Stokes’ Theorem for evaluating surface
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- Stokes’ Theorem for evaluating surface integrals Evaluatethe line integral in Stokes’ Theorem to determine the value of thesurface integral ∫∫S (∇ x F) ⋅ n dS. Assume n points in an upwarddirection. F = r/ |r|; S is the paraboloid x = 9 - y2 - z2, for 0 ≤ x ≤ 9(excluding its base), and r = ⟨x, y, z⟩ .arrow_forwardStokes’ Theorem for evaluating surface integrals Evaluate the line integral in Stokes’ Theorem to determine the value of the surface integral ∫∫S (∇ x F) ⋅ n dS. Assume n points in an upward direction. F = ⟨y, z - x, -y⟩; S is the part of the paraboloidz = 2 - x2 - 2y2 that lies within the cylinder x2 + y2 = 1.arrow_forwardStokes’ Theorem for evaluating surface integrals Evaluate the line integral in Stokes’ Theorem to determine the value of the surface integral ∫∫S (∇ x F) ⋅ n dS. Assume n points in an upward direction. F = ⟨ex, 1/z, y⟩; S is the part of the surface z = 4 - 3y2 thatlies within the paraboloid z = x2 + y2.arrow_forward
- Let F=xi+yj+zk and let f(x,y,z)=x^2e^(y-z). a) Describe a surface S together with orientation so that double integral (F.dS)<0 b) Explain why you cannot find a surface S so that double integral f(x,y,z)dS<0.arrow_forwardx = y + y3, 0 ≤ y ≤ 4 Set up and evaluate an integral for the area of the surface obtained by rotating the curve about the x-axis then the y-axisarrow_forwardUse only the Stokes’ theorem to evaluate integralC(−z − y)dx + (x − z)dy + (x − y)dz where C is the intersection of z = x^2 + y^2 and z = y, oriented counterclockwiselooking down from the positive z-axis. Sketch the surface σ and the region Raccurately. Simplify your answer.arrow_forward
- Not using divergence theorem. I just need I don't need to solve. Just need the surface integral for S1 and S2arrow_forwardNo integrals Let F = ⟨2z, z, 2y + x⟩, and let S be the hemisphereof radius a with its base in the xy-plane and center at the origin.a. Evaluate ∫∫S (∇ x F) ⋅ n dS by computing ∇ x F and appealing to symmetry.b. Evaluate the line integral using Stokes’ Theorem to check part (a).arrow_forwardStokes’ Theorem for evaluating surface integrals Evaluatethe line integral in Stokes’ Theorem to determine the value of thesurface integral ∫∫S (∇ x F) ⋅ n dS. Assume n points in an upwarddirection. F = ⟨x + y, y + z, z + x⟩; S is the tilted disk enclosed byr(t) = ⟨cos t, 2 sin t, 13 cos t⟩ .arrow_forward
- I. Use the double integral to check that the moments of inertia in the region with respect to the axes are those illustrated in the figure. Assume that the rho density of the lamina is 1 gram per square centimeter. Then calculate the radii of rotation with respect to each axis: Determine the Surface área for f (x, y) =13+x2 -y2 on the region R= {(x, y); x2 +y2 ≤ 4arrow_forwardFlux Integral, Evaluate double integral S of sin(y)*cos(z)i +e^x*cos(z)j+cos(y)*ln(1+x^2)k)·NdS, where S is the sphere x^2+y^2+z^2=1 oriented outwards.arrow_forwardEvaluate the surface integral. The double integral over S (xz) dS, S is the part of the plane 2x+2y+z=4 that lies in the first octant.arrow_forward
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