Verify that the divergence theorem is true for the vector field f on the regione. give the flux. f(x, y, z) = x2i + xyj + zk, e is the solid bounded by the paraboloid z = 25 − x2 − y2 and the xy-plane.
Accepted Solution
A:
Surface integral: Parameterize the closed surface by
with [tex]0\le u\le5[/tex] and [tex]0\le v\le2\pi[/tex], where [tex]\mathbf s_1[/tex] defines the paraboloid part and [tex]\mathbf s_2[/tex] the planar part of the total surface [tex]\mathcal S[/tex].
where [tex]\mathcal V[/tex] denotes the space enclosed by the surface [tex]\mathcal S[/tex]. Converting to cylindrical coordinates lets us write the integral as