So I didn't finish really the first lecture,

so I'm trying to pick up in the middle of that.

That's what I call this, all this writing.

I'm just trying to catch up to where I was.

So we were talking about this piezoelectric block,

and the idea is that it has a thickness b, some area.

We have metal plates on the top and bottom

that can have charge plus sigma and minus sigma on them.

There's a voltage difference between the top and bottom b.

And we allow external stress, t plus on the top

and t minus on the bottom.

And we had to solve the boundary conditions

and the equations of motion all at the same time.

You remember that that gave us this kind

of complicated relationship where

you picked a set of independent variables

and tried to figure out what the dependent variables were

in terms of those.

And I came up with this result at the end of the last lecture

already.

So this is giving the stress on the top and bottom

and the voltage in terms of the displacement at the top

and bottom and the charge.

Now, all of these terms are oscillating at the same frequency.

These are just the amplitudes.

All I'm doing here is trying to get the amplitudes to work.

Now, what I said was that given this, if we, for instance,

say what the stress and the voltage is on the applying

externally to this block of material,

we can figure out how it responds.

And the way you do that is you take this relationship

and you have the inverse of this matrix.

And I'm not going to write down what the inverse looks

like.

It's really a mess.

But the idea is that we set the next term like t plus,

1 minus, and d, and get u plus, u minus, and sigma by n.

But again, choosing particulars like this,

they're going to have no external stress.

You set t plus, u minus, equal to 0, and then solve for u plus,

u minus, and sigma in terms of d.

So what that then gives you is it gives you sigma is equal to d

3, 3, sorry, f1, 3, 3 over d times kv,

put it down in terms of u, plus kv,

so ckm av divided by minus g plus kv, plus kv,

so ckm av times the voltage v.

So this is a big step forward towards what we want

because the charge density is, of course,

related to the current that we would have to feed into the metal

sheets on top and bottom and relate it back