Divergence Theorem: If S is a closed surface in R that bounds a solid region B and if the boundary S is oriented via an outward unit normal vector n, and if F = (P(x, y,z),Q(x, y,z), R(x, y, z)) is a...

Extracted text: Divergence Theorem: If S is a closed surface in R that bounds a solid region B and if the boundary S is oriented via an outward unit normal vector n, and if F = (P(x, y,z),Q(x, y,z), R(x, y, z)) is a vector field defined and differentiable throughout B (and its boundary) with continuous first partial derivatives. Then: net flux of F outward $e Fyds = F-ndS =|.F-dS = [[[, div(F)dV F- ndS =1Ws=ôB F-dS = |||, div(F)dV S-OB В across S = ôB In problems 13 and 14, verify that the Divergence Theorem is true for the vector field F on the region E. Problem 13. (13.8/2) F(x, y, z) = x² i+ xyj+ zk, E is the solid bounded by the paraboloid z = 4– x² – y² and the xy-plane. Problem 14. (13.8/4) F(x, y, z) = (x²,-y,z), E is the solid cylinder y +z?<9,><><2. -y,="">