Chapter 13: Vector-Valued Functions and Motion in Space - Section 13.4 - Curvature and Normal Vectors of a Curve - Exercises 13.4 - Page 765: 15

$${\bf{T}} = {\text{sech}}\left( {\frac{t}{a}} \right){\bf{i}} + \tanh \left( {\frac{t}{a}} \right){\bf{j}},\,\,\,\,\,\,{\bf{N}} = - \tanh \left( {\frac{t}{a}} \right){\bf{i}} + {\text{sech}}\left( {\frac{t}{a}} \right){\bf{j}},\,\,\,\,\,\,\,\kappa = \frac{1}{a}{\text{sec}}{{\text{h}}^2}\left( {\frac{t}{a}} \right)$$

$$\eqalign{ & {\bf{r}}\left( t \right) = t{\bf{i}} + \left( {a\cosh \left( {t/a} \right)} \right){\bf{j}},\,\,\,\,\,\,\,\,a > 0 \cr & {\text{Calculate }}{\bf{v}}\left( t \right).{\text{ Use }}{\bf{v}}\left( t \right) = {\bf{r}}'\left( t \right) \cr & {\bf{v}}\left( t \right) = {\bf{r}}'\left( t \right) = \frac{d}{{dt}}\left[ {t{\bf{i}} + \left( {a\cosh \left( {t/a} \right)} \right){\bf{j}}} \right] \cr & {\bf{v}}\left( t \right) = {\bf{i}} + a\sinh \left( {t/a} \right)\left( {1/a} \right){\bf{j}} \cr & {\bf{v}}\left( t \right) = {\bf{i}} + \sinh \left( {t/a} \right){\bf{j}} \cr & \cr & {\text{We calculate }}{\bf{T}}{\text{ from the velocity vector}} \cr & {\bf{v}}\left( t \right) = {\bf{i}} + \sinh \left( {t/a} \right){\bf{j}} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {{{\left( 1 \right)}^2} + {{\left( {\sin \left( {t/a} \right)} \right)}^2}} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {1 + {{\sin }^2}\left( {t/a} \right)} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {{{\cosh }^2}\left( {t/a} \right)} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \cosh \left( {t/a} \right) \cr & \cr & {\text{use }}{\bf{T}}\left( t \right) = \frac{{{\bf{v}}\left( t \right)}}{{\left| {{\bf{v}}\left( t \right)} \right|}} \cr & {\bf{T}} = \frac{{{\bf{i}} + \sinh \left( {1/a} \right){\bf{j}}}}{{\cosh \left( {1/a} \right)}} \cr & {\bf{T}} = {\text{sech}}\left( {t/a} \right){\bf{i}} + \tanh \left( {t/a} \right){\bf{j}} \cr & \cr & {\text{Calculate }}{\bf{N}}\left( t \right){\text{ using the equation }}{\bf{N}} = \frac{{d{\bf{T}}/dt}}{{\left| {d{\bf{T}}/dt} \right|}}{\text{ }}\left( {{\text{see page 764}}} \right).{\text{ Then}} \cr & \frac{{d{\bf{T}}}}{{dt}} = \frac{d}{{dt}}\left( {{\text{sech}}\left( {t/a} \right){\bf{i}} + \tanh \left( {t/a} \right){\bf{j}}} \right) \cr & \frac{{d{\bf{T}}}}{{dt}} = - {\text{sech}}\left( {t/a} \right)\tanh \left( {t/a} \right)\left( {1/a} \right){\bf{i}} + {\text{sec}}{{\text{h}}^2}\left( {t/a} \right)\left( {1/a} \right){\bf{j}} \cr & \frac{{d{\bf{T}}}}{{dt}} = - \left( {1/a} \right){\text{sech}}\left( {t/a} \right)\tanh \left( {t/a} \right){\bf{i}} + \left( {1/a} \right){\text{sec}}{{\text{h}}^2}\left( {t/a} \right){\bf{j}} \cr & \cr & and \cr & \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \sqrt {{{\left[ { - \left( {1/a} \right){\text{sech}}\left( {t/a} \right)\tanh \left( {t/a} \right)} \right]}^2} + {{\left[ {\left( {1/a} \right){\text{sec}}{{\text{h}}^2}\left( {t/a} \right)} \right]}^2}} \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \sqrt {\frac{1}{{{a^2}}}{\text{sec}}{{\text{h}}^2}\left( {\frac{t}{a}} \right){{\tanh }^2}\left( {\frac{t}{a}} \right) + \frac{1}{{{a^2}}}{\text{sec}}{{\text{h}}^4}\left( {\frac{t}{a}} \right)} \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \sqrt {\frac{1}{{{a^2}}}{\text{sec}}{{\text{h}}^2}\left( {\frac{t}{a}} \right)\left( {{{\tanh }^2}\left( {\frac{t}{a}} \right) + {\text{sec}}{{\text{h}}^2}\left( {\frac{t}{a}} \right)} \right)} \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \sqrt {\frac{1}{{{a^2}}}{\text{sec}}{{\text{h}}^2}\left( {\frac{t}{a}} \right)\left( 1 \right)} \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \frac{1}{a}{\text{sech}}\left( {\frac{t}{a}} \right) \cr & \cr & {\bf{N}} = \frac{{d{\bf{T}}/dt}}{{\left| {d{\bf{T}}/dt} \right|}} = \frac{{ - \left( {1/a} \right){\text{sech}}\left( {t/a} \right)\tanh \left( {t/a} \right){\bf{i}} + \left( {1/a} \right){\text{sec}}{{\text{h}}^2}\left( {t/a} \right){\bf{j}}}}{{\left( {1/a} \right){\text{sech}}\left( {t/a} \right)}} \cr & {\bf{N}} = - \tanh \left( {t/a} \right){\bf{i}} + {\text{sech}}\left( {t/a} \right){\bf{j}} \cr & \cr & {\text{Calculate }}\kappa {\text{ using the equation }}\kappa = \frac{1}{{\left| {\bf{v}} \right|}}\left| {\frac{{d{\bf{T}}}}{{dt}}} \right|{\text{ }}\left( {{\text{see page 764}}} \right).{\text{ Then}} \cr & \kappa = \frac{1}{{\left| {\bf{v}} \right|}}\left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \frac{1}{{\cosh \left( {t/a} \right)}}\left( {\frac{1}{a}{\text{sech}}\left( {\frac{t}{a}} \right)} \right) \cr & \kappa = \frac{1}{a}{\text{sec}}{{\text{h}}^2}\left( {\frac{t}{a}} \right) \cr} $$