, one can express the unit vector '^ in Cartesian coordinates. 7) 0 Hints, make a gure of this vector in the x y-plane, verify that this vector is perpendicular to the position vector r and that it is of unit length. 5). 44 CHAPTER 5. 8) Hint, you have to use the derivatives @r=@x and @r=@y again.

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