How to draw a graph f(x,y) in (x,y) plane

Question

Is there possible way to draw function $\vec{f}: R^2 \to R^2 $ such as like $\vec{f}(x,y) = \sin x+ \sin y, y+\sin x\ $ and other $\vec{f}(x,y)$ in $xy$-corordinate instead of $xyz$ plane. Is it okay to do it on some online graphers like symbolab and desmos? Thank a lot!

Answer 1

Strictly speaking it's not possible without loss of information: You need 2 dimensions for the Domain (the pairs $(x,y)$ and 2 dimensions for the image (the pairs $(f_1(x,y),f_2(x,y))$, that means 4 dimensions.

For plotting (and in general ;)) you have 3 dimensions at best. A workaround would be to plot $f_1(x,y)$ and $f_2(x,y)$ separately (e.g. https://www.wolframalpha.com/input/?i=f%28x%2Cy%29%3Dsinx%2Bsiny%2Cy%2Bsinx).

You can also plot the absolute value $|f(x,y)|$: https://www.wolframalpha.com/input/?i=plot+f%28x%2Cy%29%3D%28%28sinx%2Bsiny%29%5E2+%2B+%28y%2Bsinx%29%5E2%29

One could use colours as an additional "dimension", but I've only seen this with complex functions ${\displaystyle f:\mathbb {C} \to \mathbb {C} }$: https://en.wikipedia.org/wiki/Domain_coloring. It helps in complex analysis and you are down to the xy-Plane, but otherwise it might not be helpful.