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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2015 • Question 2

Question

Math 6510 • Differentiable Manifolds Spring 2015 • Question 2

Define \(r:\mathbb {R}^2\backslash \{0\}\to \mathbb {R}^2\) by \[ r(x,y)=\left (\frac {x}{\sqrt {x^2+y^2}},\frac {y}{\sqrt {x^2+y^2}}\right ). \] Calculate \(r^*(-y\,dx+x\,dy)\) and \(r^*(x\,dx+y\,dy)\). Show that for any two form \(\omega \) on \(\mathbb {R}^2\), \(r^*\omega =0\) and conclude that for any 1-form \(\alpha \) on \(\mathbb {R}^2\), \(r^*\alpha \) is closed (even if \(\alpha \) is not).

Raw LaTeX

Define \(r:\R^2\backslash\{0\}\to\R^2\) by
\[
  r(x,y)=\left(\frac{x}{\sqrt{x^2+y^2}},\frac{y}{\sqrt{x^2+y^2}}\right).
\]
Calculate \(r^*(-y\,dx+x\,dy)\) and \(r^*(x\,dx+y\,dy)\). Show that for any two form
\(\omega\) on \(\R^2\), \(r^*\omega=0\) and conclude that for any 1-form \(\alpha\) on
\(\R^2\), \(r^*\alpha\) is closed (even if \(\alpha\) is not).

Topic Tags

Differential Forms and Exterior Calculus