Divergence and Curl In Exercises 19-26, find (a) the divergence of the
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Calculus (MindTap Course List)
- (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_forwardSplitting a vector field Express the vector field F = ⟨xy, 0, 0⟩in the form V + W, where ∇ ⋅ V = 0 and ∇ x W = 0.arrow_forwarduse 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_forward
- use 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_forwardSubject differential geometry Let X(u,v)=(vcosu,vsinu,u) be the coordinate patch of a surface of M. A) find a normal and tangent vector field of M on patch X B) q=(1,0,1) is the point on this patch?why? C) find the tangent plane of the TpM at the point p=(0,0,0) of Marrow_forwardSketching vector fields Sketch the following vector field. F = ⟨x, 0⟩arrow_forward
- F(x, y, z) = xy2z2i + x2yz2j + x2y2zk (a) Find the curl of the vector field.arrow_forwardOutward flux of a radial field Use Green’s Theorem to compute the outward flux of the radial field F = ⟨x, y⟩ across the unit circle C = {(x, y2: x2 + y2 = 1} (see figure). Interpret the result.arrow_forwardFlux of a vector field? Let S be a closed surface consisting of a paraboloid (z = x²+y²), with (0≤z≤1), and capped by the disc (x²+y² ≤1) on the plane (z=1). Determine the flow of the vector field F (x,y,z) = zj − yk, in the direction that points out across the surface S.arrow_forward
- Testing for conservative fields Determine whether the following vectorfields are conservative on ℝ2 and ℝ3, respectively.a. F = ⟨ex cos y, -ex sin y⟩ b. F = ⟨2xy - z2, x2 + 2z, 2y - 2xz⟩arrow_forwardTesting for conservative vector fields Determine whether thefollowing vector field is conservative (in ℝ2 or ℝ3). F = ⟨e-x cos y, e-x sin y⟩arrow_forwardUsing Green's Theorem on this vector field problem, compute a) the circulation on the boundary of R in terms of a and b, and b) the outward flux across the boundary of R in terms of a and b.arrow_forward
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