The transformation relating Cartesian and cylindrical coordinate is p = (x² + y²)¿ 0 = tan¬ x = p cos 0, y = p sin 0, z = z, Show that the gradient operator in cylindrical coordinates is given by z = Z 1 V= pa, + ÷@aq + 20z

Consider that a particular physical phenomenon can be modeled, in steric coordinates, by the scalar potential 0, q1 = r, q2 = 0, 93=0, find V, r(r, 0, 0) = r cos tan 0. If ê₁ = f, ê₂ = 4, 3 = knowing that, in generalized coordinates, V4(91, 92, 93) = 3 i=1 1 მს hi dqi

(a) Let F₁ = x² 2 and F₂ = x x + y ŷ + z 2. Calculate the divergence and curl of F₁ and F₂. Which one can be written as the gradient of a scalar? Find a scalar potential that does the job. Which one can be written as the curl of a vector? Find a suitable vector potential. (b) Show that the field F3 = yz î + zx ŷ + xy 2 can be written both as the gradient of a scalar and as the curl of a vector. Find scalar and vector potentials for this function.

I cant seem to come to (3/2)a . I set up my integral like this: (4/a^3)  ∫ r^2 e^[ (-e/a)^2 ] dr is this correct?