The vector field F is shown in the xy-plane and looks the same in all other horizontal planes (in other words, F is independent of z). УА 0 The curl F is in the direction of i -i -j k -k None of the above x

Sketch the vector field F(x,y) → xi + 3yi - a. C. 'ீ, -10 10 -10 -5- L -10- 10- -10- 10 A R -10 10 10 10 b. d. -10 -10 10- IDA -10- וווו -10 10 10 d ரு VD

you are given a vector field ~F . Use graphical reasoning to decide whether the div ~F (1, 1) is positive, negative, or zero. Justify your answer with a complete sentence explanation.

Let F and G be vector fields with differentiable components. Express curl (F x G) in term of div and dot products.

F = (x² - y)i + + (4z)j + (x²)k Find the curl (curl calculation) of the vector field?

Express (5x + 7y, 8x + 5y, 0) as the sum of a curl free vector field and a divergence free vector field. (5x + 7y, 8x + 5y, 0) = 1 + where the first vector in the sum is curl free and the second is divergence free. (For this problem, enter your vectors with angle-bracket notation: , not in ijk- notation.)

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

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.

The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. (In other words, F is independent of z and its z-component is 0.) (a) Is div(F) positive, negative, or zero at P? Explain. div(F) is ---Select--- because the vectors that start near P are ---Select--- those that end near P. (b) Determine whether curl(F) = 0. If not, in which direction does curl(F) point at P? curl(F) + 0. At P the curl(F) points in the direction of positive x. curl(F) + 0. At P the curl(F) points in the direction of negative x. curl(F) + 0. At P the curl(F) points in the direction of positive y. curl(F) + 0. At P the curl(F) points in the direction of negative y. curl(F) + 0. At P the curl(F) points in the direction of positive z. curl(F) + 0. At P the curl(F) points in the direction of negative z. curl(F) : = 0

Express (5x + 7y, 8x + 5y, 0) as the sum of a curl free vector field and a divergence free vector field. (5x + 7y, 8x + 5y, 0) = where the first vector in the sum is curl free and the second is divergence free. (For this problem, enter your vectors with angle-bracket notation: , not in i jk-notation.) +