Real analysis - sketching graphs of function in $\mathbb{R}^{3}$ or graphs of functions in $\mathbb{R}^2$ given parametrically

Question

Haven't found the above tag properly formalised, hence my idead to create a post regarding the issue of:

• general methods for sketching (manually) graphs of functions of 2 variables in $\mathbb{R}^3$

• general method for sketching (also manually) graph of function in $\mathbb{R}^2$ given its parametric form

I state these issues because of the fact that different websites, as well as textbooks provide only smart tricks to tackle the only specific cases of issues mentioned above, but none provide even comprehensive "catalogue " for various cases.

Answer 1

Regarding sketching graph in $\mathbb{R}^{3}$ manually:

• applying methods from $descriptive\space geometry$, especially the method of analysis of behaviour of graph in planes where 1 of independent\dependent variable equals $0$ (behaviour of graph in the planes $x=0$, $y=0$, $z=0$ separately leads to the analysis of the behaviour of $2-dimensional$ function). In most cases the information of the pattern in these planes, combined with continuity/symmetry of function enables to extend sketch outside the planes