A cylinder is $(x-a)^2+(y-a)^2=r^2$ with axis at $z$. I don't see where the '$z$' is in the equation. The book (calc 3) I'm using mentions the equation works for any $z$, but I don't see where the $z$ output is in the equation
Here is the excerpt from the book, the only part referencing to the basic equation of the cylinder as I've read 1-2 chapters beyond this point (it is a 4 chapter vector calc text)
If you really want an equation with z that support any cylinder position or rotation you can find it on another thread here :
$$(y−z)^2+(z−x)^2+(x−y)^2=3R^2$$
The link : Formula for cylinder
I searched it for while so I leave it here to help any other person searching for it !
The equation of an object is a way of telling whether a point is part of an object -- if you substitute the coordinates of the point into the equation and the equation is true, then the point is on the object; if the equation is not true for that point, then the point is not on the object. There is no $z$ because the z-coordinate is not part of the decision of whether a point is on the cylinder.
The fact that there is no z tells you that all points where the x- and y-coordinates satisfy the equation are part of the cylinder, regardless of the value of z. For example, if the equation is $(x-1)^2 + (y+2)^2 = 4$, then one of the points on the cylinder is (1,0,0), but so is (1,0,1) and (1,0,-1) and (1,0,5) and (1,0, -789) and so on. This means that the cylinder goes on forever both up and down.
Another way to think about this is the equations of the axes themselves. For instance, take the z-axis. All points on it have x=0 and y=0. Clearly it is a line with any z-allowed.
You could write the equation of the axis as $x^2+y^2=0$, because in the reals the only solution to this equation is $x=0$ and $y=0$.
This shows you that the z-axis is like an infinitely thin cylinder (with a point cross section / 0 radius). If you allowed a positive number $a^2$ on the right side, $x^2+y^2=a^2$ all you do is increase the radius / cross-sectional area. However, you still have an equation satisfied for any value of z, just like the z-axis itself.
People tend to over-complicate when there is already intuition for this concept. Take for example a 2D graph, and you want to plot $x = 2$. You will draw the vertical line that passes at $x = 2$, since you learned in school that it is implicitly true for all values of $y$. Note that you do not see the $y$ in this equation. Similarly, when dealing with 3D and you want to plot something like $x^2 + y^2 = 1$, you assume that this holds for all $z$ values, even though you do not see it in the equation.
This is the implicit equation of a cylinder: a point $(x,y,z)$ lies on the cylinder if it satisfies the equation. The important thing here is in fact that $z$ does not occur in the equation. Which means that if some point $(x,y,z)$ lies on the cylinder, then another point $(x,y,z')$ which differs from the first only by a change in its $z$ coordinate will satisfy the same equation and therefore lie on the same cylinder. That's where the $z$ axis comes into play.
An alternative to these implicit equation sowuld be parametric equations, which describe how you compute the coordinates of points using three parameters, e.g. $(r\sin\varphi+a,r\cos\varphi+b,z)$ for $\varphi\in[0,2\pi),z\in\mathbb R$.
An equation for a smoothed cylinder can be :
$$ x^2+y^2+z^{2n} = 1 $$
with $n\in\mathbb{N}$, $n\neq0$.
The greater the $n$ the smoother the top and bottom edges resulting from the "cut" of the infinite cylinder.
See for example https://www.wolframalpha.com/input/?i=plot+x%5E2%2By%5E2%2Bz%5E20+%3D+1
The equation $(x-a)^2 + (y-b)^2 = r^2$ describes a circle on the x/y plane; of radius $r$ and centre $(a, b)$. It's solutions include all the combinations of $x$ and $y$ that make up a two dimensional circle, and no other points.
Now for 3d space we have a z-axis too, For which values of $z$ is the equation true?
Since $z$ is not mention in the equation, its true for all $z$, positive, zero or negative.
The equation therefore describes an infinite cylinder.
