Menu Outline Exercise Questions Appendices Lectures Student Projects Books Sections Introduction Atoms Molecules Crystal Structure Crystal Diffraction Crystal Binding Photons Phonons Electrons Band Model Crystal Physics Semiconductors

PHY.K02UF Molecular and Solid State Physics

Numerical integration and differentiation

This page contains some programs for integrating and differentiating numerical data. A function $f(x)$ is can be specified either by inputting a formula at the top or by pasting two columns of data in the textbox at the top-left. When the "calculate from formula" button is pressed, the formula is used to fill the table with 1000 equally spaced values of $f(x)$ equally spaced between $x_1$ and $x_2$. When the "calculate from table" button is pressed, the data is plotted on the right. Below the data and plot of $f(x)$, the derivative $\frac{df}{dx}$ and the second derivative $\frac{d^2f}{dx^2}$ are tabulated and plotted. Below the derivatives, the integral of $f(x)$ is shown as well as the integral of the integral. The integration routines assume that the measurements are equally spaced with an interval $\Delta x$.

$f(x)=$
in the range from $x_1=$  to $x_2=$ .

$x$   $f(x)$

 $f(x)$ $x$

The dervative
The dervative of $f(x)$ is calculated as,

$\Large \frac{df}{dx}\approx \frac{f(x+\Delta x)-f(x)}{\Delta x}.$

$x$   $\large \frac{df}{dx}$

 $\large \frac{df}{dx}$ $x$

The second dervative
The dervative of $f(x)$ is calculated as,

$\Large \frac{d^2f}{dx^2}\approx \frac{\frac{df}{dx}(x+\Delta x)-\frac{df}{dx}(x)}{\Delta x}.$

$x$   $\large \frac{d^2f}{dx^2}$

 $\large \frac{d^2f}{dx^2}$ $x$

The integral of $f(x)$

$\large I_1(x)=\int\limits_{x_1}^{x} f(x')dx' +I_1(x_1)$.

Here $I_1(x_1)$ is the integration constant.

$I_1(x_1)=$

The integral of $f(x)$ is calculated numerically using a method called Simpson's rule.

$\large \int \limits_a^b f(x) dx \approx \frac{b-a}{6}\left(f(a) + 4f\left(\frac{a+b}{2}\right)+f(b)\right).$

$x$   $I_1(x)$

 $I_1(x)$ $x$

The integral of $I_1(x)$
Simpson's rule was used a second time to calculate the integral of the integral of $I_1(x)$.

$\large I_2(x) = \int \limits_{x_1}^{x} I_1(x')dx' + I_2(x_1).$

$I_2(x_1)=$

$x$   $I_2(x)$

 $I_2(x)$ $x$

If the data points you have are not equally spaced in $x$, you can use either of the apps linear interpolation, or cubic spline to generate data points that are equally spaced in $x$.