5. (a) Show that every curve a in R that does not pass through the origin has an (orientation-preserving) reparametrization in the polar form a(t) = (r(t) cos Oft), r(t) sin v(t)). (Hint: Use Ex. 12 of Sec. 2.1.) If the curve a: [a, b] R? 0 is closed, prove: (b) wind(a) = 0(b) 0(a) is an integer. 21 This integer, called the winding number of a about 0, represents the total algebraic number of times a has gone around the origin in the counter- clockwise direction. (Note that wind(a) = wind(0)||0|l).) (c) If y is the 1-form in Exercise 4, then wind(a) = 21 (d) If a = (f, g), then fg'- gf' windla) dt f? + g? 21 Ja alt). alt) 1 det(a(t), o't) dt. = 21 (The determinant is of the 2x2 matrix whose rows are alt) and a'(t).) 5. (a) Show that every curve a in R that does not pass through the origin has an (orientation-preserving) reparametrization in the polar form a(t) = (r(t) cos Oft), r(t) sin v(t)). (Hint: Use Ex. 12 of Sec. 2.1.) If the curve a: [a, b] R? 0 is closed, prove: (b) wind(a) = 0(b) 0(a) is an integer. 21 This integer, called the winding number of a about 0, represents the total algebraic number of times a has gone around the origin in the counter- clockwise direction. (Note that wind(a) = wind(0)||0|l).) (c) If y is the 1-form in Exercise 4, then wind(a) = 21 (d) If a = (f, g), then fg'- gf' windla) dt f? + g? 21 Ja alt). alt) 1 det(a(t), o't) dt. = 21 (The determinant is of the 2x2 matrix whose rows are alt) and a'(t).)