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a) K (representing kappa or curvature)=|rr''|/|r'|³ by definition.

The cross-product of the two vectors is given by the determinant:

i      k|

| f'  g'  0 | =

| f'' g'' 0 |

<0,0,f'g''-g'f''> which has magnitude |f'g''-g'f''|.

r'=<f',g',0> so |r'|=√((f')²+(g')²).

Therefore K=|f'g''-g'f''|/((f')²+(g')²)^(3/2).

b) f=tan⁻¹(sinh(t)), g=ln(cosh(t)),

f'=cosh(t)/(1+sinh²(t))=cosh(t)/cosh²(t)=sech(t),

f''=-sech(t)tanh(t);

g'=sinh(t)/cosh(t)=tanh(t),

g''=sech²(t).

|f'g''-g'f''|=|sech³(t)+sech(t)tanh²(t)|.

(f')²+(g')²=sech²(t)+tanh²(t)=1

Note: cosh²(t)-sinh²(t)=1,

1-tanh²(t)=sech²(t), hence sech²(t)+tanh²(t)=1.

Therefore K=|sech³(t)+sech(t)tanh²(t)|=|sech(t)|.

When t=ln(5), sinh(t)=(5-⅕)/2=12/5, cosh(t)=(5+⅕)/2=13/5,

tanh(t)=12/13, sech(t)=5/13.

Curvature K=5/13.

by Top Rated User (825k points)

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