What is the difference between curvature and concavity in one or two dimensions?

[wil3]

Hello- would someone mind clarifying the distinction between curvature of a function and the concavity? I would prefer if you could keep it to the one or two dimensional case, since my math background is just multivariable. Thank you very much.

 

[HallsofIvy]

A graph is either "concave upward", "concave downward", or has an inflection point at a given point. "Curvature" of a graph, at a given point, is a number reflecting how far the graph differs from a straight line. In particular, the reciprocal of the curvature is the radius of the "osculating circle", the circle that best approximates the graph at that point.

 

[wil3]

Thank you very much for your reply.

Okay, so the numerical value of concavity is meaningless, and the sign is the only important feature?

I've been reading about "mean curvature" in the context of how objects with zero mean curvature are minimal surfaces. If curvature is related to the osculating circle radius, then can it be signed? Because I'm a little confused how a non-planar surface can have a mean curvature of zero unless it somehow has negative curvature.

 

[Skyler0114]

Thank you very much for your reply.

Okay, so the numerical value of concavity is meaningless, and the sign is the only important feature?

I've been reading about "mean curvature" in the context of how objects with zero mean curvature are minimal surfaces. If curvature is related to the osculating circle radius, then can it be signed? Because I'm a little confused how a non-planar surface can have a mean curvature of zero unless it somehow has negative curvature.

The numerical value of concavity does have meaning, the farther away from 0 the value is the greater the change of rate of the derivative. It's like if g = 5 instead of 9.8, we would fall slower, thus taking longer to hit ground.

Also, since you get the magnitude of your numerator and denominator even if they were negative they would become positive.

What I don't get is the difference between curvature and concavity in 2 dimensions. Can anybody elaborate on that.
EDIT: Sorry for necroposting

 

[wil3]

EDIT: Sorry for necroposting

no worries, I'm still alive. Three years of physics later, I feel much better about this distinction, but thanks for clarifying.

 

[Skyler0114]

no worries, I'm still alive. Three years of physics later, I feel much better about this distinction, but thanks for clarifying.

Would you care to elaborate on that distinction a bit more in 2d, that's the only part that seems to have me a bit confused. I'm also a physics major so getting this distinction down now would be pretty nice.

 

[HallsofIvy]

Remarkably, I'm still alive too (its a reunion!). I have never seen "concavity" given as a number. I have only ever seen "concavity" stated as "concave upward" or "concave to the right". The number you are attributing to "
concavity" probably is the "curvature". Can you give a specific example of "concavity" as a number?

 

[wil3]

We mean something like "The concavity of f at x_0 is the value of the second derivative of f at x_0."

 

[HallsofIvy]

Well, that is certainly NOT true. It can be said that the concavity at x_0 is determined by the second derivative but not that it is equal to it.

But I did not ask what you meant- I asked if you could tell where, in a textbook or otherwise you saw a statement assigning a number to "concavity".

 

[Redbelly98]

I, too, have only seen concavity used to indicate a direction for a function's curvature, never with a numerical value associated to it. That being said...

If we are going to use "concavity" simply to mean the 2nd derivative, then it's pretty straightforward to see what it is different than curvature.

The concavity of a function y(x) is simply y'', the 2nd derivative of y.

The curvature of a function is the reciprocal of the radius of curvature, and you can find the formula for these in a calculus textbook or a web search:
$$\text{curvature }= \frac{1}{\text{r.o.c.}}=\frac{y''} {(1+y'^2)^{3/2}}$$
These are not the same, in general, though they are equal wherever y' is zero.

p.s. Remarkably, I was alive when this discussion was first started. However, proof of my existence lies outside the boundaries of this thread.