When I compute the boundary of a 3D domain, I always do it in a sort of intuitive way, if I can visualize it. But how does one proceed in general to compute the boundary of an abstract set?
Are there some useful tricks?
For example, if I have something like
$$B=\{(x,y,z):x^2+y^2+(z-1)^2<4, \ z\geq 0\}$$ I picture $B$ as a sphere centered at $(0,0,1),$ cut by the plane $z=0,$ and we consider only the portion of sphere lying above the plane. So the boundary $\partial B$ should be given by the disk of radius $\sqrt{3} $ at the base and the $z>0$ part of the boundary of the sphere $$\partial B =\{x^2+y^2\leq 3, \ z=0\}\cup\{x^2+y^2+(z-1)^2=4, \ z>0\}$$
Even if this is correct, how would you prove it formally?
Next, consider some more complicated thing, for example take $$V=\{(x,y,z)\in\mathbb{R}^3: 1\leq x^2+y^2+z^2 \leq 2, \ z^2 \geq x^2+y^2 \}$$
Here we have the intersection of a cone with a sphere of radius $2$ which has an empty hole inside given by the sphere of radius $1.$
The boundary of the cone is $\partial C= \{x^2+y^2=z^2\},$ while the boundary of the other surface should be $$\partial(\{1\leq x^2+y^2+z^2 \leq 2\})= \{x^2+y^2+z^2=2\}\cup \{x^2+y^2+z^2=1\}$$
Right?
Now,
- How do I compute the boundary of $V$ ?
- Given the boundary $\partial V,$ how do I compute the portion of $\partial V$ lying on the cone? Is it $\{ (x,y,z): z=x^2+y^2, 1 \leq 2(x^2+y^2)\leq 2 \}$ ?
- Given the boundary $\partial V,$ how do I compute the portion of $\partial V$ lying on $\{1\leq x^2+y^2+z^2 \leq 2\}$ ?
Here is another related question of mine.