Integrate the vector field A = ze z over a sphere with radius a, centered at the

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Integrate the vector field A = zez 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 eθ̂ and a sin θd e∅̂. Here eθ̂ 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


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


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(c) Explain why (e , eθ̂ , 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.

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