~ f'(x) = divF. Repeat that the vector field F(x, y) = (0, g(y)) (g is a differentiable function of one variable) %3D AxAy At and show that the total outflow is again approximately equal to divF. %3D 18. What are the flow lines of the vector field F(x, y) = (-x,-y)? Determine geometrically the sign of its divergence. %3D 19. It can be easily checked that curl r = 0, where r = xi + vj + zk. Interpret this result physically, by visualizing r as the velocity vector field of a fluid. %3D r%3D Consider the vector fields F = –yi + xj, G = F//x² + y², and H = F/(x² + 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. 20. %3D %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 field F 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 %3D 21. F(x, y, z) = cos yi + sin xj+ tan zk 22. F(x, y, z) = -y²zi+ (3y²/2 – 2xyz)j – xy²k 23. F(x, y) = 3x²yi + (x³ + y³)j 24. F(x, y, z) = xi + y²j+ zk %3D 25. F(x, y, z) = -yi – xj – 3k 26. Check whether the vector field F(x, y) = i/(x In xy) + j/(y In xy) is conservative for x, y > 0, and if so, find all functions f such that F = grad f. %3D 27. Verify that curl (grad f) = 0 for the function f(x, y, z) = (x² + y² + z?)-!. %3D 28. Verify that (curl F)1/əx + a(curl F)2/ây + a(curl F)3/dz = 0 for the F(x, y, z) = 3x3 y²i + yx²j – x³z°k, where (curl F),, (curl F)2, and (curl F)3 are the components of curl F. vector field %3D %3D 29. Is there a C² vector field F such that curl F = xy²i+ yz²j+ zx²k? Explain. %3D 30. Is there a C² vector field F such that curl F = 2i + j + 3k? If so, find such a field. 31. A vector field F is irrotational if curl F = 0. Show that any vector field of the form F(x, y, z) = f(x)i+ g(y)j + h(z)k, where f, g, and h are differentiable real-valued functions of one variable, is irrotational. %3D 32. A vector field Fis incompressible if div F = 0. Show that any vector field of the form F(x, v. z) = 6a and h are differentiable real-valued functions of two %3D