So last time we have developed a week of variation formulation for at the example of the Poisson's

equation with homogeneously boundary conditions and one point which where we made a first

attempt which was not yet feasible as we figured out is to formulate a basic space, a basic

vector space in which we look for the solution of the equation. The first trial what we did

is we said okay we allow roughly spoken for kinks in the solution so we only require that

the solution is on some tessellation of the domain is only piecewise continuously differentiable.

But we have seen that is not enough because this space at least in the norms which are

so to speak the ones we have to use then is not a complete space, is not a Banach space.

And you said you know as we have seen the variational equation is equivalent to a minimization

problem you can imagine that not every minimization problem on a non-complete space has a minimum.

For example if even if the function has all the ingredients so to speak which are there

for have to have a minimum. Take parabola opened to the above so of course we know there

is a minimum there is a unique minimum but it is only on R on the complete Banach space

R the real numbers. If you would say okay we do we shift the parabola such that square

root of 2 is the minimum and we look only at the minimization problem on Q then we don't

have a minimum. And that is the same situation we are now encountered with our space that

is our space is not large enough. So another possibility would be to say okay we need first

order derivatives so let just say we take functions which have first order derivatives

point wise. Point wise always means almost everywhere in the sense of the Rebeck measure

without any requirements on what kind of functions those derivatives should be. But that is also

not a good idea because in real analysis there are some quite strange functions for example

there is Cantor's function you might have heard of it. Cantor's function is a function

on in the real so on the interval 0 1 and it has the properties a continuous function

it is not 0 it is not constant it is differentiable and the derivative is 0 almost everywhere

but what does that mean? It just means that the basic theorem of the calculus of analysis

is not valid anymore in the sense that we can regain in 1D the function by integrating

the derivative because here the derivative is just 0 here we could get a 0 but this function

is not 0. But this is again the basis in 1D and if it does not work in 1D it cannot work

in more than 1D. This is the basis again for partial integration so we have no chance to

do partial integration and that was our starting point to derive the weak formulation. So this

way is not feasible so we need some requirements on those derivatives and to do so we need

a new notion of derivative and to develop that let's first introduce a few a little

bit of notation the first thing is the so called multi index notation which I am going

to introduce so we have D variables and we want to look at various partial derivatives

in those D variables and of course we can write it down in the way you might have used

in analysis or in mathematics for engineers or you can we can do it also a little bit

more compact probably you did this or this more compact notation already so what we do

is the amount of derivatives we would like to have in the first in the second in the

D coordinate so these are non natural numbers these are natural numbers including 0 these

we put together to a vector and this vector we call it multi we call multi index. So this

might the so to speak the L1 norm of this vector we call the order or the length of

alpha this is the total amount of derivatives we would like to do and then on the basis

of this notation first of all we can introduce a shorthand notation for polynomials but now

not in one variable but in D variables the corresponding analog for a polynomial in one

variable would be a linear combination of functions of this types of products of powers

of the components and this thing here we just write in a shorthand notation as x the vector

x to the vector alpha this is the shorthand notation for this quantity and now we can

transfer this shorthand notation for polynomials to differential operator is if we do the formal

substitution in the sense that we substitute the vector formally by the vector of the partial

derivatives that means that alpha of u means take alpha one derivatives in the first coordinate