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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2014 • Question 3

Question

Math 6510 • Differentiable Manifolds Spring 2014 • Question 3

Let \(S^2=\{(x,y,z):x^2+y^2+z^2=1\}\) be the standard 2-sphere in \(\mathbb {R}^3\) and \(f:S^2\to \mathbb {R}\) the restriction of the function \((x,y,z)\mapsto x^2+2y^2+3z^2\) to \(S^2\). Find the critical points of \(f\) and show that they are non-degenerate. (Recall that a critical point is non-degenerate if in some coordinate chart the determinant of the Hessian is non-zero.)

Raw LaTeX

Let \(S^2=\{(x,y,z):x^2+y^2+z^2=1\}\) be the standard 2-sphere in \(\R^3\) and
\(f:S^2\to\R\) the restriction of the function \((x,y,z)\mapsto x^2+2y^2+3z^2\) to
\(S^2\). Find the critical points of \(f\) and show that they are non-degenerate.
(Recall that a critical point is non-degenerate if in some coordinate chart the
determinant of the Hessian is non-zero.)

Topic Tags

Morse Theory and Critical Points