My lecture notes talk about position vectors and surfaces but I don't understand it and it is holding up my learning taking up too much time so can you please help me.
I understand that the following is a cylinder :
$$r(u,v) = 3sinui + 3costujk + vk$$
$$ x/3 = sinu$$ and $$y/3 = cosu$$ and $$z= v $$
I really don't understand intuitively how that is a cylinder or what it is even telling me.
If they're saying that x/3 = sinu then thats a nice statement but how does it help me ?
I am struggling to make the connection in my mind about how these things for surfaces.
Is it just a case of graphing the different points with different values for u ? 3 times the sine of some angle gets you to a point as does 3cos of some angle and it ends up look like a cylinder ?
Okay, that is nice but how do I know that from looking at the equation ?
General - hard to say.
In this case:
$x^2+y^2 = (3\sin u)^2+(3\cos u)^2 \Rightarrow x^2+y^2 = 9$ - equation of a circle in plane xy, $z = v \Rightarrow$ z from any independent of x, y $\Rightarrow $ equation of the cylinder centered at the origin and the axis coincident with the z axis.
But in general it can be quite a challenge.