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I have a function in 3-D. The morse index for one of its critical points is 2 and the other is 3. Which one is the local maximum and which is the saddle point?

Artemisia
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1 Answers1

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A morse index of 3 means (depending on your definition) that there are three orthogonal directions in which the function is concave down. That makes it a local maximum, assuming that your function is indeed a Morse function (i.e., all critical points nondegenerate).

To put it differently, a Morse index of 3 means that there are local coords $xyz$ on a neighborhood of $P$ such that $x(P) = y(P) = z(P) = 0$ and your function, $f$, in these coordinates, looks like $f(x, y, z) = -x^2 -y^2 -z^2$ to second order.

John Hughes
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  • Yes the function is a morse function. What about the critical point having morse=2? Is it a point that is neither maximum nor minimum? – Artemisia Nov 19 '13 at 16:15
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    Yep. Because in local coordinates as above, the 2nd-order taylor series for your function looks like $f(x, y, z) = x^2 - y^2 - z^2$. If you move in the $x$-direction, the function increases; in the $y$ and $z$ directions, it decreases. Hence it's neither a max nor a min, so it's saddle-like. – John Hughes Nov 19 '13 at 19:10