MATH1025 AF

FW00F

Tutorial 1 - week of Sept. 18

1. Find the polar coordinates of the points whose Cartesian coordinates are:

(a) (21/2, 21/2); (b) (31/2, -1); (c) (0, 2); (d) (-3,-6).

2. Find the Cartesian coordinates of the points whose polar coordinates are:

(a) (1, 3pi/4); (b) (2, pi); (c) (0, pi/3); (d) (5, 7pi/6).

3. Identify the following curves:

(a) r =; (b) r = .

4. Convert the following Cartesian coordinates in 3-dimensions into both cylindrical and spherical polar form:

(a) (1,-1, 21/2); (b) (-2, 0, (12)1/2); (c) (31/2, 1,-2).

5. Convert the following cylindrical polar coordinates into Cartesian form:

(a) (4, 3pi/2,-2); (b) (6, 7pi/6, 3); (c) (1, pi, -5); (d) (0, pi/4, -5).

6. Convert the following spherical polar coordinates into Cartesian form:

(a) (1, pi/3, 4pi/3); (b) (5, pi/2, 0); (c) (2, pi/6, 2pi/3); (d) (8, 0, 5pi/4).

7. Describe the following curves or surfaces given in cylindrical polar coordinates:

(a) (b)

(c) z = 2r; (d)

8. Describe the following curves or surfaces given in spherical polar coordinates:

(a) (b)

(c) (d)

9. If z1 = 2 - 2i and z2 = 31/2 + i find z1z2, z12 and z1/z2.

10. Convert the complex numbers z1 and z2 given in question 9 to polar form and evaluate the expressions given there.

11. Use Demoivre's formula to find an expression for in terms of .

12. Find the cube roots of 1 + i.