If the definitions are:
$$D=\{(x,y) \in\mathbb{R}^2 \mid x^2+y^2<4\}$$
$$f:D\rightarrow\mathbb{R}$$
$$f(x,y)=\sqrt{4-x^2-y^2}$$
How do I graph the contour line (subset of $D$) where $f(x,y)=c$ with a real constant $c > 0$?
I'm using a mac, so I've got the Grapher app at my disposal. Any help on how I can do this would be very helpful.
Let $c\in(0,2]$. The equation $f(x,y)=c$ is:
$$\sqrt{4-x^2-y^2}=c.$$
Squaring both sides:
$$4-x^2-y^2=c^2.$$
Rearranging $$x^2+y^2=4-c^2.$$
This is the circle of radius $\sqrt{4-c^2}$ centred at the origin.