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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2014 • Question 2

Question

Math 6510 • Differentiable Manifolds Fall 2014 • Question 2

Define \(\phi :\mathbb {R}^2\to \mathbb {R}^2\backslash \{0\}\) by \(\phi (x,y)=(e^x\cos y,e^x\sin y)\). Let \(\omega \) be a 1-form on \(\mathbb {R}^2\backslash \{0\}\) such that \(dy=\phi ^*\omega \). Show that \(\omega \) is closed but not exact on \(\mathbb {R}^2\backslash \{0\}\).

Raw LaTeX

Define \(\phi:\R^2\to\R^2\backslash\{0\}\) by
\(\phi(x,y)=(e^x\cos y,e^x\sin y)\). Let \(\omega\) be a 1-form on
\(\R^2\backslash\{0\}\) such that \(dy=\phi^*\omega\). Show that \(\omega\) is closed
but not exact on \(\R^2\backslash\{0\}\).

Topic Tags

Differential Forms and Exterior Calculus de Rham Cohomology