Chapter 12

1. Section 12.1: Vector-Values Functions
   • Define a vector values function
     r(t) = f(t)i +g(t)j
     r(t) = f(t)i +g(t)j +h(t)k
   • Note the curve is not only the set of points but the manner in which they are traced.
   • http://cs.jsu.edu/~leathrum/Mathlets/parapath.html
   • Limits of vector valued function: for `bbr(t) = f(t)bbi + g(t)bbj`
     
     for `bbr(t)= g(t)bbi +h(t)bbj + h(t)bbk`
     `lim_(t->a)bbr(t)= [lim_(t->a)f(t)]bbi+[lim_(t->a)g(t)]bbj+[lim_(t->a)h(t)]bbk `
   • Continuity at a point
     
   • Continuity on an interval
2. Section 12.2: Differentiation and Integration of Vector-Values Functions
   • Differentiation of vector-values Functions
   • Definition
     
   • for `bbr(t) = f(t)bbi + g(t)bbj` we have `bbr^'(t) = f^'(t)bbi + g^'(t)bbj`, and
     for `bbr(t) = f(t)bbi + g(t)bbj+ h(t)bbk` we have `bbr^'(t) = f^'(t)bbi + g^'(t)bbj+ h^'(t)bbk`
   • Higher order: r', r'', etc.
   • Properties:
     1. D_t[ cr(t) ] = cr'(t)
     2. D_t[ r(t) ± u(t) ] = r'(t) ± u'(t)
         
     3. D_t[ w(t) r(t) ] = w(t) r'(t) + w'(t) r(t)
     4. D_t[ r(t) • u(t) ] = r(t) • u'(t) + r'(t) • u(t)
     5. D_t[ r(t)  u(t) ] = r(t)  u'(t) + r'(t)  u(t)
         
     6. D_t[ r( w(t) ) ] = r'( w(t) ) w'(t)
     7. If r(t)•r(t) = c, then r(t) • r'(t) = 0
   • Indefinte integral
     
   • Definite integral
     
   • The anti-derivative
     
3. Section 12.3: Velocity and Acceleration
   • Position given as r(t) =x(t)i + y(t)j
   • Velocity given as v(t) = r'(t) = x'(t)i +y'(t)j
   • Acceleration given as: a(t) = v'(t) = r''(t) = x''(t)i +y''(t)j
   • Speed given as: s(t) = ||v(t)|| = ||r'(t)|| = sqrt( [(x'(t)]² + [y'(t)]² )
      
   • Position for a projectile: