What is the exact definition of a differential? In physics I have seen that they are taken as very small quantities but in mathematics I think it is not so.
Differential in calculus
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derivatives
1 Answers
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This is the definition provided in the book Analysis I by Vladimir A. Zorich:
Definition. A function $f: E \to \mathbb{R}$ defined on a set $E \subseteq \mathbb{R}$ is differentiable at a point $a \in E$ that is a limit point of $E$ if there exists a linear function $A(x - a)$ of the increment $x - a$ of the argument such that $f(x) - f(a)$ can be represented as $$f(x) - f(a) = A(x - a) + o(x - a) \qquad \text{as } x \to a, x \in E$$ The linear function $A(x - a)$ is called the differential of the function $f$ at $a$.
The mathematical way of formalizing "very small quantities" is via limits which is reflected in the use of the small $o$-notation in the definition of the differential.