Let us now talk about derivatives for functions with multiple arguments at this lecture is about the
so-called partial derivatives.
And before we can define what partial derivatives are in multi argument functions, that was first considered
the one dimensional case.
So here we have the function F of X, which could, for example, be this blue function that you can
see here, which is a Kubic function.
Now, if you want to to find the derivative, we have to look at the Points X, which is, for example,
there's Redpoint here.
So we take the value of this function F of X, and then we also consider different point X plus H.
So this could, for example, be this orange point here where the distance in the X direction is H.
And so we take the value of the function at this point, which is does one here take the difference?
And we divide by the distance in X direction, which is H, and the result is the slope of this orange
function here, which goes through both of these points.
Now this is not yet the derivative.
We first have to make h smaller and smaller.
We have to take the limit of H to zero so that the Orange Point moves along the blue curve and approaches
the Red Point here.
And if we do this, we get a red function here, which has to slope, which is the derivative of the
function at this point.
And this is exactly the same definition as the conventional derivative in one dimension.
And alternatively, you could also use this definition where you go into the negative direction with
the slope.
And you could also take this one here in the limit of zero.
All of these expressions would get this would give you the same value.
You can just take whatever you prefer.
And as you may notice, this is exactly the same as the conventional derivative in one dimension.
So this partial derivative, which is denoted by this this expression here is exactly the same as the
conventional derivative.
And to make it a bit easier in the notation, you could also write it down like this, and sometimes
it's also written down like this.
So all of this are just different notations for the same thing in one dimension in 2D.
This is a bit different because here we have a function that depends on multiple arguments, for example,
here, X and Y.
So we cannot simply write F and then this dash here, because then it's not really clear what we mean
by that.
So in two dimensions, we really need to consider these partial derivatives.
For example, here, if we calculate the partial derivative of F of X and Y with respect to X, then
this means we only derive with respect to this coordinate and not with respect to Y.
So this means in this limited definition here, we just add a value H to the X variable and not to Y,
and the rest is then the same as previously.
So what this means is that when we have such a function here where we have X and Y at these X's and
the height of this plot is the value of F.
So this is something like.
Yeah, like Mount.
And you could imagine where you have two coordinates and the third coordinate the height, then the
value of the function.
So if we want to calculate here the slope along the X direction at some point, for example here, then
we have to calculate the partial derivative of F with respect to X.
And alternatively, we could, of course, also calculate the partial derivative with respect to why
and in this case, we have to add this small value of age, which we make small and smaller to the Y
argument.
And then it gives us the slope of this line here, which measures the slope of this mountain along the
Y direction.
Now, here we had only shown what the derivative means for one specific point.
But of course, like in one dimension, we can write down the derivative in a general way where we are
not restricted to a single point.
So this derivative then becomes a function itself.
So let's consider this as an example here.
So let's take this mountain that we also had on the previous slide.
And here the function is given by the expression here we have basically a caution function in the X
direction, which is an exponential function with the argument minus X square.
So this gives us such a caution shape here and then in the right direction.
I have chosen a bit of a different profile.
I have chosen the exponential function with the argument minus Y to the power of four, which gives
us more of a peculiar shape here.
So just I wanted to make it a bit more interesting and not just a boring Gorshin function in every direction.
So now we can calculate the partial derivative with respect to X.
So what we have to do is we have to derive this expression here with respect to X and we have to leave
Y like a number.
So if we would derive this with respect to X, it would give zero.
So what we have here, we could also imagine something like minus X squared, plus two, for example.
Now, how would we derive such an expression?
First of all, we must apply the chain rule because we have an exponential function of some argument.
And then the argument itself is, again, a function.
This is this a function would be minus X squared, minus Y to the power of four.
So first we have to calculate the outer derivative and the outer derivative of an exponential function
is the exponential function because the exponential function does not change and the derivative.
But then we also have to multiply with the derivative of the inner function because of the chain rule
and the derivative of this expression.
If we derive the respect to X is minus two times X and this one is zero.
So we get as a whole partial derivative minus two times x2 in the derivative times, the outer derivative,
which is this function itself.
And if we plot this and calculate the partial derivative along the X direction, then we get something
like this.
So this HitIer tells us the value of this partial derivative and every position, X and Y.
So you could imagine like this, you, your you as a human stand here at the bottom of the mountain
and you want to go up and now you want to plot how steep is the slope of this mountain.
And of course, here in this area, it's especially steep.
So you get large value and at the top of the mountain you have a plateau.
So there is no slope at all.
Does it see this wide region in between?
And then it goes down and here it's very steep and negative.
So you get this blue area here.
And when you are, for example, here and you go just along the X direction, there is almost no slope,
just maybe a bit here.
But or if you go here and it's really zero.
So this is why this is here zero in all cases.
Now, imagine you want to take a different trip.
You don't want to stand here and go along the X direction.
You want to approach the mountain from a different side.
You stand here and you go along the Y direction.
So if you want to calculate the slope, you have to calculate the partial derivative with respect to
Y in this case.
So this is a very similar procedure.
But this time we have to consider that this one here as a constant and this one is our argument.
So our partial derivative would be again the outer derivative, which is just exponential function times,
you know, derivative, which is minus four times Y to the power of three.
So this is a partial derivative.
And if we plotted, we get something like this.
So here you can see we first approach this very, very steep slope here where this derivative has a
large peak.
And then when you approach the top of the mountain slope gets less steep.
And then at the plateau, the slope.
Zero, this is why you get here a very large region where the river turf is basically zero and then
it goes back down.
So you get this negative value.
So you can see in principle, it's the same function as here, but it's rotated and the slopes are much
more steeper because we have here this way to the poor for dependents and here we have to x to the power
of two dependents.
OK, so we have learned what partial derivatives are and what they mean in terms of approaching a mountain
along the X or the Y direction.
Now, of course, you could also go a different path.
You could stand here and go along this diagonal direction.
So how would we calculate the slope along this direction?
And here we have to use a directional derivative, which is, of course, related to these two individual
derivatives.
So let's consider this as a vector along which we go, which is yeah, it has one one component.
So it's really along the diagonal.
And then here I just normalized it so that the length of this vector is one.
And the result will look like this.
So when we go along this direction, we will get a slope profile that looks like this and how can we
calculate this?
So we have to take this directional vector Eve and we have to calculate the DOT product with this vector
of the partial derivatives.
So here we have the partial derivative, which is pegged to X, Y and Z of our function F.
Now, what we can also do is we can take this to here and write it down like this, so this is an abbreviation
of this whole expression and this triangle here is called Nabala Operator.
And we will come to this operator in the next lecture in detail.
But here, it's also important to realize that this directional vector eve is a unit vector, because
if it would have a larger length, then we would artificially scale up our slope.
So it's important to normalize this vector here to the length of one that I already told you this.
This is called the Doppler operator.
This is this vector after partial derivatives that you have to let's act on the function F so in general,
our function F can depend on multiple arguments, for example, here on three three dimensions.
And then it could of course also depend on another argument which is not included here in this Knobler
vector.
And then this knobler acting on F is also called the gradient.
But once again we will come to this later.
So let's go back to our case here and let's calculate this DOT product.
So we calculate five times the gradient of F and the gradient of F is just a vector with this entry
as the first entry and this year as the second entry.
So we write one of the square root two times, one times this one, which is this plus one of a square
root two times, one times this one here.
And what this gives us, because in both cases we have this exponential function.
We can round it down like this.
We have in both cases the minus sign have in both cases the exponential function.
The only difference is the two X and the four Y to the power of three.
And if we plot this expression, we get exactly this one here, which is the directional derivative
along this direction, which is today I and in the next lecture we will discuss the KNOBLER operator
in detail and we will talk about the gradient, which you have already learned about a bit, but also
divergence.
And Karl.