3

Let $f$ be a smooth function on an open set $U\subset R^k$. For each $x \in U$ let $H(x)$ be the Hessian Matrix of $f$, whether $x$ is critical point or not. Prove that $f$ is Morse function if an only if

$$det(H)^2 + \sum_{i=1}^k (\frac {\partial f}{\partial x_i})^2>0$$

=>

Assume that $f$ is a Morse function, meaning every critical point $x$ is a non-degenration point. so at these point $x$ , $det(H) \not =0$, thus $det (H)^2 >0$.

If $x$ is a critical point then $\frac {\partial f}{\partial x}=0$ so $\sum_{i=1}^k (\frac {\partial f}{\partial x_i})^2=0$

if $x$ isn't a critical point, $\frac {\partial f}{\partial x} \not =0$ so $\sum_{i=1}^k (\frac {\partial f}{\partial x_i})^2>0$.

Since $det (H)^2 >0$, either $\sum_{i=1}^k (\frac {\partial f}{\partial x_i})^2=0$ or $\sum_{i=1}^k (\frac {\partial f}{\partial x_i})^2>0$. We still have

$$det(H)^2 + \sum_{i=1}^k (\frac {\partial f}{\partial x_i})^2>0$$

<=

I assume that $$det(H)^2 + \sum_{i=1}^k (\frac {\partial f}{\partial x_i})^2>0$$ I need to show that $det (H) \not =0$ for all critical point of $f$. For $x$ are critical point of $f$ then $\frac {\partial f}{\partial x}=0$ so $\sum_{i=1}^k (\frac {\partial f}{\partial x_i})^2=0$ . This implies that $det(H)^2 >0$, thus $det (H) \not =0$ for every critical point $x$ of $f$. Therefore, every critical point $x$ of $f$ is non-degeneration point. Hence $f$ is Morse.

I still have the feeling that I missed something important. I would be very appreciated if anyone could help me check this proof.

Diane Vanderwaif
  • 4,292

1 Answers1

3

You aren't doing anything specifically wrong, it just can be written out more succinctly than how you're doing it.

If you have $n$ numbers $a_1,...,a_n$, then $a_1^2 + ... + a_n^2 > 0$ if and only if at least one $a_i$ is nonzero. So you're being asked to show the function is Morse if and only if at each point $x$ either $det(H) \neq 0$ or the vector $\nabla f$ is nonzero. This is exactly the same as the requirement that at each $x$, if $\nabla f$ is zero then $det(H)$ is nonzero. This is the definition of a Morse function.

Zarrax
  • 44,950