teaching:topics:calculus:notation
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teaching:topics:calculus:notation [2024/05/03 18:51] simon [Leibniz] |
teaching:topics:calculus:notation [2024/05/10 11:13] (current) simon [Lagrange] |
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| 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*} | ||
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