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teaching:topics:calculus:notation [2021/01/11 14:02] simon [some context ...] |
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 | ||
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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 ...==== | ||
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\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}^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) | + | \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 | ||
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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*}%% | ||
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{\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*} | ||