Surface Integrals of Vector Fields When dealing with surface integrals of vector fields we encounter the surface orientation issue. Our basic method of integration is: | [F(x, y, z)-ñ ds = ± z)-nds: ±] [F(x, y, z) - (FF) A R Where the + indicates that we need to decide which it is. Ideally we'd like to parametrize the surface so that the cross-product generated normal vectors point in the direction of desired orientation but this is quite difficult. Instead we take the approach from class - we look at the cross product XT, and examine whether it agrees with or disagrees with Σ's orientation. If it disagrees we negate. For example suppose we wish to evaluate √ √(x²î + yî + zk) • ñ dS where Σ is the part of the parabolic sheet y = x² with −2 ≤ x ≤2 and 0 ≤ z ≤ 3, oriented Σ outwards, meaning with positive y-component. We can parameterize Σ with F(x, z) = x+(4−x²)ĵ+zk with −2 ≤x≤2 and 0≤≤3. We can find FXF, with Matlab: syms x z; rbar [x,4-x^2,2]; cross(diff(rbar,x),diff(rbar,z)) This yields output: ans = [ -2*x, -1, 0] These vectors have negative y-component and hence they disagree with Σ's orientation. The final integral then needs to be negated. Here is the whole thing: syms x y z; rbar -[x,4-x^2,2]; F = [x^2,y,z]; -int (int(simplify (dot (subs (F, [x,y,z], rbar), cross (diff(rbar, x), diff(rbar, z)))),x,-2,2),z,0,3) Suppose Σ is the portion of the parabolic sheet y = x² bounded by -2≤x≤3 and 0≤z≤5 with orientation in the negative y direction. If F(x, y, z) = xî + xyj + yzk describes the flow of a fluid, find the rate of flow of F through Σ in the direction of orientation. Assign the result to q7.