Chapter 13: Vector-Valued Functions and Motion in Space - Section 13.4 - Curvature and Normal Vectors of a Curve - Exercises 13.4 - Page 765: 4

$${\bf{T}} = \cos t{\bf{i}} + \sin t{\bf{j}},\,\,\,\,\,\,\,\,\,{\bf{N}} = - \sin t{\bf{i}} + \cos t{\bf{j}},\,\,\,\,\,\,\,\,\,\kappa = \frac{1}{t}$$

$$\eqalign{ & {\bf{r}}\left( t \right) = \left( {\cos t + t\sin t} \right){\bf{i}} + \left( {\sin t - t\cos t} \right){\bf{j}},\,\,\,\,\,\,\,\,\,\,\,t > 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[ {\left( {\cos t + t\sin t} \right){\bf{i}} + \left( {\sin t - t\cos t} \right){\bf{j}}} \right] \cr & {\bf{v}}\left( t \right) = \left( { - \sin t + t\cos t + \sin t} \right){\bf{i}} + \left( {\cos t + t\sin t - \cos t} \right){\bf{j}} \cr & {\bf{v}}\left( t \right) = t\cos t{\bf{i}} + t\sin t{\bf{j}} \cr & \cr & {\text{We calculate }}{\bf{T}}{\text{ from the velocity vector}} \cr & {\bf{v}}\left( t \right) = t\cos t{\bf{i}} + t\sin t{\bf{j}} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {{{\left( {t\cos t} \right)}^2} + {{\left( {t\sin t} \right)}^2}} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {{t^2}{{\cos }^2}t + {t^2}{{\sin }^2}t} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {{t^2}\left( {{{\cos }^2}t + {{\sin }^2}t} \right)} \cr & \left| {{\bf{v}}\left( t \right)} \right| = t \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{{t\cos t{\bf{i}} + t\sin t{\bf{j}}}}{t} \cr & {\bf{T}} = \cos t{\bf{i}} + \sin t{\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( {\cos t{\bf{i}} + \sin t{\bf{j}}} \right) \cr & \frac{{d{\bf{T}}}}{{dt}} = - \sin t{\bf{i}} + \cos t{\bf{j}} \cr & \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \sqrt {{{\left( { - \sin t} \right)}^2} + {{\left( {\cos t} \right)}^2}} \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \sqrt {{{\sin }^2}t + {{\cos }^2}t} \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = 1 \cr & {\bf{N}} = \frac{{d{\bf{T}}/dt}}{{\left| {d{\bf{T}}/dt} \right|}} = \frac{{ - \sin t{\bf{i}} + \cos t{\bf{j}}}}{1} \cr & {\bf{N}} = - \sin t{\bf{i}} + \cos t{\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}{t}\left( 1 \right) \cr & \kappa = \frac{1}{t} \cr} $$