Simplifying $\left.\dfrac{\partial f}{\partial z} \right\arrowvert_{z+\Delta z} - \left.\dfrac{\partial f}{\partial z} \right\arrowvert_{z} = ?$

Question

I am deriving a partial differential equation for wave in a string. $f$ represents displacement of the string at point $z$. I am stuck at a step.

Can anyone help me, how $ \left.\dfrac{\partial f}{\partial z} \right\arrowvert_{z+\Delta z} - \left.\dfrac{\partial f}{\partial z} \right\arrowvert_{z} $ can be reduced to $ \dfrac{\partial^2 f}{\partial z^2} \Delta z$ ?

Please avoid titles that are entirely in $\TeX$. Thanks. – J. M. ain't a mathematician Oct 30 '11 at 15:33 — J. M. ain't a mathematician, Oct 30 '11 at 15:33
@ J.M.: Ok. I got it. Will take care in future. – orionphy Oct 30 '11 at 16:56 — orionphy, Oct 30 '11 at 16:56

score 1 · Accepted Answer · answered Oct 30 '11 at 18:23

1

Let $g(z)=\frac{∂f}{∂z}|_z$. So we have $g(z+\Delta{z})-g(z)=\frac{∂g}{∂z}\Delta{z}=\frac{∂^2f}{∂z^2}\Delta{z}$.

Sincerely,

Tigran

answered Oct 30 '11 at 18:23

Tigran Hakobyan

815

Simplifying $\left.\dfrac{\partial f}{\partial z} \right\arrowvert_{z+\Delta z} - \left.\dfrac{\partial f}{\partial z} \right\arrowvert_{z} = ?$

1 Answers1