Answer to Question #271916 in Differential Geometry | Topology for Angel Nodado

Question #271916

Find the curvature, the radius of curvature and the center of curvature having a parametric equations at the given point:



y=e^x , (0,1)


1
Expert's answer
2021-12-01T14:04:55-0500

Solution;

y=exy=e^x

P(x0,y0)=(0,1)P(x_0,y_0)=(0,1)

y=exy'=e^x

y(0)=1y'(0)=1

y=exy''=e^x

y(0)=1y''(0)=1

Radius of curvature;

R=(1+y(x0)2)32y(x0)R=\frac{(1+y'(x_0)^2)^{\frac32}}{y''(x_0)}

By substitution ;

R=(1+1)321=23=22R=\frac{(1+1)^{\frac32}}{1}=\sqrt{2^3}=2\sqrt2

The curvature;

k=1R=122k=\frac1R=\frac{1}{2\sqrt2}

The centre of curvature;

Let P be the center of curvature , given as (a,b)

a=xodydx[1+(dydx)2d2ydx2a=x_o-\frac{\frac{dy}{dx}[1+(\frac{dy}{dx})^2}{\frac{d^2y}{dx^2}}

a=0+e2xex=e0=1a=0+\frac{e^{2x}}{e^x}=e^0=1

b=yo+1+(dydx)2d2ydx2b=y_o+\frac{1+(\frac{dy}{dx})^2}{\frac{d^2y}{dx^2}}

b=1+1+e2xex=1+2=3b=1+\frac{1+e^{2x}}{e^x}=1+2=3

Therefore;

P=(1,3)


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