Question

Let F(x, y, z) (xr,y, z). Compute the outward flux of F: 9y2622 on the bounded region inside of S. However, you may wish to consider the region bounded between S and the sphere of radius 100.) 7/Fthrough the ellipsoid 4c2 36. (Hint: Because F is not continuous at zero, you cannot use the divergence theorem Suppose that E is the unit cube in the first octant and F(z,y, z) = (-x,y, z). Let S be the surface obtained by taking the surface of E without it's top (so S has five sides). Calculate ffg F dS in two different ways: i) First, by directly calculating a surface integral; (ii) Second, by using the divergence theorem

Answer 1

divF dv Mese We must we F.ds F.n ds F.n2 ds Fds t F ds cs Scanned with CamScanner S

F not definedl at the ongin. let S, be win Centre at e Since The sphera odi 1 Let S be the ongin aud outer norma n1 suafa ca at the ellipsetd aud let e be the Solid o betwean S anol S2 Then the outward fux of F given by outer nomal n through the elipsote Fds- divFdY F.C ds + -3 But F r So, -3 -4 -3 3 2 and S En, r Iri Thus, F.n2 ds ds t o dy SL SI the unit E Sphore Sunfaco a rea 2 lcs Scanned with Camscanner