Your favorite 3D function

mark

(5 pts)

Write out your favorite function that maps $\mathbb R^2 \to \mathbb R$ in LaTeX and plot its graph in 3D. If applicable, include the code you used or link to an online tool.

audrey

My favorite function, written in Cartesian coordinates, is
$$f(x,y) = e^{-(x^2+y^2)}.$$
Because of the circular symmetry, it has a nice representation in polar coordinates:
$$f(r\cos(\theta),r\sin(\theta)) = e^{-r^2}.$$

We can plot it in Mathematica like so:

ParametricPlot3D[{r*Cos[t], r*Sin[t], Exp[-r^2]}, 
  {t, 0, 2 Pi}, {r, 0, 2}, BoxRatios -> {1, 1, 1/2}, Ticks -> False]

Alternatively, you can plot it on Math3D.org . to produce the following version:

TheMeansOfProduction

A cool 3D function I know, written in polar coordinates, is $$f(r,\theta)=\left(\frac{J_1(r)}{r}\right)^2,$$ where $J_\alpha(r)$ is a Bessel function of the first kind. This function can be written in Cartesian coordinates by substituting $\sqrt{x^2+y^2}$ for $r$: $$f(x,y)=\frac{J_1(\sqrt{x^2+y^2})^2}{x^2+y^2}.$$ We can plot it in Mathematica with the code

Plot3D[(BesselJ[1, r]/r)^2 /. r -> Sqrt[x^2+y^2], {x, -8, 8}, {y, -8, 8}, 
  PlotRange -> All, PlotPoints -> 100, Boxed -> False, Axes -> False]

Physically, this is the intensity of the diffraction pattern created by light from a far away point source passing through a circular aperture (with all constants set to $1$ for convenience).

Will

Here's a function that's pretty neat. In Cartesian coordinates it can be written as: $$f(x,y)=\frac{-\cos(x^2+y^2)}{x^2+y^2}.$$

This function conveniently can be written in Polar coordinates as: $$f(r\cos(\theta),r\sin(\theta))=\frac{-\cos(r^2)}{r^2}.$$

If we plot it in Mathematica, we get something that looks like this:

ParametricPlot3D[{r*Cos[t], r*Sin[t], -Cos[r^2]/r^2}, 
  {t, 0, 2*Pi}, {r, 0, 4}, BoxRatios -> {1, 1, .5}, Ticks -> False]

SighHillDoveN

Oi ere's a bonkers Cartesian function:

$$f(x,y)=\frac{\sin(x^2+y^2)}{\cos(\sqrt{x^2+y^2})}$$

And this is the Polar version:

$$f(r\cos(\theta),r\sin(\theta))=\frac{\sin(r^2)}{\cos(r)}$$

If you plot it right 'n proper in Mathematica with this code:

ParametricPlot3D[{r*Cos[t],r*Sin[t],Sin[r^2]/Cos[r]},{t,0,4*Pi},{r,0,8},BoxRatios->{1,1,1/2},Ticks->False]

It looks like this:

Quite a posh function innit?

Zoë

I think the following 3D function is pretty cool. It can be written in Cartesian coordinates $$f(x,y)=\sqrt{x^2+y^2}$$ or in Polar coordinates
$$f(r\cos(\theta),r\sin(\theta))=\sqrt{r^2}.$$

If you plot it in Mathematica, it looks like an ice cream cone.

ParametricPlot3D[{r*Cos[t], r*Sin[t], (r^2)^.5}, {t, 0, 2*Pi}, {r, 0, 
  4}, BoxRatios -> {1, 1, 1}, Ticks -> False]

super_duper_late_dude

Heres my favorite 3D function written in Cartesian coordinates $$F(x,y) = e^{x^2+y^2}sin(20e^{-x^2-y^2})$$

It can also be represented in Polar coordinates as $$F(r,\theta) = e^{r^2}sin(20e^{-r^2})$$

If you plot it in Mathematica Plot3D[E^(x^2 + y^2) Sin[20 E^(-x^2 - y^2)], {x, -2, 2}, {y, -2, 2}] it looks something like this

mega_late

My " more groovy than the latter's" favorite function, written in Cartesian coordinates, is
$$f(x,y)=e^{−(x4+y2-xy^3+14x)}$$.

I plotted this function in 3D using wolfram alpha @ https://www.wolframalpha.com/

Red

This is the $Groovy$ function I chose, mostly because it looks like it'd be a cool hammock.

In Cartesian coordinates, it can be written as $f(x,y)=(x^2+y^2)^2$ which in polar is $f(r,\theta)=r^4$

Using wolfram alpha its plot is as follows

mega_late

lusta

My favorite function is

$f(x,y) = \sin(5x)\cos(5y)/5$

Its called...Rainbow Waves of Nazareth

and can be plotted with Mathematica

Plot3D[Sin[5 x]*Cos[5 y]/5, {x, -1, 1}, {y, -1, 1}, ColorFunction -> "Rainbow"]

AX_KE

A function I like is $$f(x,y) = \frac{x - 2y}{x^2 +y^2}$$

Here is the plot in Mathematica:

Plot3D[(x - 2 y)/(x^2 + y^2), {x, -3, 3}, {y, -4, 4}, BoxRatios -> Automatic]