what is a curve ? Is the concept of derivative limited to curves only?

Question

I am trying to understand derivative and I want to know intuitive and rigorous definitions for a curve and if derivative is lmited only to curves or not..

Answer 1

A curve is the graph of an application \begin{align*}\gamma :A\subset \mathbb R&\longrightarrow \mathbb R^n\\ t&\mapsto (\alpha_1(t),...,\alpha_n(t))\end{align*}

where \begin{align*} \alpha_i:A&\longrightarrow \mathbb R\\ t&\mapsto \alpha_i(t). \end{align*}

And yes, derivative are limited to curve. For a surface or any other space... you will talk about differential and not derivative. For a curve, the derivative and the differential coincident.

Answer 2

In elementary calculus context, there is no need to rigorously define what it mean by "curve". Intuitively you can think of a curve as an arbitrary line that can be drawn in one stroke; the simplest curve is a straight line.

The concept of derivative originates from the problem of finding the slope of a curve; you may imagine that, if you are not aware of any calculus, how you may solve that problem. By treating a curve as the graph of a "continuous" function, we can make mathematically rigorous what it means by the slope of a curve at a given point in terms of the derivative of the function: "the slope of a curve at a given point" is rephrased as "the slope of the tangent line to the graph of the function at the point".

Curves can be studied independently, but this is another story.