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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.
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