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.