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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2006 • Question 8

Question

Math 6510 • Differentiable Manifolds Fall 2006 • Question 8

Verify the formula \[ d\omega (X,Y)=X(\omega (Y))-Y(\omega (X))-\omega ([X,Y]) \] for any 1-form \(\omega \) on \(\mathbb {R}^n\) and any two vector fields \(X,Y\) on \(\mathbb {R}^n\). This is using Spivak’s normalization conventions \[ dx\wedge dy\left (\frac {\partial }{\partial x},\frac {\partial }{\partial y}\right )=1; \] using Guillemin-Pollack’s one should multiply the left-hand side by 2.

Raw LaTeX

Verify the formula
\[
  d\omega(X,Y)=X(\omega(Y))-Y(\omega(X))-\omega([X,Y])
\]
for any 1-form \(\omega\) on \(\mathbb{R}^n\) and any two vector fields \(X,Y\) on
\(\mathbb{R}^n\). This is using Spivak's normalization conventions
\[
  dx\wedge dy\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)=1;
\]
using Guillemin-Pollack's one should multiply the left-hand side by 2.

Topic Tags

Differential Forms and Exterior Calculus Left-Invariant Vector Fields Vector Fields and Flows