http://calc3.askbot.com/questions/Open source question and answer forum written in Python and DjangoenCopyright Askbot, 2010-2011.Wed, 02 Jul 2014 23:27:06 -0500http://calc3.askbot.com/question/53/how-do-i-accurately-visualize-and-describe-level-curves/For the 14.1 homework tonight, many of the problems ask you to describe the level curves of the function. I understand how to do the rest of the problem, but how does one visualize what the level curves will look like? Is a 3D graphing program necessary, or is it possible to look at the function and tell? For example (Exercise 14.1.1), let $ f(x,y) = (x−y)^2$. The level curves of this function looks like lines of slope $1$... how could one tell that from the function alone? *Comment*: We haven't covered level curves in class, but they will be **major** when we get back after the fourth of July!!Wed, 02 Jul 2014 14:16:17 -0500http://calc3.askbot.com/question/53/how-do-i-accurately-visualize-and-describe-level-curves/http://calc3.askbot.com/question/53/how-do-i-accurately-visualize-and-describe-level-curves/?answer=60#post-id-60Running off of Kyouko and from what I understand, the level curves take shape of their functions for that axis in the $xy$ plane. For example, $f(x,y)=x^2-y^2$ can be broke down as $y=x^2$ (easy visual) and $y=\sqrt x$ for $0 \leq x \leq \infty $ as $0 \geq x \geq -\infty$ can be represented with $-\sqrt {\mid x \mid}$ (also easily visualized). For the proximity of the lines from one to another, $f(x,y)$ is also recognized as $z$. The larger the magnitude of "$z$" from the set interval change should result in the lines appearing closer together on the contour plot. So like Kyouko stated, it's very similar to a topographic map with $z$ representing the elevation change. Wed, 02 Jul 2014 23:27:06 -0500http://calc3.askbot.com/question/53/how-do-i-accurately-visualize-and-describe-level-curves/?answer=60#post-id-60http://calc3.askbot.com/question/53/how-do-i-accurately-visualize-and-describe-level-curves/?answer=59#post-id-59This doesn't really answer your question directly, but level curves can be thought of as a value-bound portion of a 2D or 3D graph/figure where anything between two of the level curves describes the value between a certain range. To help you visualize this, think of a hiking map. When you see these, there are various enclosed figures (for simplicity sake lets call them circles) that represent a span of altitude (say 1000-1100 ft above sea level). As you look over the map, you'll see many of these circles denoting various elevations as one would climb up or down the mountain. The concept of level curves is important in computing a "steepest descent" sort of problem where you want to find the local or absolute minimum/maximum of a function. Let's assume we're looking at the bowl from yesterday's class where we have $$ f(x,y) = x^2 + y^2 $$ (i'm not sure how i am able to graph this and put it in here, and I don't have mathmatica either) We could then apply level curves along each "z" component (or in this case f(x,y)), and what you would see are rising circles that go up the sides of the bowl. I hope this helps you get an idea, but i'm not sure if this explanation is very goodWed, 02 Jul 2014 21:54:08 -0500http://calc3.askbot.com/question/53/how-do-i-accurately-visualize-and-describe-level-curves/?answer=59#post-id-59