Evaluating a Line
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Calculus
- a. Show that the outward flux of the position vector field F = x i + y j + z k through a smooth closed surface S is three times the volume of the region enclosed by the surface. b. Let n be the outward unit normal vector field on S. Show that it is not possible for F to be orthogonal to n at every point of Sarrow_forwardProve that the vector field F(x, y, z) = (x2 + yz)i − 2y(x + z)j + (xy + z2)k isincompressible, and find its vector potential function.arrow_forwarda) Find the work done by the force field F on a particle that moves along the curve C. F(x, y) = (x2 + xy)i + (y – x2 y)j C : x = t, y = 1/t (1 ≤ t ≤ 3)arrow_forward
- Using Green's Theorem, find the outward flux of F across the closed curve C.F = xy i + x j; C is the triangle with vertices at (0, 0), (4, 0), and (0, 2)arrow_forwardRain on a roof Consider the vertical vector field F = ⟨0, 0, -1⟩, correspondingto a constant downward flow. Find the flux in the downward direction acrossthe surface S, which is the plane z = 4 - 2x - y in the first octant.arrow_forwardThe figure shows a vector field F and two curves C_1 and C_2. Are the line integrals of F over C_1 and C_2 positive, negative, or zero? Explain.arrow_forward
- Line 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_forwardFind the flux of the vector field F across the surface S in the indicated direction.F = x 2y i - z k; S is portion of the cone z = 4 square root of x^2+y^2 between z = 0 and z = 1; direction is outward a)-1/24 pi b)-1/8 pi c)1/24 pi d)-1/48 piarrow_forward(a) Show that any vector field of the form h(x, Y, z) = f(x)i+g(y)j+h(z)k, where f, g, h are differentiable functions, is irrotational. (b) Determine whether there is a vector field g such that V x g = xi+yj+zk.arrow_forward
- Line 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 line segment from (1, 1) to (5, 10)arrow_forwardFind the flux of the vector field F across the surface S in the indicated direction.F = x 2y i - z k; S is portion of the cone z = 4 square root of x^2+y^2 between z = 0 and z = 1; direction is outward a)-1/24pi b)-1/8pi c)1/24pi d)-1/48piarrow_forwarduse Green’s Theorem to find the counterclock-wise circulation and outward flux for the field F and curve C. F = (x2 + 4y)i + (x + y2 )j C: The square bounded by x = 0, x = 1, y = 0, y = 1arrow_forward
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