Answer to Question #117667 in Differential Geometry | Topology for Nouhad

Question #117667
Let E be a Euclidian vector space, let (i,j) be its standard basis. Note C
the circle centered at the origin with radius a, where a is a real positive number. ⃗⃗
1. Let δ = (a cos as )i+(a sin as )j. Show that ([0, 2πa], δ) is a unit speed parametrization of C
1
Expert's answer
2020-05-24T15:08:52-0400

First, the equation of a circle centered at the origin "p.O\\left(0;0\\right)" and radius "R=a>0" has the form



"x^2+y^2=a^2"

For example, "a=3"





Secondly, from physics we know the relationship between the velocity vector and the radius vector



"\\overrightarrow{r}=x(t)\\cdot\\overrightarrow{i}+y(t)\\cdot\\overrightarrow{j}\\longrightarrow\\\\[0.3cm]\n\\overrightarrow{v}=\\frac{d\\overrightarrow{r}}{dt}=\\frac{dx(t)}{dt}\\cdot\\overrightarrow{i}+\\frac{dy(t)}{dt}\\cdot\\overrightarrow{j}\\\\[0.3cm]\n\\overrightarrow{v}=v_x(t)\\cdot\\overrightarrow{i}+v_y(t)\\cdot\\overrightarrow{j}\\longrightarrow\\\\[0.3cm]\n\\left|\\overrightarrow{v}\\right|=\\sqrt{v_x^2+v_y^2}"

In our case, the natural variable "t-" time was replaced by the general variable "s" .


"\\overrightarrow{v}\\equiv\\overrightarrow{\\delta}(s)=\\underbrace{\\left(a\\cos as\\right)}_{\\delta_x(s)}\\cdot\\overrightarrow{i}+\\underbrace{\\left(a\\sin as\\right)}_{\\delta_y(s)}\\cdot\\overrightarrow{i}\\longrightarrow\\\\[0.3cm]\n\\left|\\overrightarrow{\\delta}(s)\\right|=\\sqrt{\\delta_x^2(s)+\\delta_y^2(s)}=\\sqrt{\\left(a\\cos as\\right)^2+\\left(a\\sin as\\right)^2}=\\\\[0.3cm]\n=\\sqrt{a^2\\left(\\cos^2as+\\sin^2as\\right)}=a\\cdot\\sqrt{1}=a"

Conclusion,



"\\left|\\overrightarrow{\\delta(s)}\\right|=\\left\\{\\begin{array}{l}\n\\text{is a unit speed},\\quad\\text{if}\\quad a=1\\\\[0.3cm]\n\\text{is not a unit speed},\\quad\\text{if}\\quad a\\neq1\n\\end{array}\\right."


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