Surface
46. F = 〈e–y, 2z, xy〉 across the curved sides of the surface
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- calc 3 13.7 #6 Evaluate the surface integral ∫∫S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = y i + x j + z2 kS is the helicoid (with upward orientation) with vector equation r(u, v) = ucos(v)i + usin(v)j + v k, 0 ≤ u ≤ 5, 0 ≤ v ≤ 2?.arrow_forwardPropyl alcohol with a density of d = 0.2 g/cm2 flows over the closed curve r(t) = (sin t)i - (cos t)j, 0<= t<= 2pai, according to the vector field F = dv, where v = (x - y)i + x2j is a velocity field measured in centimeters per second. Find the circulation of F around the curve r(t).arrow_forwardLine integrals of vector fields in the plane Given the followingvector fields and oriented curves C, evaluate ∫C F ⋅ T ds. F = ⟨x, y⟩ on the parabola r(t) = ⟨4t, t2⟩ , for 0 ≤ t ≤ 1arrow_forward
- Flux integrals Compute the outward flux of the following vector fields across the given surfaces S. You should decide which integral of the Divergence Theorem to use. F = ⟨-yz, xz, 1⟩ ; S is the boundary of the ellipsoidx2/4 + y2/4 + z2 = 1.arrow_forwardFlux integrals Compute the outward flux of the following vector fields across the given surfaces S. You should decide which integral of the Divergence Theorem to use. F = ⟨x sin y, -cos y, z sin y⟩ ; S is the boundary of the regionbounded by the planes x = 1, y = 0, y = π/2, z = 0, and z = x.arrow_forwardThe heat flow vector field for conducting objects is F=−k∇T, where T(x,y,z) is the temperature in the object and k is a constant that depends on the material. Compute the outward flux of F across the given surface S for the given temperature distribution. Assume k=1. T(x,y,z,)=−7ln( x2 +y2 + z2) S is the sphere x2+y2+z2=a2arrow_forward
- The base of the closed cubelike surface shown here is the unit square in the xy-plane. The four sides lie in the planes x = 0, x = 1, y = 0, and y = 1. The top is an arbitrary smooth surface whose identity is unknown. Let F = x i - 2y j + (z + 3)k, and suppose the outward flux of F through Side A is 1 and through Side B is -3. Can you conclude anything about the outward flux through the top? Give reasons for your answer.arrow_forwardLine integrals of vector fields in the plane Given the followingvector fields and oriented curves C, evaluate ∫C F ⋅ T ds. F = ⟨-y, x⟩ on the semicircle r(t) = ⟨4 cos t, 4 sin t⟩ , for 0 ≤ t ≤ πarrow_forwardEvaluate the surface integral SS F * dS for the given vector field F and the oriented surface S. For closed surfaces, use the positive (outward) orientation.F(x,y,z) = -xi - yj +z^3k. S is the part if the cone z = (x^2+y^2)^1/2 between the planes z = 1 and z =3 with a downward orientation.I attempted to solve this with an (r, theta) parametrization. I'm wondering if I did this properly, or where I went wrong since the final answer differs from the solved solutions on this website ( https://www.bartleby.com/solution-answer/chapter-137-problem-24e-essential-calculus-early-transcendentals-2nd-edition/9781337692991/evaluate-the-surface-integral-s-f-ds-for-the-given-vector-field-f-and-the-oriented-surface-s-in/09e1d51f-05ec-4023-aa9e-a016378d0dc7 ).arrow_forward
- Flux integrals Compute the outward flux of the following vector fields across the given surfaces S. You should decide which integral of the Divergence Theorem to use. F = ⟨x2ey cos z, -4xey cos z, 2xey sin z⟩ ; S is the boundary of theellipsoid x2/4 + y2 + z2 = 1.arrow_forwardSurface integrals of vector fields Find the flux of the following vector field across the given surface with the specified orientation. You may use either an explicit or a parametric description of the surface. F = ⟨e-y, 2z, xy⟩ across the curved sides of the surfaceS = {(x, y, z): z = cos y, | y | ≤ π, 0 ≤ x ≤ 4}; normal vectorspoint upward.arrow_forwardThis is a two part problem. Compute the flux of the vector field F = 2x^2y^2(k) through the surface S, which is the cone (sqrt(x^2 + y^2) = z), with z between 0 and R, oriented downward. A. Parameterize the cone using cylindrical coordinates. B. Find the flux of F through S.arrow_forward
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