teaching:topics:calculus:notation [Simonweb wikinotes]

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=====Three notations for Calculus=====
====Why three?====
The three most commonly used come from different perspectives. [[#Newton]] was developing what is now called
classical dynamics, and concerned principally with forces and the movement of objects --- considered as
changes over time. The big idea behind calculus concerning infinitessimals and infinite sums was new, and
very contested, so he began his book with a traditional geometric analysis of planetary motions. His
notation avoids making the infinitessimals or calculus explicit, refering instead to the physics
application of them. Exploring this physics was the original motivation for developing the mathematics.

[[#Leibniz]] was primarily concerned with that mathematics, which is much more generally applicable than
just physics, and not necessarily about time. This notation  explicitly refers to the underlying
mathematical ideas, a concept quite distinct from any particilar application. It is a mathematical
language, dealing with generalised number, ratios and sums --- number which might be applied to
specific measurements while modelling in one or another field. It is a much more flexible notation, much
more extensible. The integration notation is part of Leibniz system.

[[#Lagrange]] was a century later, a mathematician. He worked on the three body problem in dynamics (in
particular the Sun - Earth - Moon behavior), and a great deal more. He created a new mathematical
formulation of dynamics that focussed on functions, and functions of functions. His notation reflects this,
it explicitly references functions and the mapping of functions to new functions, rather than the ratios or
sums of the changing values of these. It follows from the ideas developed soon after calculus and the notation developed for functions (which had an explicit differential operator symbol). An analogy here could be the difference between procedural and functional computer programming paradigms.

====some context ...====
Newton's was often used as more succinct and descriptive of the immediate application. The idea embodied in
that notation --- physical systems described by Cartesian coordinates in a euclidean space evolving
over a completely independent time in ways calculatable using the masses and the changing forces involved ---
has been inappropriate in most physics developed since 1905. This was called Dynamics. It was the fundamental insight
of their time. It had taken several generations and well over a century of pretty solid work to get there.
It remains a very good approximation for most practical purposes. Certain details of quantum mechanics
still depend on that spacial-temporal assumption (where efforts to formulate the equations in relativistic
terms have failed, and the small scale and times involved mean this is not numerically significant).

It is interesting to note that exact, closed form, solutions to most dynamic problems involving feedback
(such as forces between objects that determine their movement which in turn changes those forces) are not
possible, and that even numerical solutions (evolving a system step by step) are very
limited. 'Chaos' is the mathematics that deals with this idea, showing that the evolution of even quite
simple systems over time is overwhelmingly dependent on very small variations of the initial conditions.
There is no exact, general solution to the three body problem --- it is not possible to make equations
giving the position of three stars orbiting each other explicitly as a function of time (unless we are
looking at some special cases). This is reflected in the stars we observe. Binary stars are very common,
trinaries consisting of a close binary and another much more distant star are found,
[[https://www.bbc.com/news/science-environment-33428506|more complex pairings certainly exist]],
planets and moons around a star are stable when they are, effectively, independent binary systems. The
three- or four-star systems we know that are not pairs of pairs are in areas of intense star formation, and
presumably will split up, perhaps launching one as another of the high-velocity stars that move rapidly
through, and out of, galaxies. 

There are other very important physical models that are based on describing the whole system over a
period of time, as opposed to a snapshot in time or their continuous or stepwise evolution through time
determined by dynamically changing forces, acceleration, momentum and such. They existed
in some form in classical Greek times. They can seem rather challenging in terms of our usual ideas of
causality yet nevertheless are very useful and important. The cutting edge of physics today includes ideas that
might radically alter our sense of time and space, at least at the fundamental, philosophical level. But that is for later --- just to get started requires a very deeply mathematical view and a strong sense of
the physics up to now --- at school we have barely started that journey in either field.

====Newton====
... refers directly to rates of change over time, which is the original motivation for calculus. We can
explore these rates of change numerically (like Feynman did when introducing Newton's Dynamics),
geometrically (like Newton did with Kepler's planetary laws when introducing Dynamics originally) or
algebraically --- using calculus, finding potentially stable systems (which are specific closed form
solutions to a set of differential equations describing the feedbacks involved) then looking for examples
of these experimentally --- which was the focus a lot of 20th century particle physics.

His is the classical mechanics perspective, and a very good place to start.

Taking `x` to represent position, `v` velocity and \(\boldsymbol{v}\) the velocity vector then ...\\
\begin{align*}%%
\dot x &&& \text{is the rate of change of position (that is velocity),} \\
\ddot x &&& \text{is the rate of change of the rate of change of position (so acceleration),} \\
\dot v &&& \text{is the rate of change of velocity (and so also acceleration) and} \\
\dot{\boldsymbol{v}} &&& \text{is the same but as a vector, so no longer in just one dimension.}
%%\end{align*}

It is useful to use each of these symbols when talking about such rates of change over time, even before we
consider the calculus methods of modelling and working with these values.

====Leibniz====
... is more explicit about how this relates to equations and graphs, it is much more extendable to
many of the other fields where calculus is used. It is a more mathematical perspective. It was developed
simultaneously with Newton's notation.
\[\frac{{\rm d}y}{{\rm d}x}\]
is the ratio of the instantaneous changes in two related values,
\[\frac{\rm d}{{\rm d}t}(4t^2+3)\]
is between the values of the expression and the variable.
\[\text{repeated:}\quad\frac{{\rm d}^2}{{\rm d}x^2}(3x^3+5)\qquad\text{means:}\quad\frac{\rm d}{{\rm d}x}\left(\frac{{\rm d}}{{\rm d}x}(3x^3+5)\right).\]
\[\int_a^b\!x^2 + 4x\,{\rm d}x\]
is the definite integral of the expression taken from `a` to `b`. The sum of the infinitessimal slices under that curve.

The kind of calculus that we deal with at school treats each of these as a single symbol, but it does
make sense to break them down into their parts, considering say \(\ {\rm d}x\ \) on its own. Doing so
requires some care and a deep understanding of what is being done, but is a very powerful mathematical
technique.
====Lagrange====
... was a century later, and considers //functions// as entities which can be operated on. He used the 
function notation introduced earlier in the 18th century by Euler and others. This is another distinct 
mathematical perspective, and a language that facilitates a very different kind of thinking.

\begin{align*}%%
{\rm f}(t) &&& \text{a function with values that depend on the variable } t \\
{\rm f}'(t) &&& \text{the function that is the derivative of that function} \\
{\rm f}^{\prime\prime}(t) &&& \text{the function that is the derivative of that derivative function} \\
\text{and even} \quad {\rm f}^{(n)}(t) &&& \text{the function that is the }n \text{th derivative of }{\rm f}(t) \\
\text{or sometimes} \quad {\rm f}^{(-n)}(t) &&& \text{the  }n\text{th antiderivative or intergal of }{\rm f}(t)
%%\end{align*}
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