To my understanding (heavily biased by physics), $\Delta$ is often and seemingly only used to eventually use the fundemental theorem of calculus. For example, let's say we have an arbitrary change in temperature. Then, we denote that as $\Delta{T}$. Then, after establishing this first case for a given situation, I often see the limit being taken of $\Delta$ quantities to create a $dT$ quantity (an infinitesimal) so we can integrate and etc.
Now, I understand that the fundamental theorem of calculus explicitly uses $\Delta$ along with the limit taking mentioned above. So, it seems like it is a serious mathematical concept, but using $\Delta$ feels very hand-wavey or imprecise, if that makes sense.
Any help or pointing to resources would be much appreciated.
Best Answer
$\Delta$ isn't really a concept, it's just a symbol. For example, the definition of a derivative can be written like so:
$$f'(x) = \lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x},$$
but in this equation, $\Delta$ is not really a concept, it's just part of the symboll combination "$\Delta x$" that is just some variable denoting a real number. That's all. The above equation is equally valid if I write it as
$$f'(x) = \lim_{\mathrm{elephant} \to 0} \frac{f(x+\mathrm{elephant}) - f(x)}{\mathrm{elephant}}.$$
That said, it's a symbol that usually denotes the change of... something. And as long as it is used unambiguously and clearly, the symbol $\Delta$ can make a lot of things easier to understand. For example, one can define a shorthand notation and say that for any function $f$ and any real value $\Delta x$, the function $\Delta f$ is the function $x\mapsto f(x+\Delta x) - f(x)$. Note that this notation is usually not explicitly defined in the text, but the definition is clear and unambiguous from context.
Also note that using this notation, the definition of the derivative becomes
$$f'(x) = \lim_{\Delta x \to 0}\frac{\Delta f(x)}{\Delta x}$$
which is nice because it really gives the intuition behind why $f'(x)$ is often written as $\frac{df}{dx}$.