Hi. Welcome to the fourth
week of the calculus course. Recently, we've
covered the concept of differentiability
and a derivative of single variate function. Now, it's time to move on
to the multivariate case. It's essential since
we know quite a lot of concepts for single variate
functions such as seconds, tangent lines, linear
approximations, that we need to generalize this concept as a
multivariate functions. But it's tricky because
introducing another degree of freedom into the
system can cause us some conceptual problems here. Thus, we need to understand what concept here is
easily differentializable. This concept we are going
to call, The Dinosaur Law. Because since we've used
our approximation by straight lines as a definition of differentiability for
single variate functions. That's why we're going to
use the same definition but approximation of some
hyperlinear concept, such as hyperplane for
multivariate case. So in the rest of our week, we're going to just to gradually first of all define what
is a tangent plane or hyperplane is for
multivariate functions and understand what is
the chain rule here. Then we proceed with
derivatives of higher order, and as the concept of convexity
in multivariate case. See me the following video