teaching:topics:calculus:notation
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teaching:topics:calculus:notation [2021/05/02 12:03] simon [Leibniz] |
teaching:topics:calculus:notation [2024/05/03 18:51] (current) simon [Leibniz] |
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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. It follows from the ideas of functions developed soon after calculus and the notation that Euler used for functions and his notation which has an explicit differential operator. An analogy here could be the difference between procedural and functional computer programming paradigms. | + | 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 ...==== | ====some context ...==== | ||
| Line 81: | 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}^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} | + | \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 | ||
teaching/topics/calculus/notation.1619921011.txt.gz · Last modified: 2021/05/02 12:03 by simon
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