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