MATH1025 AF


FW00F



Answers to Tutorial 1



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

(a) ; (b) ; (c) (0, 2); (d) (-3, -6).

Ans. (a); (b); (c); (d)where the angle must be chosen in the third quadrant.


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

(a) ; (b) ; (c) ; (d) .

Ans. (a); (b) (-2, 0); (c) (0, 0); (d)


3. Identify the following curves:

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

Ans.

(a) Converting to Cartesian coordinates and completing the square yields

(x-1)2 + (y-1)2 = 2 which is a circle with centre (1, 1) and radius.

(b) This is a spiral (plot it out for angles starting at 0 and increasing indefinitely).


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

(a) ; (b) ; (c) .

Ans. (a) cylindrical , spherical;

(b) cylindrical, spherical;

(c) cylindrical, spherical


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

(a) ; (b) ; (c) ; (d) .

Ans. (a) (0, -4, -2); (b); (c) (-1, 0, -5); (d) (0, 0, -5)




6. Convert the following spherical polar coordinates into Cartesian form (NOTE CHANGES TO PARTS (a) and (d)):

(a) ; (b) ; (c) ; (d) .

Ans. (a) ; (b) (0, 0, 5); (c); (d).



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

(a) (b)

(c) z = 2r; (d)

Ans. (a) ray starting from z-axis parallel to x-y plane;

(b) changing to Cartesian coordinates gives (x - 0.5)2 + y2 = 0.25 which is a circular cylinder of radius 0.5 with axis parallel to the z-axis and passing through (0.5, 0, 0);

(c) changing to spherical coordinates yields or which is a cone with vertex at the origin and axis the z-axis;

(d) this is a three-dimensional spiral on the surface of the circular cylinder r = 2.



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

(a) (b)

(c) (d)

Ans. (a) the x-y plane;

(b) a circle, radiuswith centre on the z-axis and parallel to the x-y plane

(c) multiplying by and converting to Cartesians gives x2 + y2 + (z - 0.5)2 = 0.25 which is a sphere with centre (0, 0, 0.5) and radius 0.5.

(d) if we change to cylindrical polar coordinates we have so that this equation becomes which is the same as question 7(b), i.e. a cylinder.



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

Ans. z1z2 =; z12 = -8i; .



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

Ans.


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

Ans.



12. Find the cube roots of 1 + i.

Ans.