Question 31 and 34

2/files/folder/Homeworks?preview=39002050 en X 840b2c..: "Towar... 12. F = ( x + y, > > 13. F = (x,y - x) 28. (x, y) = 2xy, for | x| $ 2, ly| $ 2 14. F - Vety Vety 15. F = (e-, 0) 29-36. Gradient fields Find the gradient field F = Vo for the follow- ing potential functions p. e 16. Matching vector fields with graphs Match vector fields a-d with 29. 4(x, y) = x2y - yzx graphs A-D. a. F = (0,.x2 ) 30. 4(x, y) = Vxy b. F = ( x - y, x) C. F = (2x, -y ) d. F = (y.x ) 31. p(x, y) = x/y 32. 4(x, y) - tan" (y/x) 33. 4(x, y, z ) = (x] + 12 + 27) /2 34. 4(x, y, z) = In (1 + x2 + y2 + z) 35. 4(x, y, 2) = (x] + y) + 2) -1/2 36. 4 (x, y, z ) - e- sin (x + y) 37-40. Equipotential curves Consider the following potential func- tions and graphs of their equipotential curves. a. Find the associated gradient field F = Vo. (B) b. Show that the vector field is orthogonal to the equipotential curve at the point (1, 1). Illustrate this result on the figure. c. Show that the vector field is orthogonal to the equipotential curve at all points (x, y). d. Sketch two flow curves representing F that are everywhere orthogo- nal to the equipotential curves. 37. 4(x, y) - 2x + 3y 38. 4(x, y) = x ty 17-20. Normal and tangential components Determine the points (if any) on the curve C at which the vector field F is tangent to C and nor- mal to C. Sketch C and a few representative vectors of F. 17. F = (x,y), where C = { (x, y): x2 + y? = 4 } 18. F = (y, -x), where C = { (x, y): x2 + y? = 1} 19. F = (x, y), where C = { (x, y): x = 1} 39. 4 ( x, y ) = exx 40. 4(x, y) = x2 + 2y? 20. F = (y, x), where C = { (x, y): x2 + y? = 1} 21-24. Three-dimensional vector fields Sketch a few representative vectors of the following vector fields. 21. F = (1, 0, z) 22. F = (x, y, z) 23. F = (y. -x, 0) 20 ( x, y. z ) 24. F =- V.x2+ 32 + 2 20