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- Sketch the vector field represented by F (x, y) = (-x,y)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.)Let F and G be vector fields with differentiable components. Express curl (F x G) in term of div and dot products.
- Prove that the vector given by (sin y sinh x + cos y cosh x) i + (cos y cosh x – sin y sinh x) j is a gradient. Also, find the function having the given vector as its gradient.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?Find the work done by the vector field (5x + yx, x² + 3 ) or the boundary of the rectangle 0 < x< 3,0 < y < 6|in the counterclockwise direction. a particle moving along (The force is measured in newtons, length in meters, work in joules=(newton-meters).) W = joules