Let $f(x,y) = (x^2+y^2-1)(2x+y-1)$.

Then how can I sketch the level curves of $f(x,y)$?

Thank you for your help in advance.

  • 1
    $\begingroup$ Welcome to math.SE. Do you have any thoughts on the problem? It will make our job easier and more likely to answer you. $\endgroup$ – Don Larynx Dec 19 '13 at 19:16
  • $\begingroup$ Are you looking for curves defined by $f(x,y)=z$, for any given $z$, or to the specific case $z=0$, that is $f(x,y)=0$? The latter is almost trivial (union of the unit circle and a line), while the former is less obvious. $\endgroup$ – Stop hurting Monica Dec 19 '13 at 19:23

According to the definition of level curves, if we are given a function of two variables $z=f(x, y)$,the cross-section between the surface and a horizontal plane is called a level curve or a contour curve. Thus, level curves have algebraic equations of the form: $$f(x, y) =k$$ for all possible values of $k$. Now let's do this goal by using a mathematical software like Mathematica or Maple. I did it by Maple 16 for you:

enter image description here

  • $\begingroup$ the level curves should be in the plane? $\endgroup$ – ILoveMath Dec 19 '13 at 19:50
  • $\begingroup$ @AtahualpaInca: If we assume that $z$ is the same as $f(x,y)$, so they are some surfaces as you noted in 3d. $\endgroup$ – mrs Dec 19 '13 at 19:52
  • $\begingroup$ I still think they should live in the plane $\endgroup$ – ILoveMath Dec 19 '13 at 19:56
  • $\begingroup$ @AtahualpaInca: I got what you mean. Let me correct it accordingly to be out of $\mathbb R^4$. $\endgroup$ – mrs Dec 19 '13 at 19:58
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    $\begingroup$ no, it helps to me. you helped me a lot. Thanks! $\endgroup$ – math2357 Dec 19 '13 at 20:21

You have a function $f: \mathbb{R}^2 \to \mathbb{R}$. The level curves of $f$ is the set $$ \{ (x,y) \in \mathbb{R}^2 : f(x,y) = K, K \in \mathbb{R} \}$$. So, in order to find the level curves of your function, just set it equal to a constant K, and try different values of $K$. For instance

$$f(x,y) = (x^2 + y^2 -1)(2x + y -1) = K$$

Now, test values foe $K$, say $K=-1,-2,0,1,2,3$, and graph it in each different scenario.

  • $\begingroup$ I know, but can I compute this without computer? For example, when K=0, it is easy to get the curve. But K=1, K=2, or K=-1, then it seems very hard to figure out the whole level curves... $\endgroup$ – math2357 Dec 19 '13 at 20:05
  • $\begingroup$ look at B.S. Answer. $\endgroup$ – ILoveMath Dec 19 '13 at 20:06

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