I'd like to get a graphical approach to Analysis in higher dimension vector spaces, such as $\mathbb{R^n}$. To make this easier, my goal is to be able to visualize the graph of functions from $\mathbb{R^2}$ to $\mathbb{R}$. If you take one of the most basic functions in one dimension $ f(x) = x $, it's pretty straightforward to see that it's a line with slope 1. Now, consider $ f(x,y) = xy $. By considering $ y $ as a constant $y_0 $ and looking at the graph of $ f(x, y_0) $ , we see that the slope will get bigger and bigger the more we continue along the y-axis. Our functions is also symmetric for $x$ and $y$ but I don't see any other clues to get a better idea of its graph. Help me out if you've got some tricks plz.

Thanks in advance

  • You could use a 3 dimensional graph, no? – barrycarter May 22 '22 at 15:10
  • @barrycarter Yeah but can you graph any function in your head just like that ? – sleepydrmike May 23 '22 at 17:40
  • @sleepydrmike that depends on the function. You may want to investigate level sets which are often used to help understand higher dimensional functions graphically. – CyclotomicField May 23 '22 at 17:44
  • Oh, I was suggesting using software (or even pen and paper) for visualization. If you look at enough software visualizations, you might be able to build up some familiarity with these functions and be able to visualize them in your head. In general, I would suggest not doing math without writing things down unless there is a good reason not to use pen and paper (such as competing in a mental math competition or something). – barrycarter May 24 '22 at 02:24
  • @barrycarter So basically you're answer is either experience or don't do it. – sleepydrmike May 24 '22 at 10:53
  • Yup. Unless you're a super-genius, trying to learn things entirely in your mind is not a good idea. Even Newton and Einstein used paper! – barrycarter May 24 '22 at 11:29

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