1. Gradient practice. {4f} points} Compute the gradients of the following functions f in Cartesian. cylindrical~ and spherical coordinates. For the non-Cartesian coordinate systems? rst use the formula for the gradient in terms of the non-Cartesian nnit vectors+ and then use the conversions between the unit vectors to convert your answer hack to lli'j'artesian coordinates. In all cases~ you should find the same answer independent of the coordinate system! {You will nd that1 for all of these examples? the gradient is easy in one coordinate system but a mess in at least one of the others; this illustrates the value of choosing a good set of coordinates for the prohlem at hand.) [3} fl31y1zl=$2+y2+22 {h} fiI1y12}=sinz is} firvsnzl=ar+y+z . Divergence practice. {4H points} lCompute the divergences of the following vector elds v in the given coordinate systems. checking as in problem 1 that you get the same answer [which should just be a scalar function} in all coordinate systems by converting your nal answer hack to Cartesian coordinates. [a]: v = 3:): + y? + si: lCartesian1 cylindrical~ spherical {h} v = : Cartesian, cylindrical {c} v = Q: Cartesian, spherical Rotational invariance of the divergence. {2f} points} In index notation with the sum- mation convention? the divergence of a vector eld 11" can he written as rich Show that the divergence is incorrect under a rotation of the coordinate system. To do this+ use the coor- dinate transformation dened in Part 3 of Discussion 9, If = Raf-it which can he inverted to give 1\"" = [IiiWide". In other words? a vector with an upper index transforms with a factor of fit1+ while the gradient with a lower index transforms with a factor of R [as shown in Discussion '3}. Showing the invariance of the divergence is now just an exercise in careful index placement: make sure not to repeat any dummy indices in more than one summation