Fundamental TheoremVf• år = f(B) – f(A)of Line IntegralsGreen's Theorem(8x - f,) dA =f dx + g dy(Circulation form)Stokes' Theorem(V X F) • n dS =Evaluate the line integral for the following problems over the given regions. Use the abovetheorems if the problem meets the criteria. If the criteria is not met, use Section 17.2 methods forevaluating the line integral. F = (x² – y²,x, 2yz); C is the boundary of the plane z = 6 – 2x – y in the first octant.

Given, F=x2-y2,x,2yz Find the line integral ∮CF·dr in the following. Here C is the boundary of the plane z=6-2x-y in the first octant.Now using Stokes' Theorem, ∫∫S∇×F·n dS=∮CF·dr Then, ∇×F=ijk∂∂x∂∂y∂∂zx2-y2x2yz =i2z-0-j0-0+k1+2y =2z i+1+2yk Consider gx,y,z=2x+y+z-6So, n=∇g∇g =2 i+j+k22+12+12 =2 i+j+k6 Then, ∮CF·dr=∫∫S∇×F·n dS =∫∫S2z i+1+2yk·2 i+j+k6 dS =16∫∫S4z+1+2y dS In yz-plane, z=6-yTherefore, ∮CF·dr=16∫y=06∫z=06-y4z+1+2y dzdy =16∫y=062z2+z+2yzz=06-y dy =16∫y=0626-y2+6-y+2y6-y dy =16∫y=0678-13y dy =1678y-13y22y=06 =1678×6-13622-78×0-13022 =16×234 =396Hence, the value of the line integral is ∮CF·dr=396