I am reading "Calculus Metric Version 7th Edition International Edition" by James Stewart.
On p.879, Stewart wrote as follows:
“It is often useful to parametrize a curve with respect to arc length because arc length arises naturally from the shape of the curve and does not depend on a particular coordinate system.”
I cannot understand why arc length does not depend on a particular coordinate system.
I cannot understand even what the sentence "something does not depend on a particular coordinate system" means precisely.
What is a curve?
Does a curve exist independently without a coordinate system?
I think a curve itself depends on a particular coordinate system.
When we define a curve, we always use a particular coordinate system.
The curve is itself a geometrical object, existing in space, now to describe it we put numbers at each point in the space (coordinates) , so we can distinguish here from there quantitatively.
Usually when we learn calculus, we work from functions to seeing they have curves as representation but here we work backwards, we see curves and think about what kind of functions they will be represented by. In this case, it is not good to talk about the curve in an ambient setting with coordinates like (x,y,z), it would be best to describe the curve by parameters intrinsic to itself.
So, how do we achieve coordinate independent quantities? Vectors and tensors. For a more elaborate write up, refer Pavel Grinfeld's Tensor calculus book. It may sound out of reach but it's written for undergraduate who wish to learn the 'invariance'/ geometrical property of the vector derivatives.