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Question Number 3693 by Rasheed Soomro last updated on 19/Dec/15

Every equation of x,y has a curve in a plane. Does every curve in a plane has an equation?

$$\mathcal{E}{very}\:{equation}\:{of}\:{x},{y}\:{has}\:{a}\:{curve}\:{in}\:{a}\:{plane}. \\ $$$$\mathcal{D}{oes}\:{every}\:{curve}\:{in}\:{a}\:{plane}\:{has}\:{an}\:{equation}? \\ $$

Commented by 123456 last updated on 19/Dec/15

i think the answer is yes lets f:R^2 →R a curvw can be given by f(x,y)=k,k∈R what i think that can happen is that the function f(x,y) is so hard to be given in some cases

$$\mathrm{i}\:\mathrm{think}\:\mathrm{the}\:\mathrm{answer}\:\mathrm{is}\:\mathrm{yes} \\ $$$$\mathrm{lets}\:{f}:\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R} \\ $$$$\mathrm{a}\:\mathrm{curvw}\:\mathrm{can}\:\mathrm{be}\:\mathrm{given}\:\mathrm{by} \\ $$$${f}\left({x},{y}\right)={k},{k}\in\mathbb{R} \\ $$$$\mathrm{what}\:\mathrm{i}\:\mathrm{think}\:\mathrm{that}\:\mathrm{can}\:\mathrm{happen}\:\mathrm{is}\:\mathrm{that} \\ $$$$\mathrm{the}\:\mathrm{function}\:{f}\left({x},{y}\right)\:\mathrm{is}\:\mathrm{so}\:\mathrm{hard}\:\mathrm{to}\:\mathrm{be}\:\mathrm{given} \\ $$$$\mathrm{in}\:\mathrm{some}\:\mathrm{cases} \\ $$

Commented by Rasheed Soomro last updated on 19/Dec/15

If we draw a curve with our hand in a plane, must it have an equation(theoretically)?

$$\mathcal{I}{f}\:\:{we}\:{draw}\:{a}\:{curve}\:{with}\:{our}\:{hand}\:{in} \\ $$$${a}\:{plane},\:{must}\:{it}\:{have}\:{an}\:{equation}\left({theoretically}\right)? \\ $$

Commented by 123456 last updated on 19/Dec/15

yes, can simple take f(x,y)= { (0,((x,y)∈A)),(1,((x,y)∉A)) :} and drawn f(x,y)=0 A⊂R^2 its look fine, but it depends of how you describe it, the circle of radius r with center in (0,0) can be given in cartesian by x^2 +y^2 =r^2 or in polar by r(θ)=r note that in the second case its more simple, so there are curve that can be easy given in one coordinate system than other

$$\mathrm{yes},\:\mathrm{can}\:\mathrm{simple}\:\mathrm{take} \\ $$$${f}\left({x},{y}\right)=\begin{cases}{\mathrm{0}}&{\left({x},{y}\right)\in{A}}\\{\mathrm{1}}&{\left({x},{y}\right)\notin{A}}\end{cases} \\ $$$$\mathrm{and}\:\mathrm{drawn} \\ $$$${f}\left({x},{y}\right)=\mathrm{0} \\ $$$${A}\subset\mathbb{R}^{\mathrm{2}} \\ $$$$\mathrm{its}\:\mathrm{look}\:\mathrm{fine},\:\mathrm{but}\:\mathrm{it}\:\mathrm{depends}\:\mathrm{of}\:\mathrm{how}\:\mathrm{you} \\ $$$$\mathrm{describe}\:\mathrm{it},\:\mathrm{the}\:\mathrm{circle}\:\mathrm{of}\:\mathrm{radius}\:{r}\:\mathrm{with} \\ $$$$\mathrm{center}\:\mathrm{in}\:\left(\mathrm{0},\mathrm{0}\right)\:\mathrm{can}\:\mathrm{be}\:\mathrm{given}\:\mathrm{in}\:\mathrm{cartesian} \\ $$$$\mathrm{by} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} ={r}^{\mathrm{2}} \\ $$$$\mathrm{or}\:\mathrm{in}\:\mathrm{polar}\:\mathrm{by} \\ $$$${r}\left(\theta\right)={r} \\ $$$$\mathrm{note}\:\mathrm{that}\:\mathrm{in}\:\mathrm{the}\:\mathrm{second}\:\mathrm{case}\:\mathrm{its}\:\mathrm{more} \\ $$$$\mathrm{simple},\:\mathrm{so}\:\mathrm{there}\:\mathrm{are}\:\mathrm{curve}\:\mathrm{that}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{easy}\:\mathrm{given}\:\mathrm{in}\:\mathrm{one}\:\mathrm{coordinate}\:\mathrm{system}\:\mathrm{than} \\ $$$$\mathrm{other} \\ $$

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