Chapter 13 - Vector Functions - Review - Concept Check - Page 921: 6

(a) The notion of curvature measures how sharply a curve bends. We would expect the curvature to be $0$ for a straight line , to be very small for curves which bend very little and to be large for curves which bend sharply.Its direction changes if the curve bends .The more the curve bends , the more the direction of the tangent vector will change. It is defined as $\kappa=||\frac{dT}{ds}||$ Here, $T$ represents a unit tangent vector. (b) From part (a), we have $\kappa=||\frac{dT}{ds}||$ Using chain rule of differentiation, we get $\frac{d}{dt}[u(f(t))]=f'(t)u'(f(t))$ $\kappa=||\dfrac{dT/dt}{ds/dt}||$ Since, $ds/dt=||r'(t)||$ Thus, $\kappa=||\dfrac{dT/dt}{r'(t)}||$ (c) A formula for curvature in terms of $r'(t)$ and $r''(t)$ is: $\kappa(t)=\dfrac{|r'(t) \times r''(t)|}{|r'(t)^3|}$ (d) A formula for the curvature of a plane with equation $y=f(x)$ is: $\kappa(t)=\dfrac{|f''(x)|}{[1+(f'(x))^2]^{\frac{3}{2}}}$

(a) The notion of curvature measures how sharply a curve bends. We would expect the curvature to be $0$ for a straight line , to be very small for curves which bend very little and to be large for curves which bend sharply.Its direction changes if the curve bends .The more the curve bends , the more the direction of the tangent vector will change. It is defined as $\kappa=||\frac{dT}{ds}||$ Here, $T$ represents a unit tangent vector. (b) From part (a), we have $\kappa=||\frac{dT}{ds}||$ Using chain rule of differentiation, we get $\frac{d}{dt}[u(f(t))]=f'(t)u'(f(t))$ $\kappa=||\dfrac{dT/dt}{ds/dt}||$ Since, $ds/dt=||r'(t)||$ Thus, $\kappa=||\dfrac{dT/dt}{r'(t)}||$ (c) A formula for curvature in terms of $r'(t)$ and $r''(t)$ is: $\kappa(t)=\dfrac{|r'(t) \times r''(t)|}{|r'(t)^3|}$ (d) A formula for the curvature of a plane with equation $y=f(x)$ is: $\kappa(t)=\dfrac{|f''(x)|}{[1+(f'(x))^2]^{\frac{3}{2}}}$