1.Draw these vector fields by a computing software. (Hint: if you use matlab you can use meshgrid and quiver functions)
V=yi, v=yi-xj
2. Find a parametric representation r(t) of the straight line through points P = (0, 3,−5) and Q=(−7, -2, 1). Orient the line so that it goes from P to Q with increasing t. It should lie on point P at t = 0.
3. Let curve C be described parametrically by r(t) =[3 cos(πt) 3t + 1 2t2 + 1 ] (a) Find an expression for the tangent vector.
(b) Find an expression for the line tangent to C at point r(2).
4. Ten turns of copper wire are to be wrapped helically around a cylinder of length 20mm and
radius 2.5mm. The turns are to be wrapped evenly from one end of the cylinder to the other.
(a) Assume that the axis of the cylinder is on the +z-axis with the bottom at z = 0 and the top at z= 20. Find a parameterization r(t) =[x(t) y(t) z(t)] that describes the path taken by the copper wire.
Hints: (1) You will need the parameterization formula for a helix discussed in class. (2) Since you want ten turns of wire, it is convenient to let the range of parameter t be 0 ≤ t ≤ 20π radians, which is ten complete 2π-radian circles.
(b) Use the arc length formula from section 9.5 along with your answer to part (a) to calculate the length of copper wire that will be required.
5. An object has velocity vector v(t) that is described parametrically by v(t) =[3t t2 t3-t] (a) Calculate the acceleration a(t) at time t = 2.
(b) Calculate the tangential (to the velocity) acceleration vector a∥ at time t = 2.
(c) Calculate the normal (to the velocity) acceleration vector a⊥ at time t = 2.
0 comments