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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2024 • Question 3

Question

Math 6510 • Differentiable Manifolds Fall 2024 • Question 3

Let \(\omega =dx\wedge dy\) be the 2-form on \(\mathbb {R}^3\). Let \(h:\mathbb {R}^3\to \mathbb {R}\) be smooth so that 0 is a regular value. Let \(Z=h^{-1}(0)\). Assume that \(\frac {\partial h}{\partial z}\ne 0\) at all points of \(Z\). Show that the restriction of \(\omega \) to \(Z\) is a volume form on \(Z\).

Raw LaTeX

Let \(\omega=dx\wedge dy\) be the 2-form on \(\mathbb{R}^3\). Let \(h:\mathbb{R}^3\to\mathbb{R}\) be smooth so that 0 is a regular
value. Let \(Z=h^{-1}(0)\). Assume that \(\frac{\partial h}{\partial z}\ne 0\) at all points of \(Z\). Show that the restriction
of \(\omega\) to \(Z\) is a volume form on \(Z\).

Topic Tags

Differential Forms and Exterior Calculus Orientability and Volume Forms Regular Values and Level Sets