Question

please solve the problem instead of copying from another posted answer.

#11 Compute the outward flux of F(xil, z) = (x+1) e + y ez zez through the Surface s of a tetrahedson whose vertices are at (0,0,0), (1,0,0), (0,1,0), (90,1), a) do me thing the appropriate fly intepral b) Applying an appropriate intepral theorem. Sketch S.

Answer 1

Solution. Let the gires Vestire be 0 = (0,0,0), A = (1,0,0), B- (0,1,0), c= (0,0,1) Then the detrahedson, forured in by the given Vertices is OA B C A which & shown in the figure. It has four triangular faces naurely in OAB OBC, OAC and ABC Sied Straight liner edges and four Vaeltices. I -> la) Given I = (2+1) e te zez to find out, fruga of a closed surfaces, Assume i represents Velocity of the frid particle. Formula! If I geprakents & veloutſ. of a &hid particle, then this across a closed surface S (Tetrahedron) is - ( T n de (where in is ont crood) is normal to the set fare) Id - Indst findet (fikidet (Finnds-o овоосоосто се the saleface OABO; Here sa o to I, 420701, 2 = 0 and ating ds - dado - daday (lei) = 1)

Solention to ch! By Gruss divergence there we Lowe Stonds - dividv. - 8 ra the given tetrahedson, x=0 to 1, y =oto 1, 2= o tol.. we have divi - D. F2 = 1 +/-1 = 1 again dy =dx dydze Substitute above information in RoHS of Egr 8 we get Sinds = ss su dady dz x=0 450 2=0 = (x) (8) (2) = 10) 0 = 1 - 6 from Exer & , o Gauss divergence theoreur is verified.

No mal to this surface is Toote 28 + hot =ept 2(1) + ez (1) = e, tez tez Merit nolonal = n = To .. Let R be the projection of the plane. OAB CA In dy-plane. than ds = dadya da daya & dady Iñine v also Fin= [(a+1) 8,460,- 2033t Cesteste) -t [arity-2] [+1++4+9=] ( from cano) to (2x +2y] (+y) s finds a S S 2 (1+y). 13 dxdag ABCA X-0 Y=0 = 2 (apt 2 dan = 28 (2 + 2 ) dx 200 = 2(22+4)!=(***)=2 substitute the values of Egh's , G , 4 and 6 in Ear Then the flun across the Surface sa SFinds=0+6D+0+2 = 1 0 2279-) ya x=0

also finto [(x+1), tv-2 t]. (-e) e z = 0 Corso ethir Surface) Inds= (oods = o- SABO OA Bo o n For the susface of co n there fotol, 20to), a-0, h e de a dyde dag daz ne) and senn – – (x + 1) – – 1 ( x=0 SI inds= 1 (1-1) dry chez = - (-y) (2) ' OBCO Y=0 250 = -0.) = -1 - s for the surface oCAD or there a=0 to 1, y=0,2 =otol in=- ds - dadz - dada and do Fin = – 8 = 0 (y=0 OCAO SCAD For the surface ABCA: 2 n=0 to 1, 4= 0 to 1 - 0 to 1 equation of the plane to is alty + ² = 1 j.e x+y+z-1=0 het p = x+y+2-1