teaching:topics:calculus:notation
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teaching:topics:calculus:notation [2021/01/11 14:07] simon [Lagrange] |
teaching:topics:calculus:notation [2024/05/10 11:13] (current) simon [Lagrange] |
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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 80: | Line 79: | ||
| many of the other fields where calculus is used. It is a more mathematical perspective. It was developed | many of the other fields where calculus is used. It is a more mathematical perspective. It was developed | ||
| simultaneously with Newton's notation. | 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 that ratio but 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 between those values. | ||
| - | \begin{align*}%% | ||
| - | \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} \\ | ||
| - | \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) | ||
| - | %%\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 | ||
| make sense to break them down into their parts, considering say \(\ {\rm d}x\ \) on its own. Doing so | make sense to break them down into their parts, considering say \(\ {\rm d}x\ \) on its own. Doing so | ||
| Line 91: | Line 93: | ||
| technique. | technique. | ||
| ====Lagrange==== | ====Lagrange==== | ||
| - | ... was a century later, and considers //functions// as entities which can be operated on. He used function | + | ... was a century later, and considers //functions// as entities which can be operated on. He used the |
| - | notation introduced earlier in the 18th century by Euler and others. This is another distinct mathematical | + | function notation introduced earlier in the 18th century by Euler and others. This is another distinct |
| - | perspective, and a language that facilitates a very different kind of thinking. | + | mathematical perspective, and a language that facilitates a very different kind of thinking. |
| + | \[{\rm f}(t)\] | ||
| + | a function with values that depend on the variable `t`, | ||
| + | \[{\rm f}'(t)\] | ||
| + | the function that is the derivative of that function, | ||
| + | \[{\rm f}^{\prime\prime}(t)\] | ||
| + | the function that is the derivative of that derivative function, | ||
| + | \[\text{and even} \quad {\rm f}^{(n)}(t)\] | ||
| + | the function that is the `n`th derivative of \({\rm f}(t),\) | ||
| + | \[\text{or sometimes} \quad {\rm f}^{(-n)}(t)\] | ||
| + | the `n`th antiderivative or indefinite integral of \({\rm f}(t).\) | ||
| - | \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) | ||
| - | %%\end{align*} | ||
teaching/topics/calculus/notation.1610334431.txt.gz · Last modified: 2021/01/11 14:07 by simon
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