2. Consider the vector field F = cos xi + sin xj. (a) Sketch the vector field F = cos xi + sin xj. (b) Sketch the field line passing through (0, 0). (May be on the same graph as in 2a). (c) Calculate the divergence of F (d) Calculate the curl of F (e) Does F represents an incompressible, expanding, contracting or irrotational vector field answer.

only solute question c ， please

12. (a) Consider again the vector field in Exercise 8 and its curl. Sketch the vector field and use your pic- ture to explain geometrically why the curl is as you calculated. (b) Use geometry to determine ▼ × F, where F = (xi+ yj + zk) Vx² + y² + z² (c) For Fas in part (b), verify your intuition by explic- itly computing V × F. 8. F = xi+ yj + zk

Sketch and describe the vector field F (x, y) = (-y,2x)

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--- (b) Determine whether curl(F) = 0. If not, in which direction does curl(F) point at P? O curl(F) + 0. At P the curl(F) points in the direction of positive x. O curl(F) + 0. At P the curl(F) points in the direction of negative x. O curl(F) + 0. At P the curl(F) points in the direction of positive y. O curl(F) + 0. At P the curl(F) points in the direction of negative y. O curl(F) + 0. At P the curl(F) points in the direction of positive z. O curl(F) + 0. At P the curl(F) points in the direction of negative z. O curl(F) = 0 those that end near P.

1# 2: For the vector field F = xî a) Plot the vector field on the xy Cartesian coordinate system. b) Find the divergence of the vector field. c) Find the curl of the vector field.

Cal 3

The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus the vectors in a vector field are tangent to the flow lines. (a) Use a sketch of the vector field F(x, y) = xi − yj to draw some flow lines. From your sketches, can you guess the equations of the flow lines?   (b) If parametric equations of a flow line are x = x(t), y = y(t), explain why these functions satisfy the differential equations dx/dt = x and dy/dt = −y.   (c) Solve the differential equations to find an equation of the flow line that passes through the point (x, y) = (−1, −1).

(a) Calculate the flux of the vector field F(x, y, z) = 275k through a sphere of radlus 4 centered at the origin, oriented outward. Flux = (b) Calculate the flux of the vector field F(x, y, z) = -53 +5 through a cube of side length 4 with sides parallel to the axes, oriented OL Flux =

Find the directional derivative of the scalar field þ= x¹y + 4xz at the point (2, -2, 4) along the direction vector (2, -1, -2):