Exercise 3. Consider the torus T given by the parametrization f : R2 + TCR3,...
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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 be a 1-form on T defined by 0 = zdx +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 be a 1-form on T defined by 0 = zdx +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) ?)
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