# Bijection between parameters in integral formula

1. Sep 8, 2014

### pierebean

Hello,

To measure the atomic force with an AFM. One can use the frequency shift of a cantilever. This change of frequency is linked to the atomic force by what we called the Franz J. Giessibl formula in the community.

z is the AFM tip-sample distance. frequency(z) is the change of frequency of the cantilever vs the distance. Forces(z) is the atomic force versus the distance. a is a parameter linked to the drive of the cantilever.

If I put all the useless constants equals to 1, I have:

frequency (z)=(1/a)*Integrale[ Force[z+a(1+u)]*u / Sqrt[1-u^2] , du from -1 to 1]

I use u for the integration

This integral is some kind of functional linking frequency(z) and Force(z).

The Force(z) function looks like this: http://www.teachnano.com/education/i/F-z_curve.gif [Broken]

I would like to proove that for any function Force[z] that is real and ( z is positif ) and that goes through a negative minimum called Fmin, there is a frequency(z) function that goes through a minimum which is freq_min. It works numerically.

I think there is one-to-one relationship between F_min and freq_min but I'd like it to be more than a feeling.

Any ideas?

Last edited by a moderator: May 6, 2017
2. Sep 8, 2014

### WWGD

That looks more like physics than like Math. Maybe you can post it in the Physics forum.

3. Sep 8, 2014

### pierebean

The physics is just to illustrate the use of the formula. But I would like to prove mathematically the one-to-one relationship between the two minimums.

Maybe physicists knows the answer to this calculus question but it's still calculus now?

If not, in what section of physics should I put this question? I need to find a community that uses similar integral.