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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2013 • Question 4

Question

Math 6510 • Differentiable Manifolds Spring 2013 • Question 4

For \((x,y)\in \mathbb {R}^2-\{(0,0)\}\), let \[ \omega (x,y)=\frac {-y}{x^2+y^2}\,dx+\frac {x}{x^2+y^2}\,dy. \] Find \(\int _{C(r)}\omega \) where \(C(r)\) is the circle of vectors in \(\mathbb {R}^2\) of norm equal to \(r>0\) oriented counterclockwise. Find a smooth function \(f:\mathbb {R}^2-\{(0,0)\}\to \mathbb {R}\) such that \(\omega =df\) or explain why there is no such \(f\).

Raw LaTeX

For \((x,y)\in\R^2-\{(0,0)\}\), let
\[
  \omega(x,y)=\frac{-y}{x^2+y^2}\,dx+\frac{x}{x^2+y^2}\,dy.
\]
Find \(\int_{C(r)}\omega\) where \(C(r)\) is the circle of vectors in \(\R^2\) of norm
equal to \(r>0\) oriented counterclockwise. Find a smooth function
\(f:\R^2-\{(0,0)\}\to\R\) such that \(\omega=df\) or explain why there is no such
\(f\).

Topic Tags

Differential Forms and Exterior Calculus Orientability and Volume Forms