Motion of a Liquid In Exercises 17 and 18, the motion of a liquid in a cylindrical container of radius 3 is described by the velocity field F(x, y, z). Find
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- Flux across the boundary of an annulus Find the outward flux of the vector field F = ⟨xy2, x2y⟩ across the boundary of the annulusR = {(x, y): 1 ≤ x2 + y2 ≤ 4}, which, when expressed in polar coordinates, is the set {(r, θ): 1 ≤ r ≤ 2, 0 ≤ θ ≤ 2π}arrow_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_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
- 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_forwardFind the flux of the field F =(x + y)i -(x2+y2)j outward across the triangle with vertices (1, 0), (0, 1), (-1, 0).arrow_forwarda. 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_forward
- Flux across curves in a vector field Consider the vector fieldF = ⟨y, x⟩ shown in the figure.a. Compute the outward flux across the quarter-circleC: r(t) = ⟨2 cos t, 2 sin t⟩ , for 0 ≤ t ≤ π/2.b. Compute the outward flux across the quarter-circleC: r(t) = ⟨2 cos t, 2 sin t⟩ , for π/2 ≤ t ≤ π.c. Explain why the flux across the quarter-circle in the third quadrant equals the flux computed in part (a). d. Explain why the flux across the quarter-circle in the fourth quadrant equals the flux computed in part (b).e. What is the outward flux across the full circle?arrow_forwardcheck the stokes theorem for vactor field A=(x)i+(y)j+(2xy)k where S is the lower hemisphere x2+y2+z2=4 and z<=0arrow_forwarduse Green’s Theorem to find the counterclock-wise circulation and outward flux for the field F and curve C. F = (x - y)i + ( y - x)j C: The square bounded by x = 0, x = 1, y = 0, y = 1arrow_forward
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