Using arrows, clearly sketch the vector field u(r) = (x + yz, xz+y, xyz) in the (x, y)-plane.
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- A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by v=(x-y,z+y+7,z2) and the net is decribed by the equation y=1-x2-z2, y20, and oriented in the positive y- direction. (Use symbolic notation and fractions where needed.)(b) Using arrows, clearly sketch the vector field u(r) = (y², x + y, 0) in the (x, y)-plane.Sketch some vectors in the vector field F(x, y) = −yi + xj.
- A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by v = (x-y, z + y + 3, z²) and the net is decribed by the equation y = √1-x²-2², y ≥ 0, and oriented in the positive y- direction. (Use symbolic notation and fractions where needed.)Sketch the vector field F = ⟨x, y⟩ .Calculate the curl of the vector g = (x−y) ax+(y+x) ay, and visualize it. Explain what does "curl of vector field" mean, in your own words. When does it equal zero?
- The figure shows the vector field F (x,y) = {2xy,x2 } and three curves that start at (1,2) and end at (3,2). (a) Explain why∫c F. dr has the same value for all the three curves. (b) What is this common value?Sketch some vectors in the vector field F(x, y) = 2xi + yj.Show that V² (v-x) = 2V.v+x-V²v, where v is a vector field and x is the position vector of a point in a 3D space.
- Suppose that r1(t) and r2(t) are vector-valued functions in 2-space. Explain why solving the equation r1(t)=r2(t) may not produce all the points where the graphs of these functions intersect. Please Provide Unique Answer. Thank you!Sketch the vector field F(x, y) = (0, -y) Sketch the vector field F(x, y) = (-y, x).Please solve (b) by finding a vector valued function f(t)=(x(t), y(t)) whose image set is the one given