[MUSIC] So since we've covered the idea
of single variate functions, it's time to think about the limits for
multivariate functions. Because let us face it, in real life,
we do not normally think about that real life function is a function
of a single variate, right? Because, for example, the price of
your apartment does not depend only on the address of your apartment. It depends, for example, on number
of rooms, number of rats to hunt, number of ghost stories in it, right? So basically, we get as a result function
of quite a number of degrees of freedom. The thing is that we are going to
consider only function of several or finite number of variables, right? So it doesn't mean that that's the only
case and one cannot consider, for example, the function of
infinite number of variables. But it's kind of unrealistic, and we are going to stay in the ground,
at least a little bit. So what is a multivariate function? Multivariate function is a mapping
from some n-dimensional vector or an n-dimensional set of real
numbers to real numbers. As notation written here, basically what is real
numbers power at n is the cut product of n real number sets,
okay? That's an easy one. Basically what is, for example, R squared, it's a real plane where
R is multiplied by R. Or just you are looking at all possible
pairs where the first element is real number and
the second element is real number. The same applies, for
example, for R power 3 or three-dimensional real space. You are looking at the set of three
real numbers, where first element is all possible real numbers, second
element is all possible real numbers, and this can be all possible real numbers. So usually we will consider only
the case of a function of two variables, firstly, because it's easily
generalizable from this point on to also finite number of variable case. And it's quite easy to draw,
it's basically the only thing that's easy to draw in multivariate case, so
we will consider only this case. So firstly, we need to state that
all the terms that we've defined for single variate functions or functions at
all pretty much hold, R remains the same. So we know what is a domain,
it's a set of all possible variables, what is range or codomain, and
what is support of the function. But we're going to say that from now on,
the graph of the multivariate function or, for example, the graph of two variable
function is called a surface. Basically what it is, formally, it is a set of all three-dimensional points as x, y and x towards x, y here. Which is, well, for example consider
that you are taking a sheet, a clean sheet, and
then just throw it into the air, and it is somehow curved into the air while
it's flying, and that's what is surfaces. Okay, so
the trick here is that it's hard for our brain to imagine a surface
just by the equation of, just at all understand how it looks like. Because you need not only to draw it,
but to imagine it and find the idea of how to rotate it,
what turned into what, what's the result in different sections. Well, that's totally a nightmare,
in a shot of word. So we've come up with another idea. Basically the idea is that
if we have a surface, we kind of want it to be described
by our plane picture, plane figure. In order to do so, we introduced the
concept of function's level, or C-level. Basically what it is, consider that
we an equation f towards x and y, basic our function equals to C. We are interested in all possible pairs,
all possible real dots x and y, which satisfies our
equation f equals to C. This curve on the plane is
called function's level. Okay, obviously function has a lot of
levels since function have a lot of different values, right? Let's try to prove some basics here. For example, everybody understands that
different levels do not intersect, right? Because if two different levels intersect, it basically means that they do
have some common point x and y. And thus, at this very point, function is
supposed to have two different levels, two different values, which is not possible
for the functional mapping, right? Okay, and the second one, for example,
each point in the function domain belongs to some level,
which is kind of pretty much the same. Assume that we have some
point in the domain x, y, if it is in the domain,
thus we can apply a function to it. Basically we can consider function's
value at this very point. Thus we have a level which is
determined by the value of the function at this very point. Okay, fine, let us consider some example. For example, let us took pretty
awesome linear function x + 2y, and we are aiming to draw it levels. First, let's start with
writing down an equation, x + 2y equals some level C, right? So let us just move x to the right part, divide both sides by 2. And thus we get our usual
equation over straight line, except we need to write it carefully down. So what do we have? We do have a straight line with
different vertical shifts, which is related to different levels, and the same slope minus one-half, right? So we need to draw
several lines with minus one-half slope here, that's our levels. And to finish it off,
it's usually common to draw an arrow, which implies that going
from one level to another in the direction of this as the level rises. Basically since our C-level is included
as positive value into vertical shift, the higher the level, the higher
the value of the function on it. So level rises into the upwards direction. Okay, so let us move on to other thing,
which is, well, another example,
which is the maximum value of two absolute values of coordinates, right? So it's kind of important for,
for example, functional analysis. Because the idea of maximum
of two absolute values is, for example,
one way to determine the distance between two dots in two-dimensional space or
whatsoever. So what we're going to do,
we're going to write an equation down. And basically what it means,
it basically means that we have to choose, firstly, which variable
has the greatest value, and then we are going to say that
is this variable is fixed with C. Since we are looking at
absolute value of x and y, thus the picture is going to
be pretty much the same for all quadrants here, right? So we are going to to consider only
positive x's and positive y's. Thus we are looking only for max value of x and y equal to C. Assume that x is greater than y, thus basically it means that x all points lies below this y equal
to x straight line. Thus we need to draw a line
which coincides with the idea that x equals to C. This is a vertical line. Okay, well, and as you probably all
understand, the same applies for the upper part, which coincides that y
is the greatest value, y holds maximum. Thus we got this right angle here, and by the symmetricity of absolute value, we got such a nice square,
with a center in 0. Let us draw several levels, and then let us state that
we are talking about expanding levels from
center to the outer space. Okay, and as a final touch,
let's consider the graph here itself. And what we are basically looking at, we are basically off
looking at several sections of this graph by horizontal planes,
such as z equals to C. So to finish it off,
let us consider some other example, which is x squared plus y squared. And it is extremely easy, but this is kind of we are going to
to look at quite a lot. So the levels are quite straightforward. Firstly, let us state that this
function doesn't have negative levels, all values are positive. And this is our equations of
circle with a center at 0, thus we get the following levels, concentric circles. And as in previous examples, level rises right from
the center to the outside. Okay, the surface of this function looks
something like this, this is a paraboloid. And the last thing I'm going to see here,
basically what we're looking at in the function which is
not directly from two variables. Because as you can understand,
we can add a square root and return it by adding power 2 here, right? So basically the thing which is written under the outer power here is the distance between a point x, y and
point 0, 0, right? Basically it is the distance
towards to the 0 value, right? So this function does not
actually depends onto x and y separately, it depends only
on its distance to the 0 point. Thus basically it means that on
the circle with the same radius, well, if we just fix the radius,
the distance towards 0 point, this function remains the same. Or basically you can just
take a function with, for example, y equal to 0 and
then rotate it by its o, z axes. And then you will get
this pretty nice surface. [MUSIC]