Consider the one-form 1 (-y dx + x dy) w = (x² + y²)ª on U = R² \ {(0,0)} , and the smooth function o : V → U, with V = {(r, t) E R² |r > 0}, given by $(r, t) = (r cos(t), r sin(t)). Find the pullback one-form ø* w on V. d*w =| dr + dt Note that o defines the well known change of coordinates from Cartesian to polar coordinates on U, where U is the xy-plane with the origin removed. We are using the convention here that the angle t can take any values, so $(r, t) = ¢(r,t +2rn) for any integer n E Z. In other words, our map ø is not injective .

Consider the one-form 1 (-y dx + x dy) w = (x² + y²)ª on U = R² \ {(0,0)} , and the smooth function o : V → U, with V = {(r, t) E R² |r > 0}, given by $(r, t) = (r cos(t), r sin(t)). Find the pullback one-form ø* w on V. d*w =| dr + dt Note that o defines the well known change of coordinates from Cartesian to polar coordinates on U, where U is the xy-plane with the origin removed. We are using the convention here that the angle t can take any values, so $(r, t) = ¢(r,t +2rn) for any integer n E Z. In other words, our map ø is not injective .

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter9: Multivariable Calculus

Section9.CR: Chapter 9 Review

Problem 4CR

See similar textbooks