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Your favorite 3D function
(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.
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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:
Alternatively, you can plot it on Math3D.org . to produce the following version:
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
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).
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:
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:
It looks like this:
Quite a posh function innit?
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.
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 thisMy " 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/
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
My favorite function is
$f(x,y) = \sin(5x)\cos(5y)/5$
Its called...Rainbow Waves of Nazareth
and can be plotted with Mathematica
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]