Visualization of Functions in the Space of Real Numbers

In this chapter the means of simulation are used to represent different types of functions graphically and to visualize them in their two-or three dimensional context. In most cases the functions have up to four parameters that can be varied.

Physical quantities like mass or length are always associate with a dimension. It is our goal to convey an impression of the character and of the type of the functions $y=f\left(x\right)$. If one considers functions of physical quantities $A$, for example temperature $T$, voltage $U$ and mass $M$, one has to make the argument of the function dimensionless as a rule. Thus the $x$ in $f\left(x\right)$ is then understood as $T\u2215$K, $U$/V , $M$/kg and so on, where K stands for Kelvin, V for Volt and kg for kilogram. The physical quantities appearing in the following section are thus assumed to be made dimensionless in this way. Hint: if the unit is changed then $x$ also changes, for example we have $L$/cm$\phantom{\rule{0em}{0ex}}=\phantom{\rule{0em}{0ex}}100L$/m.

For some simulation files, in particular those of functions of three variables, is changed in the simulations, one parameter is changed as function of time periodically. The animation achieved in this way enhances the spatial impression and rapidly conveys a sense of the influence of that parameter. Animation is also used for the representation of parameter functions as paths on the plane and in space.

Each file contains a description and hints for experiments.

In a selection window a rather large number of standard functions $y=f\left(x\right)$ is listed after its type (for example Poisson distribution, surface wave). In a text window the formula of the selected function is shown and can be edited or even rewritten from scratch. Changes of the formula have to be confirmed with enter.

The command panel below the plot mostly looks the same for all simulations, with selection window and formula display, with four sliders for adjusting the parameters either continuously or as integer values, with input fields for scales etc. and option fields where needed for showing or suppressing additional functions such as derivative or the integral. In the following first example this is discussed in detail.

6.1 Standard functions
$y=f\left(x\right)$

6.2 Some Functions $y=f\left(x\right)$ that are important in Physics

6.3 Standard Functions of two variables $z=f\left(x,y\right)$

6.4 Waves in Space

6.5 Parameter Representation of Surfaces: $x={f}_{x}\left(p,q\right);\phantom{\rule{0em}{0ex}}y={f}_{y}\left(p,q\right)z={f}_{z}\left(p,q\right)$

6.6 Parameter representation of curves, space paths $x={f}_{x}\left(t\right);\phantom{\rule{0em}{0ex}}y={f}_{y}\left(t\right)z={f}_{z}\left(t\right)$

6.2 Some Functions $y=f\left(x\right)$ that are important in Physics

6.3 Standard Functions of two variables $z=f\left(x,y\right)$

6.4 Waves in Space

6.5 Parameter Representation of Surfaces: $x={f}_{x}\left(p,q\right);\phantom{\rule{0em}{0ex}}y={f}_{y}\left(p,q\right)z={f}_{z}\left(p,q\right)$

6.6 Parameter representation of curves, space paths $x={f}_{x}\left(t\right);\phantom{\rule{0em}{0ex}}y={f}_{y}\left(t\right)z={f}_{z}\left(t\right)$