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
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- Subject 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_forwardUsing 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_forwardUsing Stoke’s theorem, evaluate the circulation of the field F( x, y, z )=x ^2i +2xj+ z ^2k around the ellipse 4x^2+y^2=4 in the xy plane, counterclockwise when viewed from abovearrow_forward
- Using 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_forwardGradient fields in ℝ3 Find the vector field F = ∇φ for thefollowing potential functions. φ(x, y, z) = 1/ | r | , where r = ⟨x, y, z⟩arrow_forwardF(x, y, z) = xy2z2i + x2yz2j + x2y2zk (a) Find the curl of the vector field.arrow_forward
- 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_forwardConsider the simple, piecewise, oriented, closed curve C shown in the figure: the figure is in the first attached image Consider the nonconservative vector field given by F (x, y) = (3xy - y2 , y + 2x2) the answers are in the second imagearrow_forwardUse Stoke's theorem to evaluate F.dr for the vector field F(x,y,z) = -3y2i+4zj+6xk; C is the triangle in the plane z=x/2 with vertices (2,0,0) , (0,-2,1) and (0,0,0) with anti-clockwise orientation looking down the positive z-axis. Text book: Early Transcendentals 10th Edition by Howard Anton; Chapter: 15.8Please provide the graph figurearrow_forward
- Flux Consider the vector fields and curve. a. Based on the picture, make a conjecture about whether the outwardflux of F across C is positive, negative, or zero.b. Compute the flux for the vector fields and curves. F and C givenarrow_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_forwardStreamlines and equipotential lines Assume that on ℝ2, the vectorfield F = ⟨ƒ, g⟩ has a potential function φ such that ƒ = φxand g = φy, and it has a stream function ψ such that ƒ = ψy andg = -ψx. Show that the equipotential curves (level curves of φ)and the streamlines (level curves of ψ) are everywhere orthogonal.arrow_forward
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