What if Archimedes would have known
functions? The story which I'm going to
tell you borrows from a theory taught
at single variable calculus courses or
at the extension school. There is no new
math. It is part of what one calls quantum
calculus or calculus without limits. We
are going back 20,000 years, when
mathematicians were writing on bones
like the Ishango bone. I want you to
think about these marks as a function. As 
the constant function 111111.
Introducing the numbers is a summation
process, like 4, which is 1 plus 1 plus
1 plus 1. We can go backwards by
taking differences, 4-3 is one, and we can
continue that and can add up to these
numbers and get triangular numbers.
This is what Gauss did as a six-year-old boy or
we can add these numbers up,
stacking triangles on top of each
other and get a volume, called tetrahedral
numbers. Here, you see the Pascal
triangle again. Let's boldly call this
rows a function like 1, x, x^2 over
two factorial or x to the power 3 over 3 factorial etc. We
have a summation, which goes down and the
differentiation we 
goes up. This is a quantum information of
the algebra we know, but it's done in
such a way that the derivative of  x to
the n is n times X to the n-1
a formula we know. You can also take the
exponential function, take the compound
interest formula 1 plus a to the power x, where
a is the interest and x is the number of years. That
is your fortune. You take the
derivative. What you gain is a times the
exponential function. This is already the
fundamental theorem of calculus.
Summing the difference gives f(x) minus f(0).
Taking the difference of the sum is f(x).
The picture is already approved
without words but it's such an important
result that we are going to prove it.
Here is the proof of the first result we
take the sum of the differences and you
see that the middle terms cancel away.
This is a telescopic sum and only the
boundary terms
survive. This is a very important
principle which goes over to multivariable
calculus or differential geometry.
The second proof is even
easier:
we have two terms we take the difference
of the sun and only one term survives
and you get the function back.
This is a very important principle: the
fundamental theorem of calculus links
something easy with something hard and
because we have a link between these two
things the hard, the hard becomes easy.
A generalization: we took the step 1. You
can take the step size h which is
positive number. 
Everything stays the same
you can also change the notation to like
fix the notation and then we have the
fundamental theorem of calculus as we
are all know it. But we don't take
any limits. This is true for all functions.
You can even get more functions. We
have already seen the exponential
function. If you take a imaginary
interest rate, we get the real and
imaginary part because the cosine and
sine. That's the definition and
we immediately to get the result of the
derivative of sine is cosine and
the derivative of the cosine is minus the sine.
These are defirmed functions not the function
you're familiar but very close.
Now, we can harvest the fruits. You can
get things like sum the power of squares
Of course these are the deformed
function. Here we have on the left side
the old squares we can usm
them up and get them immediately a formula
because we know how to integrate.
Or can sum up exponential and get explicit
formulas. You want more functions. We can
multiply two functions again. These are
functions, they don't even have to be
continuous.
It is the Leibniz rule. It is an exact
role. You can look at it. Tt's perfectly
fine for every function f. You get Abel summation
if you turn it around,
which is a fantastic tool. The chain hold
also. It very nice. I actually just noticed that
a h is missing.  But this
formula is exactly also for every
function f and g this is true and if you
look at this formula you see why the
chain rule is true. So,
let's shift gears and go to math 1b. Taylor's
theorem. Of course
the Taylor theorem holds exactly the same way. The function
are general functions and x^n are deformed
functions and f is any function and
these power series actually has been
known by Newton and Gregory already. I only
started with proving it here. It is a nice
exercise to prove this. It is a finite sum.
You can quickly prove it.
It is fantastic even applied to the
exponential function. We have the normal formula,
we are all familiar with. Of
course these are deformed functions. And
here's an example. An arithmetic result: 32 is
1 plus 5 plus 5 over 2 etc.  Thats a 
a  Taylor series expansion.
We can apply these two day meeting here
I took the 20 last years of the Dow
Jones index and just computed the taylor
series and got the function. No linear
algebra or statistics needed.
It's much much faster than linear
algebra. We can even shift more gears.
And go to multi. Oh no, I also
wanted to cover differential equations.
For differential equations: also here, we
don't have to rewrite the books.
Everything you see the books is true.
Just replace the derivative with the D
and deform the functions. This is the harmonic
oscillator. Of course the course are
the deformed functions. Everything
is the same. as we  know it.
Multivariable calculus: a function is a
function is a function on the vertices
of the graph. A vector field is a
function of the edges of the graph. The
gradient is the difference between the
function values and if you add up the
gradient along the curve we just have
the boundary terms which survive.
That's the fundamental theorem of line
integrals and it's the same
cancellation process. For Stokes theorem:
the curl of a vector field is just a
function on triangles and if you look at
triangles fit it together.
There are cancellations between every
match and only the boundary terms survive
and we have the Stokes theorem as we know it.
Would Archimedes have being able
to find the fundamental theorem of
calculus?
I think the answer is yes if he would
not have been even the sword but the concept
of a function.
Thank you.
