teaching:topics:calculus:notation
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teaching:topics:calculus:notation [2020/07/27 21:38] simon [Newton] |
teaching:topics:calculus:notation [2024/05/03 18:51] (current) simon [Leibniz] |
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| language, dealing with generalised number, ratios and sums --- number which might be applied to | 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 | specific measurements while modelling in one or another field. It is a much more flexible notation, much | ||
| - | more extensible. | + | 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 | [[#Lagrange]] was a century later, a mathematician. He worked on the three body problem in dynamics (in | ||
| Line 19: | Line 19: | ||
| formulation of dynamics that focussed on functions, and functions of functions. His notation reflects this, | 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 | it explicitly references functions and the mapping of functions to new functions, rather than the ratios or | ||
| - | sums of the changing values of these. An analogy here could be the difference between procedural and | + | 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. |
| - | functional computer programming paradigms. | + | |
| ====some context ...==== | ====some context ...==== | ||
| Line 26: | Line 25: | ||
| that notation --- physical systems described by Cartesian coordinates in a euclidean space evolving | 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 --- | over a completely independent time in ways calculatable using the masses and the changing forces involved --- | ||
| - | has been inappropriate in most physics since 1905. This was called Dynamics. It was the fundamental insight | + | 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. | 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 | It remains a very good approximation for most practical purposes. Certain details of quantum mechanics | ||
| Line 51: | Line 50: | ||
| determined by dynamically changing forces, acceleration, momentum and such. They existed | 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 | in some form in classical Greek times. They can seem rather challenging in terms of our usual ideas of | ||
| - | causality, nevertheless be very useful and important. The cutting edge of physics today includes ideas that | + | 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 | 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, and at school we have barely started that journey in either field. | + | the physics up to now --- at school we have barely started that journey in either field. |
| ====Newton==== | ====Newton==== | ||
| Line 68: | Line 67: | ||
| \begin{align*}%% | \begin{align*}%% | ||
| \dot x &&& \text{is the rate of change of position (that is velocity),} \\ | \dot x &&& \text{is the rate of change of position (that is velocity),} \\ | ||
| - | \ddot x &&& \text{is the rate of change of rate of change of position (so acceleration),} \\ | + | \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 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.} | \dot{\boldsymbol{v}} &&& \text{is the same but as a vector, so no longer in just one dimension.} | ||
| Line 82: | Line 81: | ||
| \begin{align*}%% | \begin{align*}%% | ||
| - | \frac{{\rm d}y}{{\rm d}x} &&& \text{is the ratio of the instantaneous changes in two related values} \\ | + | \frac{{\rm d}y}{{\rm d}x} &&& \text{is the ratio of the instantaneous changes in two related values} \\ \\ |
| - | \frac{\rm d}{{\rm d}t}(4t^2+3) &&& \text{is between the values of the expression and the variable} \\ | + | \frac{\rm d}{{\rm d}t}(4t^2+3) &&& \text{is between the values of the expression and the variable} \\ \\ |
| - | \text{repeated:}\quad\frac{{\rm d}^2y}{{\rm d}x^2} &&& \text{meaning:}\quad\frac{\rm d}{{\rm d}x}\left(\frac{{\rm d}y}{{\rm d}x}\right) | + | \text{repeated:}\quad\frac{{\rm d}^2}{{\rm d}x^2}(3x^3+5) &&& \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 &&& \text{is the definite integral of the expression taken from }a\text{ to }b | ||
| %%\end{align*} | %%\end{align*} | ||
| The kind of calculus that we deal with at school treats each of these as a single symbol, but it does | The kind of calculus that we deal with at school treats each of these as a single symbol, but it does | ||
| Line 91: | Line 91: | ||
| technique. | technique. | ||
| ====Lagrange==== | ====Lagrange==== | ||
| - | ... was a century later, and considers //functions// as entities which can be operated on. This is a | + | ... was a century later, and considers //functions// as entities which can be operated on. He used the |
| - | distinct mathematical perspective, a language that facilitates a very different kind of thinking. | + | 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*}%% | \begin{align*}%% | ||
| Line 98: | Line 99: | ||
| {\rm f}'(t) &&& \text{the function that is the derivative of that function} \\ | {\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} \\ | {\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{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*} | %%\end{align*} | ||
teaching/topics/calculus/notation.1595849899.txt.gz · Last modified: 2020/07/27 21:38 by simon
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