Exercise 3. Consider the torus T given by the parametrization f: R2 + TCR3, f(t, s) = ((4 + cos(t)) cos(s), (4 + cos(t)) sin(s), (sin(t))) Let o be a 1-form on T defined by o = zd.+ ydz. (This is a 1-form on R3 that we restrict to T.) 1. What is the 1-form f*o on R? 2. Consider the 2-form do on T. What is its pull-back f* (do)? (Hint: you don't need to use charts!) 3. What is f* (do) (1, 0) (0,0), (0, 1)(0,0)) ? 4. What is do(0,0,1)(5.0.0), (0,1,0)(5,0,0)) ? (Hint: you don't need to use the definition of exterior derivative. Just use the definition of pull-back! What is df (1, 0) (0,0) ?) Exercise 3. Consider the torus T given by the parametrization f: R2 + TCR3, f(t, s) = ((4 + cos(t)) cos(s), (4 + cos(t)) sin(s), (sin(t))) Let o be a 1-form on T defined by o = zd.+ ydz. (This is a 1-form on R3 that we restrict to T.) 1. What is the 1-form f*o on R? 2. Consider the 2-form do on T. What is its pull-back f* (do)? (Hint: you don't need to use charts!) 3. What is f* (do) (1, 0) (0,0), (0, 1)(0,0)) ? 4. What is do(0,0,1)(5.0.0), (0,1,0)(5,0,0)) ? (Hint: you don't need to use the definition of exterior derivative. Just use the definition of pull-back! What is df (1, 0) (0,0) ?)