So first of all, we need to sort of define
what we are talking about. Since we are going
to fit our curves, our surfaces with
a linear objects, let us as always assume
as a two-variable case. Firstly, what we've known for a single variable
k is that we need to find the best linear
approximation which is a tangent line. Then we are going to generalize it's struggled to
tangent plane or hyper plane for the case of functions of more
than two variables. So what is a tangent plane? Well, it's kind of
essential that since we define the tangent line
as a limit of all secant, then we need to define the tangent plane in
the same fashion. We don't need to
define second plane, which is the plane just
which goes through our defined given chosen point. Then actually it's not the best approximation
and we're seeking for the fittest one. Codes and Charles Darwin here. So since it's quite easy to
define this in this way, we can proceed as follows. But as you can
understand that's not to constructive way to
define our tangent plane. So we need to elaborate
on this further.