Firstly, let's start with
the idea of the directional derivative.
We need to be honest with each other.
When we define partial derivatives,
we actually define
the derivatives of function restrictions,
restrictions towards some preferable directions.
Firstly, what we've done,
we've considered only function which
is restricted by the plane.
Well, in horizontal projection,
with this horizontal section
and this vertical section. All right.
That was how we defined our partial derivatives.
But what if we're thinking
about the growth of function
towards some other direction,
some other plane, same as a screen which is not
horizontal or vertical on X or Y plane.
What is the speed of
a ghost but how should one properly define?
The answer is, as you
might guess, the directional derivative.
Well, so grossly a directional derivative has two parts,
which is familiar for us.
We need to understand what is
the direction properly defined for mathematics?
Well, obviously the direction
on the plane is a vector, right?
Because if you have some just a little truly point and
you have to show some direction,
you just go into [inaudible] goes in.
That's a simple error.
So, first, we need to understand terms.
The direction doesn't have
a length in a way except, for example,
a segment has a length or the way
that a dot goes from one to another,
a curve has a length.
The direction has only the idea of where to go,
where to [inaudible] so say for example,
axis of the plane.
So there is an assumption set the directional vector
is always normalized or in
certain levels it's length equal to one.
That's fine. So interdimensional cases
basically is dimensional,
a directional vector is the vector which has
cosine and sine functions of the angle
with for example, ox axis.
So let us consider some simple graph here,
we have our axis,
we have our zero point, and then what?
Then we have directions for
partial derivative towards x,1,0.
Direction for partial derivative towards y,
0,1 and some other direction.
I'm going to call it L for most of the slides.
So what is this direction?
This direction is basically a vector with unified length.
That's by the Pythagoras theorem
and by definition of cosine and sine functions.
So we can actually find its coordinates if we do know
the angle between the spectrum and x-axis.
So if this angle equals two, example Gamma,
then we can proceed with
coordinates cosine Gamma and sine Gamma respectively.
Sometimes, these coordinates are called
cosine of directional angles because as you may obviously
understand that the other angle here
is p divided by
2 minus Gamma and its cosine is basically sine of Gamma.
That's a very basic trigonometric rules.
Now, we know what is a direction.
So let us try to define a derivative from here.
So what is the most basic knowledge of
the derivatives that we do have?
The derivative of single variate function is the limit
of function change towards the argument change right?
So let's try to write it down.
Firstly, formally, we have two variable function,
f towards X and Y,
given point and the direction.
We are going to use annotation
as for the partial derivatives here.
So basically, we are writing
that in search for the spin of change
over function towards direction L. Then we are
just writing a very basic thing
that's our relation of function change.
So function argument as well
with two in derivative definition.
But the thing here is,
how one should define
a function change and the change of his argument.
Well firstly let us consider some basic setup here.
Here is our point AB here is
our direction and assumes that the second point,
the point we are comparing ours with,
since we are considering the change of
function and the change of argument,
we need to consider some other point and compare
this other point for example x, y with ours.
But that's not an arbitrary point.
That's the point on our directions,
the point on the straight line
which starts from the point
AB and follows the directional vector L. So
since it is like that then the change between
x-coordinate of the initial point and
the x-coordinate with point x,
y is basically should be proportional
to the x-coordinate of the direction.
The same applies for
the change of the second variable or the y change.
That's the change of a function
is basically the difference
between the value and the initial point is a point a,
b and the value at point which we have
got as a result of applying
this directional vector to the point a, b,
t times basically multiplying our direction by
t. In my search here is
that the distance between
the initial point and the point x,
y is point we comparing other function with is
proportional to t is actually is t
because the length of our direction equals to one.
So we have pretty much all the setup we need.
We just need to write the result down.
So as a result,
we get that our directional derivative
is the limit towards t approaches zero,
the change of the function divided by t. Once again,
I'm going to stress this out.
You need to normalize your interaction here.
Otherwise, your answer will be
drawn exactly in the length of the direction times.
One more thing I'm going to
give you this notation once again except
that set notation written on the slide is operational
and the second notation
is the usual one for us with the lower index.
So that's what is directional
derivative through its definition.
Let us consider some final example here.
First of all let's just stop for a second at
this very functions and
understand why it is so disturbing here.
We are going to proceed as follows,
we are not going to
consider whether differentiable or not.
It's north by the matter of fact.
We just going to think about whether or not it has
a limit at the point 0,0.
Here, where it gets tricky because let us assume that we
are looking at some approach.
For example x equals t,
y equals zero quite common how horizontal approach.
We did this on the second week.
How function turns into, well,
zero divided by t power four plus zero, right?
So basically it is zero.
So if there is a limit of this function at zero point,
it's equals to zero.
But if you can see the non-straight parabolic,
quadratic approach for example x equals t,
y equals t squared,
then the function turns into one-half
and one-half actually approaches
one-half whether or not you are taking any limit.
So this function doesn't have
any limit at initial point
because it's not continuous, it's not differentiable.
It's utterly disastrous.
But what about our derivative? What we should do?
We should consider for example some direction right.
We need to understand what the direction is all about.
So let us just take arbitrary direction,
cosine Alpha, sine Alpha, right?
Who cares? We can do just in general case.
Well, since we are talking about a,
b equals to 0,0,
you need to compute the limit of t approaches zero.
The change of function minus
its value at zero point which is zero,
divided by t. I'm going to change x and y
straight out by a t multiplied by L to make it simpler.
So what we are going to have here,
we're going to have t squared cosine
squared multiplied by t sine.
This is a disaster.
T powered four, cosine powered four
plus t powered two sum sine power two.
These frightens us but we can work this out.
First of all, we can just
divide and eliminate denominator by t
squared and we're left with a t
in denominator and t in
the denominator of the whole fraction here.
So we can divide it also and thus we
get some pretty nice expression here.
We get sine of Alpha multiplied by
cosine squared divided by t squared.
Well, then goes cosine powered four plus sine squared.
If t approaches zero,
this whole term approaches
zero and thus we have our answer,
cosine squared divided by sine alpha.
Well and in case alpha is not zero,
we can actually count on this very answer.
So let us just take a moment here.
This function is bad.
This function is not continuous.
This function is not differentiable but it has
directional derivatives for most of the directions right?
Okay. This is funny
because even if you
can't say anything good about this function,
it doesn't have decent linear approximation.
It still grows towards
various directions can be pretty
much assessed easily and fully.