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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2005 • Question 8

Question

Math 6510 • Differentiable Manifolds Winter 2005 • Question 8

Let \(f:\mathbb {R}^n\to \mathbb {R}\) be a smooth function. Show that for almost every \(a\in \mathbb {R}^n\) the function \[ f_a:\mathbb {R}^n\to \mathbb {R} \] defined by \[ f_a(x)=f(x)+x\cdot a \] is Morse.

Raw LaTeX

Let \(f:\mathbb{R}^n\to\mathbb{R}\) be a smooth function. Show that for almost
every \(a\in\mathbb{R}^n\) the function
\[
  f_a:\mathbb{R}^n\to\mathbb{R}
\]
defined by
\[
  f_a(x)=f(x)+x\cdot a
\]
is Morse.

Topic Tags

Morse Theory and Critical Points