Is not a space curve. It is a "space curve," ie, called a space curve. "Distance," "Tangent," "Normal," (& "BiNormal" in 3d) and "curvature" are defined mathematical operations which have their origin in and can potentially be interpreted as real quantities. A space curve can be created by assigning 1 to a distance on a piece of paper and assuming it is the unit of x and y. The space curve, Distance, Tangent, Normal, (& BiNormal in 3d), and curvature are then relative to and exist only on the piece of paper.
Because mathematicians teach math to engineers and physicists, the distinctions can become confusing. The physicists and engineers have the double burden of not only memorizing definitions, rules, and formulas, and being adept at their manipulation, but simultaneously creating some kind of real meaning to them in their minds. The result, sadly, is to cull out of potential engineers those who tend to think about the physical meaning of things, ie, potential engineers.
Re: Space Curves
I don't know why you posted that- I see no question. I would, in any case, disagree with most of what you said. "x= x(t), y= y(t)" does NOT define a "space curve", it defines a "plane curve". Space requires three dimensions: x= x(t), y= y(t), z= z(t). And we cannot create even a plane curve by just assigning "1 to a distance". You must have an entire coordinate system. In addition to distance you need to establish two distances as coordinate axes. (If we agree to use only "right hand Cartesian coordinate systems", an serious restriction for some purposes, we would only need to designate a single axis.)
And I don't understand why requiring people to "create some kind of real meaning" would tend to cull precisely those who "tend to think about the physical meaning". I would think that those who thought about such things would be best at assigning physical meaning to symbols, definitions, etc.
I wonder if you aren't referring to people who consider "physical meanings" so apparent they don't have to learn precise statements of definitions, rules, etc. But I would not call them "potential engineers', I would call them "sloppy engineers" and hope they will not be designing any bridges, railroads, etc. in an area where I might be.
Re: Space Curves
I used a 2d "space curve" simply as a matter of semantics because it's easier to talk about plotting a curve on a piece of paper than in a 3d system defined by a cardboard box. To plot a 3d curve by marking off a distance of 1 somewhere to define units on the axes of the box is much more difficult to do. You have to suspend pencil points inside the box (a wire). In either case, the resulting curve exists as a curve in space only as a plot in the coordinate system.
As for intuition, the meticulous, tortuous, abstract development of something on a level with Rudin is useless when the rubber hits the road. The difficult question for most partial differential equations occurrs when talking about functions such as temperature defined on closed real bodies. In this case, "Analysis" doesn't even address defining and integrating partial derivatives up to and on a boundary or directional derivatives on a boundary, only the case where the closed "volume" is inside of a domain so that everything is defined up to, on, and past, the boundary- the trivial case for engineering science and much of physics. Then analysis presumes to deal with equations of science which it can neither define or prove, and then comes up with classical engineering and physics solutions (solved by intuition and insight) using abstract notation as if it had solved it using “analysis.” And the Engineers are sloppy?
Rudin gives an abstract definition and derivation of Stoke’s theorem, starting with the assumption that the “volume” is inside a domain, which is the trivial and uninteresting case because now you don’t have to think about partial derivatives in the context of a boundary, the real world for science. Amazingly, Rudin gives a 1-dim example of his proof as the integral from 0 to 1 of a continuously differentiable function on the closed interval [0,1], which is exactly the case which the whole theorem and proof avoids because it can’t deal with it, except in the 1-dim case. The true 1-dim example of his theorem and proof is the integral from 0 to 1 for f on the open interval (0,1). I believe the mistake was intentional, to imply that you really don’t have to wade through his proof which, as you can "see" from his “1-d case,” applies to partial differential equations of science in general.
I don't advocate being sloppy, I advocate being precise and clear. I don't consider abstraction (functional analysis symbols in a calculus proof, for ex) to be synonomous with precision.