Exercises 4.6 291 Exercises 7 to 10: Find an example of a vector field (write down a formula, or make a sketch) that satisfies the following requirements. 7. curl F = 0 and div F = 0 %3D 8. curl F + 0 and div F = 0 9. curl F = 0 and div F 0 10. curl F + 0 and div F # 0 11. Sketch a vector field in R² whose divergence is positive at all points. 12. Sketch a vector field in R² whose divergence is zero at all points. Exercises 13 to 16: Find the curl and divergence of the vector field F. 13. F(x, y, z) = y²zi – xzj + xyzk 15. F(x, y, z) = (x² + y² + z²)(3i +j- k) %3D 14. F(x, y, z) = (In z + xy)k %3D 16. F(x, y, z) = e*Yi+ ev*j+ etk %3D %3D 17. Let F(x, y) = (f(x), 0), where f(x) is a differentiable function of one variable. Show that the total outflow from a rectangle R with sides Ax and Ay placed in the flow (as in Figure 4.31) is given by O(AI) = (f(x + Ax) – f(x))A1AY. Conclude that the calculation with the vector field F(x, y) = (0, g(y)) (g is a differentiable function of one variable) and show that the total outflow is again approximately equal to divF. %3D O(AI) AxAy At ~ f'(x) = divF. Repeat %3D 18. What are the flow lines of the vector field F(x, y) = (-x, -y)? Determine geometrically the sign of its divergence. 19. It can be easily checked that curl r = 0, where r = xi + yj + zk. Interpret this result physically, by visualizing r as the velocity vector field of a fluid. %3D 20. Consider the vector fields F = -yi + xj, G = F//x² + y², and H = F/(x? + y²). Compare their divergences and curls. Show that circles centered at the origin are the flow lines for all three vector fields. Describe their differences in physical terms. %3D Exercises 21 to 25: It will be shown in the next chapter that a vector field F defined on all of R' (or all of R²) is conservative if and only if curl F = 0. Determine whether the vector fieldF is conservative or not. If it is, find its potential function (i.e., find a real-valued function V such that F= -grad V). %3D 21. F(x, y, z) = cos yi+ sin xj + tan zk FL Y. Z) = - y²zi+ (3y²/2 – 2xyz)j – xy²k 22 Fr, Y. 7) = xi + y²j+ zk 24