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October 16

Interval (mathematics), Curve

How do you write down a curve interval in $\mathbb {R} ^{2}$ ? Could you writh For example This way $\{x^{2}:x\in [0,1]\}$ ? math graph. יהודה שמחה ולדמן (talk) 12:11, 16 October 2016 (UTC)[reply]

To define a curve in

\mathbb {R} ^{2}

you need to define a set of points where each point has two co-ordinates. One co-ordinate may be defined as a function of the other, for example

\{(x,x^{2}):x\in [0,1]\}

. Or each co-ordinate may be a function of a third parameter, for example

\{(\sin \theta ,\cos \theta ):\theta \in [0,2\pi )\}

. Gandalf61 (talk) 17:12, 16 October 2016 (UTC)[reply]

There are multiple solutions, due to the fact that a "curve" is not a really well-defined mathematical topic. We have an article on Path (topology) for curves that can be described by a moving point (simplifying a bit here). You could also be interested in our article on connected spaces (although it looks horribly formal for a notion that is intuitive at first).

This area is full of nontrivial mathematics. For example, if you draw a line on a sheet of paper and come back to your starting point without the line crossing itself, it defines an "inside" and an "outside", right? Well, that is what the Jordan curve theorem says, but it is relatively hard to prove. Tigraan^{Click here to contact me} 17:59, 16 October 2016 (UTC)[reply]