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- Zero divergence of the rotation field Show that the generalrotation field F = a x r, where a is a nonzero constant vectorand r = ⟨x, y, z⟩ , has zero divergence.A two-dimensional vector field describes ideal flow if it has both zero curl and zero divergence on a simply connected region.a. Verify that both the curl and the divergence of the given field are zero.b. Find a potential function φ and a stream function ψ for the field.c. Verify that φ and ψ satisfy Laplace’s equationφxx + φyy = ψxx + ψyy = 0. F = ⟨x3 - 3xy2, y3 - 3x2y⟩A two-dimensional vector field describes ideal flow if it has both zero curl and zero divergence on a simply connected region.a. Verify that both the curl and the divergence of the given field are zero.b. Find a potential function φ and a stream function ψ for the field.c. Verify that φ and ψ satisfy Laplace’s equationφxx + φyy = ψxx + ψyy = 0. F = ⟨ex cos y, -ex sin y⟩
- Computing the divergence Compute the divergence of the followingvector fields.a. F = ⟨x, y, z⟩ (a radial field)b. F = ⟨ -y, x - z, y⟩ (a rotation field)c. F = ⟨ -y, x, z⟩ (a spiral flow)Stream function Recall that if the vector field F = ⟨ƒ, g⟩ is source free (zero divergence), then a stream function ψ exists such that ƒ = ψy and g = -ψx.a. Verify that the given vector field has zero divergence.b. Integrate the relations ƒ = ψy and g = -ψx to find a stream function for the field. F = ⟨-e-x sin y, e-x cos y⟩A two-dimensional vector field describes ideal flow if it has both zero curl and zero divergence on a simply connected region.a. Verify that both the curl and the divergence of the given field are zero.b. Find a potential function φ and a stream function ψ for the field.c. Verify that φ and ψ satisfy Laplace’s equationφxx + φyy = ψxx + ψyy = 0.
- vector field F = ⟨b cos(ab) + 2a + b, a cos(ab) + 2b + a⟩. State if it is conservative with reasons.Flux integrals Assume the vector field F = ⟨ƒ, g⟩ is source free (zero divergence) with stream function ψ. Let C be any smooth simple curve from A to the distinct point B. Show that the flux integral ∫C F ⋅ n ds is independent of path; that is, ∫C F ⋅ n ds = ψ(B) - ψ(A).Conservative fields Use Stokes’ Theorem to find the circulationof the vector field F = ∇(10 - x2 + y2 + z2) around anysmooth closed curve C with counterclockwise orientation.
- Inverse square fields are special Let F be a radial field F = r/ | r | p, where p is a real number and r = ⟨x, y, z⟩. With p = 3, F is an inverse square field.a. Show that the net flux across a sphere centered at the origin isindependent of the radius of the sphere only for p = 3.b. Explain the observation in part (a) by finding the fluxof F = r/ | r | p across the boundaries of a spherical box{(ρ, φ, θ): a ≤ ρ ≤ b, φ1 ≤ φ ≤ φ2, θ1 ≤ θ ≤ θ26 forvarious values of p.Divergence from a graph To gain some intuition about the divergence,consider the two-dimensional vector field F = ⟨ƒ, g⟩ = ⟨x2, y⟩ and a circle C of radius 2 centered at the origin (see figure).a. Without computing it, determine whether the two-dimensional divergence is positive or negative at the point Q(1, 1). Why?b. Confirm your conjecture in part (a) by computing the two-dimensional divergence at Q. c. Based on part (b), over what regions within the circle is the divergence positive and over what regions within the circle is the divergence negative?d. By inspection of the figure, on what part of the circle is the flux across the boundary outward? Is the net flux out of the circle positive or negative?Stream function Recall that if the vector field F = ⟨ƒ, g⟩ is source free (zero divergence), then a stream function ψ exists such that ƒ = ψy and g = -ψx.a. Verify that the given vector field has zero divergence.b. Integrate the relations ƒ = ψy and g = -ψx to find a stream function for the field. F = ⟨y2, x2⟩
