1. 1. In cylindrical coordinates, the position vector has the following Cartesian -- -- components:

      

      where we have used the definitions of the cylindrical coordinate unit vectors in Cartesian components,

      

      These unit vectors have time derivatives

      

      Thus particle velocities are given by

      

      and particle accelerations by

      

      Substituting for the unit vector time derivatives gives

      

   2. In spherical coordinates, the position vector has the following Cartesian components:

      

      where we have used the spherical coordinate unit vectors in Cartesian components, as derived in Griffiths section 1.4,

      

      These unit vectors have time derivatives

      

      This leads to particle velocity

      

      as you will show in assigned problem 1(a), and particle acceleration

      

      Substituting for the unit vector time derivatives gives

      

2. 1. To keep , must vanish. But

      

      Thus we must have a centripetal force directed radially inward, .

   2. Since ,

      

      For constant , reduces to . So we have solutions

      

      

3. Solution appears in Griffiths Example 1.3.
4. 1. In spherical coordinates, from Griffiths section 1.4, infinitesimal increases in our variables correspond to infinitesimal physical displacements of as follows:

      

      By the chain rule, . By definition of the gradient, . Setting the two expressions for df equal gives

      

      Similarly, in cylindrical coordinates, infinitesimal coordinate changes cause infinitesimal physical displacements of as follows:

      

      So (in cylindrical -- -- coordinates). By the chain rule, . The same df is given by . Setting both expressions equal gives

      

   2. 
      

      

      

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