Curves in mathematics

Question

I am coming from a physics background. I am trying to understand the algebraic formulations of curves. For example: if we have a curve $\Gamma(t)$ = {$\gamma(t,s), s \in [0,1] $} where $s$ seems to be the curvilinear abscissa. What is exactly the difference between $\Gamma$ and $\gamma$? Why do we need other parameters ($\gamma$ and $s$) to define the curve? This makes it confusing for me about the exact definition of the curve. If I want to search for an online course that deals with curves in terms of algebra, what specific domain I should choose?

Answer 1

If you vary the time coordinate $t$, the curve $\Gamma$ deforms; if you vary the abscissa $s$, you move along the curve.

Suppose I move and bend a wire with my hands while using a third hand (pretend I have that many) to move a ring up and down its length. The former task causes the ring's position $\gamma$ to be $t$-dependent; the latter causes it to be $s$-dependent.

A physicist would argue this experiment causes $s$ to be a function of $t$, and hence reduces $\gamma$ to a function of $t$ alone. We can of course describe this in terms of the chain rule$$\frac{d\gamma}{dt}=\frac{\partial\gamma}{\partial t}+\frac{ds}{dt}\frac{\partial\gamma}{\partial s}.$$But the curve $\Gamma$ doesn't care about this (after all, every point on the curve can be endowed with its own right).