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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1996 • Question 3

Question

Math 6510 • Differentiable Manifolds Fall 1996 • Question 3

Let \(F:\mathbb {R}^2\to \mathbb {R}^3\) be a mapping given by the formula \[ F(u,v)=(e^u\cos (v),e^u\sin (v),0). \] Compute the pull-back \(F^*(\omega )\) of the differential form \[ \omega =xy\,dx\wedge dy+x\,dy\wedge dz\in \Omega ^*(\mathbb {R}^3). \]

Raw LaTeX

Let \(F:\mathbb{R}^2\to\mathbb{R}^3\) be a mapping given by the formula
\[
  F(u,v)=(e^u\cos(v),e^u\sin(v),0).
\]
Compute the pull-back \(F^*(\omega)\) of the differential form
\[
  \omega=xy\,dx\wedge dy+x\,dy\wedge dz\in\Omega^*(\mathbb{R}^3).
\]

Topic Tags

Differential Forms and Exterior Calculus Pullbacks and Wedge Products Tangent Spaces and Differential Maps