Integrate the vector field A = ze_z over a sphere with radius a, centered at the origin of the Cartesian coordinate system (i.e., compute A · d∑).

(a) Introduce spherical polar coordinates on the sphere, and construct the vectorial integration element d∑ from the two legs adθ e_θ̂ and a sin θd_∅ e_∅̂. Here e_θ̂ and e_∅̂ are unit-length vectors along the θ and ∅ directions. (Here as in Sec. 1.6 and throughout this book, we use accents on indices to indicate which basis the index is associated with: hats here for the spherical orthonormal basis, bars in Sec. 1.6 for the barred Cartesian basis.) Explain the factors adθ and a sinθd∅ in the definitions of the legs. Show that

(b) Using z = a cos θ and ez = cos θe_r̂ − sinθe_θ̂  on the sphere (where e_r̂ is the unit vector pointing in the radial direction), show that

(c) Explain why (e_r̂ , e_θ̂ , e_∅̂) = 1.

(d) Perform the integral ∫A · d∑ over the sphere’s surface to obtain your final answer (4π/3)a³. This, of course, is the volume of the sphere. Explain pictorially why this had to be the answer.

d = , e, e)a sin @dodo. (1.31)