teaching:topics:calculus:notation
Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
teaching:topics:calculus:notation [2021/05/02 12:12] simon [Why three?] |
teaching:topics:calculus:notation [2024/05/10 11:13] (current) simon [Lagrange] |
||
|---|---|---|---|
| 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. It follows from the ideas of functions 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. | + | 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 79: | 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) \\ | ||
| - | \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*} | ||
| 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 94: | Line 96: | ||
| function notation introduced earlier in the 18th century by Euler and others. This is another distinct | 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. | 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) \\ | ||
| - | \text{or sometimes} \quad {\rm f}^{(-n)}(t) &&& \text{the }n\text{th antiderivative or intergal of }{\rm f}(t) | ||
| - | %%\end{align*} | ||
teaching/topics/calculus/notation.1619921551.txt.gz ยท Last modified: 2021/05/02 12:12 by simon
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 3.0 Unported
