term for 3d 'point of inflection'?

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Mindor
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term for 3d 'point of inflection'?

Postby Mindor » Sun Nov 01, 2009 7:26 pm UTC

It has been quite a while since I've had Calc 2 and 3 where we analyzed functions of multiple variables, and my meager searches have not come up with anything useful (perhaps for lack of the correct term(s))

In 2 dimensions, (i.e. y = function of x) when a curve changes from concave up to concave down, you get a point of inflection.
In 3 dimensions,(i.e.z = function of x and y) it is still possible this happens at a single point or points, or even a single point may be inflective in one plane and not in others, but when there is a continuous line/curve of inflective points(even if it is bounded), is there a formal term for that?
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acb
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Re: term for 3d 'point of inflection'?

Postby acb » Sun Nov 01, 2009 7:40 pm UTC

I think saddle point is the term you are looking for.

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NathanielJ
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Re: term for 3d 'point of inflection'?

Postby NathanielJ » Mon Nov 02, 2009 2:08 am UTC

acb wrote:I think saddle point is the term you are looking for.


No, that's a critical point (because the first partial derivatives are all equal to zero). I'm not sure of any special name for points where the second partials are all equal to zero.
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acb
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Re: term for 3d 'point of inflection'?

Postby acb » Mon Nov 02, 2009 8:50 am UTC

Of course, reading fail!

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antonfire
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Re: term for 3d 'point of inflection'?

Postby antonfire » Mon Nov 02, 2009 5:52 pm UTC

I don't think there's a standard term, but googling "curve of inflection" yields some papers which use that phrase, and it seems to be as good a name as any.
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crazydave
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Re: term for 3d 'point of inflection'?

Postby crazydave » Mon Nov 02, 2009 9:48 pm UTC

You are thinking of critical points or stationary points. I would suggest looking at the second derivative test to determine what the point is.

Second Derivatives Test: Suppose the second partial derivatives of f are continuous on a disk with center (a,b), and suppose that fx(a,b)=0 and fy(a,b)=0 [that is, (a,b) is a critical point of]. Let
D = D(a,b) = fxx(a,b)*fyy(a,b)-[fxy(a,b)]2

(a) If D > 0 and fxx(a,b) > 0, then f(a,b) is a local minimum.
(b) If D > 0 and fxx(a,b) < 0, then f(a,b) is a local maximum.
(c) If D < 0, then f(a,b) is a saddle point.
(d) if D = 0, then the test gives no information.


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