To solve problems 1 through 5, it is helpful to choose a special coordinate system to define the angular variables (in particular, to carefully choose the vector by which the polar angle is calculated). None of the results requested in these problems depend on the choice of this coordinate system. You may make use of the fact that f() = f(, ) is a periodic function of with period 2, i.e. f(, ) = f(, + 2).

If f() is a symmetric (even function) of , i.e. f() = f() or [ f(, + ) = f(, )]

then show that for any constant vector A:

integral f()d for A<0 >0,

and integral f()d for 4pi = 2 integral A<0 f()d = 2 integral a>0 f()d