Evaluating a Line
In Exercises 29-34, evaluate
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Calculus: Early Transcendental Functions
- Using Gauss' theorem to calculate the flow of the vector field 3x3 F: F (x, y, z) = (x^2z, 2x^2, 3z^2) exiting the cylinder defined from the relations x ^2+y ^2<=1, 1<= z <= 2.arrow_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_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
- The 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_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 line segment from (1, 1) to (5, 10)arrow_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
- Rain 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_forwardUsing 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_forwarduse Green’s Theorem to find the counterclock-wise circulation and outward flux for the field F and curve C. F = (x + y)i - (x2 + y2 )j C: The triangle bounded by y = 0, x = 1, and y = xarrow_forward
- use Green’s Theorem to find the counterclock-wise circulation and outward flux for the field F and curve C. F = (y2 - x2 )i + (x2 + y2 )j C: The triangle bounded by y = 0, x = 3, and y =x.arrow_forwardSurface integral of a vector field? Let T be the upper surface of the tetrahedron bounded by the coordinate planes and the plane x + y + z = 4. Calculate the integral of the image below, where S is the face of T that is in the xy plane.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
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning