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I want to gather the various concepts in one place, to reveal the similarities between them, as they are often hidden by the serial nature of a curriculum.
There are many terms and special cases, which deal with the process of differentiation. The basic idea, however, is the same in all cases: something non-linear, as for instance a multiplication, is approximated by something linear which means, by something we can add.
This reveals already two major consequences: addition is easier than multiplication, that's why we consider it at all, and as an approximation, it is necessarily a local property around the point of consideration.
Thus the result of our differentiation should always be a linear function like the straight lines we draw in graphs and call them tangents.
And our approximation will get worse the farther away we are from the point we considered. That's the reason why these ominous infinitesimals come into play. They are nothing obscure, but merely an attempt to quantify and get a hand on the small deviations of our linear approximation to what is really going on.
To begin with, let's clarify the language:
\begin{align*}
\textrm{differentiation} &- \textrm{certain process to achieve a linear approximation}\\
\textrm{to differentiate} &- \textrm{to proceed a differentiation}\\
\textrm{differential} &-\textrm{infinitesimal linear change of the function value only}\\
\textrm{differentiability}&-\textrm{condition that allows the process of differentiation}\\
\textrm{derivative}&-\textrm{result of a differentiation}\\
\textrm{derivation}&-\textrm{linear mapping that obeys the Leibniz rule}\\
\textrm{to derivate}&- \textrm{to deduce a statement by logical means}
\end{align*}
All these terms are context-sensitive and their meanings change, if they are used, e.g. in chemistry, mechanical engineering, or common language. But even within mathematics, the terms may vary among different authors. E.g. differential has two meanings, as an adjective or as notation for df, i.e. the infinitesimal linear change on the function values. Differentials are used in various applications with varying meanings and even with different mathematical rigor. This is essentially true in calculus where ##\int f(x)dx## and ##\frac{df(x)}{dx}## is only of notational value. The most precise meaning of the term can be found in differential geometry as an exact ##1-##form.
As a thumb rule might serve: diff... refers to the process, derivative to the result.
As differentiability is a local property, it is defined on a domain ##U## which is open, not empty, and connected, at a point ##x_0## or ##z_0## in ##U##. I will not mention these requirements every time I use them. They are helpful, as one doesn't have to deal with isolated points or the behavior of a function on boundaries and one always has a way to approach ##x_0## from all sides. So with respect to the approximation which is intended, they come in naturally. I also won't distinguish between approximations from the left or from the right, since this article is only an overview. So it is always meant as identical from both sides. Moreover, a function is said to be in ##C(U)=C^0(U)## if it is continuous, in ##C^n(U)## if it is ##n-##times continuously differentiable, and in ##C^\infty(U)=\bigcap_{n\in \mathbb{N}}C^n(U)## if it is infinitely many times continuously differentiable. The latter functions are also called smooth.
Continue reading ...
There are many terms and special cases, which deal with the process of differentiation. The basic idea, however, is the same in all cases: something non-linear, as for instance a multiplication, is approximated by something linear which means, by something we can add.
This reveals already two major consequences: addition is easier than multiplication, that's why we consider it at all, and as an approximation, it is necessarily a local property around the point of consideration.
Thus the result of our differentiation should always be a linear function like the straight lines we draw in graphs and call them tangents.
And our approximation will get worse the farther away we are from the point we considered. That's the reason why these ominous infinitesimals come into play. They are nothing obscure, but merely an attempt to quantify and get a hand on the small deviations of our linear approximation to what is really going on.
To begin with, let's clarify the language:
\begin{align*}
\textrm{differentiation} &- \textrm{certain process to achieve a linear approximation}\\
\textrm{to differentiate} &- \textrm{to proceed a differentiation}\\
\textrm{differential} &-\textrm{infinitesimal linear change of the function value only}\\
\textrm{differentiability}&-\textrm{condition that allows the process of differentiation}\\
\textrm{derivative}&-\textrm{result of a differentiation}\\
\textrm{derivation}&-\textrm{linear mapping that obeys the Leibniz rule}\\
\textrm{to derivate}&- \textrm{to deduce a statement by logical means}
\end{align*}
All these terms are context-sensitive and their meanings change, if they are used, e.g. in chemistry, mechanical engineering, or common language. But even within mathematics, the terms may vary among different authors. E.g. differential has two meanings, as an adjective or as notation for df, i.e. the infinitesimal linear change on the function values. Differentials are used in various applications with varying meanings and even with different mathematical rigor. This is essentially true in calculus where ##\int f(x)dx## and ##\frac{df(x)}{dx}## is only of notational value. The most precise meaning of the term can be found in differential geometry as an exact ##1-##form.
As a thumb rule might serve: diff... refers to the process, derivative to the result.
As differentiability is a local property, it is defined on a domain ##U## which is open, not empty, and connected, at a point ##x_0## or ##z_0## in ##U##. I will not mention these requirements every time I use them. They are helpful, as one doesn't have to deal with isolated points or the behavior of a function on boundaries and one always has a way to approach ##x_0## from all sides. So with respect to the approximation which is intended, they come in naturally. I also won't distinguish between approximations from the left or from the right, since this article is only an overview. So it is always meant as identical from both sides. Moreover, a function is said to be in ##C(U)=C^0(U)## if it is continuous, in ##C^n(U)## if it is ##n-##times continuously differentiable, and in ##C^\infty(U)=\bigcap_{n\in \mathbb{N}}C^n(U)## if it is infinitely many times continuously differentiable. The latter functions are also called smooth.
Continue reading ...
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