University of Toronto at Scarborough Department of Computer and Mathematical Sciences MAT B42 Winter 2016
Assignment # 2
You are expected to work on this assignment prior to your tutorial in the week of January 18th, 2016. You may ask questions about this assignment in this tutorial. At the beginning of your tutorial in the week of January 25th, 2016 you will be asked to hand in the assignment. NO ASSIGNMENTS will be accepted at the end of the tutorial or after the tutorial. Assignments MUST BE submitted ONLY in the tutorial you are registered for.
A. Readings: 1. Lecture notes: LW2 2. Readings: Marsden & Tromba, Vector Calculus, 6e. Ch 4.3, Ch 7.1- 7.2. 3. Vector Fields (Notes).
B. Problems: 1. Sketch the vector fields
a) ! F(x, y) = ! 1 2 y ! i + x ! j
b) ! F(x, y) = (x, x 2 )
Hint: Plot vectors at several key points along y = ±x and scale the vector field.
2. Show that the curve ! c(t) = (t 2 , 2t !1, t),t > 0 is a flow line of the velocity vector field ! F(x, y,z) = (y +1, 2, 1 2z )
3. Find and draw the flow lines of the velocity vector field ! F(x, y) = (!2y, 1 2 x) Hint: Write the Cartesian equation of the family of flow lines, eliminating the parameter.
4. Determine whether or not each of the given vector fields is conservative. If the vector field is conservative, find a potential function for the field. a) ! F(x, y) = (x 2 + y 2 ) ! i ! 2x ! j b) ! F(x, y) = (2x + ycos xy, x cos xy !1)
5. The conservative field is also called the gradient field. Prove that direction!f is the direction in which f is increasing the fastest.
6. Find the mass of the thin wire if its linear density is given by f. a) The wire has the form of a semicircle ! c(t) = (a, asint, acost), a > 0, 0 ! t ! !, f (x, y,z) = xy b) The wire is the parabola y = x 2 ,z = 0; from A(0, 0, 0), to B(1,1, 0), f (x, y,z) = x + y + z c) The wire has been bent in the shape of the polar graph r =!, 0 !! ! ” 2 And the density at the point (!,! ) is 2!
7. Find the work done by the force field ! F(x, y,z) = (x, y) when a particle is moved along the straight-line segment from (0,0,1) to (3,1,1).
8. Evaluate each of the following Integrals a) x dy ! ydx, ! c(t) = (cost,sint), 0 ” t ” 2! C # b) ! F d ! s, C ! ! F(x, y,z) = y ! i + 2x ! j + y ! k, ! c(t) = t ! i + t 2 ! j + t 3 ! k, 0 ” t “1