I have a firm understanding when it comes to 2-D graphs. However 3-D plots/graphs are confusing to me. I know there exists several software packages which neatly does the job. I need to sketch it by hand in order to understand it. I know 3-d graph is an extension to 2-D with an addition of $z$-axis, I am not sure where lies the $x$-axis, where lies the $y$-axis and the $z$-axis. Here's the situation.
Given the surface $f(x,y) = z = x^2 + y^2$. I have been told to determine the nature and sketch the surface after determining the partial derivatives with respect to $x$ and $y$ respectively. I obtained the critical point $(0,0)$. I also could determine that its minimum at that point by evaluating delta. But I'm not sure how to sketch this. Please help.
By the way, what exactly is contour map? Please help me sketch that one too.
If you are curious to know from where I am solving this problem, it's from John Bird's higher engineering math, page 359.