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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2010 • Question 2

Question

Math 6510 • Differentiable Manifolds Spring 2010 • Question 2

Find the Lie bracket \[ \left [2\frac {\partial }{\partial x}+z\frac {\partial }{\partial y}, x\frac {\partial }{\partial x}+y^2\frac {\partial }{\partial z}\right ]. \] Is the plane field \[ \Delta _{(x,y,z)} =\operatorname {Span}\left \{2\frac {\partial }{\partial x}+z\frac {\partial }{\partial y}, x\frac {\partial }{\partial x}+y^2\frac {\partial }{\partial z}\right \} \] integrable?

Raw LaTeX

Find the Lie bracket
\[
  \left[2\frac{\partial}{\partial x}+z\frac{\partial}{\partial y},
  x\frac{\partial}{\partial x}+y^2\frac{\partial}{\partial z}\right].
\]
Is the plane field
\[
  \Delta_{(x,y,z)}
  =\spanop\left\{2\frac{\partial}{\partial x}+z\frac{\partial}{\partial y},
  x\frac{\partial}{\partial x}+y^2\frac{\partial}{\partial z}\right\}
\]
integrable?

Topic Tags

Distributions, Plane Fields, and Frobenius Integrability Left-Invariant Vector Fields Lie Algebras and Exponential Maps