Level curves sketching

Question

Can someone tell me how to compute the values to sketch level curves with two variable variable, for example $f(x,y) = x^2 + y^2$ or $x^2 + y^2 = 9 $. What is the right way to compute values to sketch graph with hand.

Answer 1

Consider some level $c$, then the level curve is given by $$f(x,y) = c. $$ Now the next goal is to get it to a standard form $y = g(x)$ so you can draw it in the $(x,y)$-plane. Let's take your example where $f(x,y) = x^2 + y^2$ and $c = 9$, then $$x^2 + y^2 = 9 $$

Isolating for $y$ yields $$ y^2 = 9-x^2 \implies y = \pm \sqrt{9-x^2}. $$

Hence the level curve of $f(x,y) = 9$ is given by the two curves in $(x,y)$-plane $$y = \sqrt{9-x^2} \quad\text{and}\quad y = -\sqrt{9-x^2}.$$

You should note that these are only defined in the real numbers when $-3\leq x \leq 3$.

For a general level curve $c$ it is easy to see that you get the curves $$y = \pm\sqrt{c-x^2}, $$ with $-\sqrt{c}\leq x \leq \sqrt{c}.$

If you draw it correctly, you will actually see that $f(x,y) = c$ represents a circle with center $(0,0)$ and radius $\sqrt{c}$.

Answer 2

If you're function if $\mathrm f(x,y)= x^2+y^2$, then the level-sets $\mathrm f(x,y)=k$ are given by $x^2+y^2=k$.

You should just choose values for $k$, e.g. $k=-2,-1,0,1,2$, etc.

When $k=-2$ or $k=-1$, the level-sets are empty since $x^2+y^2 \ge 0$.

When $k=0$, we have $x^2+y^2=0$, and so $x=y=0$, and the level-set is a single point.

If $k>0$, say $k=1$, then we get a circle, e.g. $x^2+y^2=1$.

When $k>0$, the level-sets $x^2+y^2=k$ are circles, centre the origin, with radius $\sqrt k$.