Question

What does parametrization of a curve in vector calculus define? What does parametrization of a curve in vector calculus define? @Mathematics

Answer 1

It's a way of defining a curve in such a way that you can 'move' along the curve as a function of one variable, that you can think of as time.

The best way to think about this is through examples. Here's a parameterization of a circle, where you continue to go round and round the circle:

r(t) = (cos(t),sin(t))

Clearly this is a circle because with x = x(t) = cos(t) and y = y(t) = sin(t)
x^2 + y^2 = 1

Here's a parameterization of the circle where you go around 5 times as fast:

r(t) = (cos(5t),sin(5t))

And here's a parameterization of a spiral in the three dimensional space:

r(t) = (cos(t), sin(t), t)