Let : [0, 1] R 2 \ {0} be a closed path with winding number k. Determine the winding number of the following paths : [0, 1] R 2 \ {0} by using the formula w(, 0) = Z y x 2 + y 2 dx + x x 2 + y 2 dy. Then explain your answer by appealing to geometric intuition. (i) (t) = (1 t) (ii) (t) = k(t)k 1(t) (iii) (t) = ((t)) where (x, y) = (y, x) (iv) (t) = ((t)) where (x, y) = (y, x)

6. Let y: [0, 1] + R2 \ {0} be a closed path with winding number k. Determine the winding number of the following paths : [0, 1] + R2 \ {0} by using the formula -Y dx + dy. x2 + y2 x2 + y2 Then explain your answer by appealing to geometric intuition. 1 (i) (t) = Y(1 t) (ii) (t) = |1y(t)||-1y(t) (iii) (t) = $((t)) where $(x, y) = (y,x) (iv) f(t) = $((t)) where o(2, 3) =(-, 2) 6. Let y: [0, 1] + R2 \ {0} be a closed path with winding number k. Determine the winding number of the following paths : [0, 1] + R2 \ {0} by using the formula -Y dx + dy. x2 + y2 x2 + y2 Then explain your answer by appealing to geometric intuition. 1 (i) (t) = Y(1 t) (ii) (t) = |1y(t)||-1y(t) (iii) (t) = $((t)) where $(x, y) = (y,x) (iv) f(t) = $((t)) where o(2, 3) =(-, 2)