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?

## term for 3d 'point of inflection'?

**Moderators:** gmalivuk, Moderators General, Prelates

### term for 3d 'point of inflection'?

You get the 'Best Newbie (Nearly) Ever" Award. -Az

Yay me.

Yay me.

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

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

- NathanielJ
**Posts:**882**Joined:**Sun Jan 13, 2008 9:04 pm UTC

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

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.

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

Of course, reading fail!

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

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.

Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

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

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 f

D = D(a,b) = f

(a) If D > 0 and f

(b) If D > 0 and f

(c) If D < 0, then f(a,b) is a saddle point.

(d) if D = 0, then the test gives no information.

Second Derivatives Test: Suppose the second partial derivatives of f are continuous on a disk with center (a,b), and suppose that f

_{x}(a,b)=0 and f_{y}(a,b)=0 [that is, (a,b) is a critical point of]. LetD = D(a,b) = f

_{xx}(a,b)*f_{yy}(a,b)-[f_{xy}(a,b)]^{2}(a) If D > 0 and f

_{xx}(a,b) > 0, then f(a,b) is a local minimum.(b) If D > 0 and f

_{xx}(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.

### Who is online

Users browsing this forum: No registered users and 8 guests