CHAPTER 3 DIFFERENTIATION VECTORS OF  If a vector depends on a scalar variable t, then can be represented as and A A At t t is then said to be a function of and is called a parameter. In the component A form we write t iˆ t  t kˆ;    A A1 A2 A3 ˆj Differentiating , we get w r t At d dt dt dt d dt ˆ d  Since is itself a vector depending on t, we can consider its derivative At dt  d A 2 6uiˆ  6iˆ  24ukˆ w r t du 2 . If this derivative exists it is denoted by t  2 order derivatives are described. d 2 ˆj dt Example 3.1 2  3u 2u 12u If find   4iˆ  5  4u Au ˆj 2 3kˆ,  . In like manner higher At d A 2 2 kˆ d A du 2 u d A If we wanted these derivatives at u=2, then we would have obtained d A and d A 2 6iˆ 48kˆ   u2 du 2 Example 3.2 If , find    1ˆj tan A iˆ Int t kˆ 2 d A d A d A 2 2 , , 0  at t d A , dt dt dt 2 2 dt 0   kˆ ˆj 2t d A t 0 dt t
CHAPTER 3 DIFFERENTIATION OF VECTORS If a vector A depends on a scalar variable t, then A can be represented as At  and A is then said to be a function of t and t is called a parameter. In the component form we write A  A1 t iˆ  A2 t  ˆj  A3 t kˆ; Differentiating w r t , we get d dt At ˆ d dt dt dt Since d dt At  is itself a vector depending on t, we can consider its derivative w r t t . If this derivative exists it is denoted by order derivatives are described. Example 3.1 If Au  3u 2  4iˆ  2u 5 ˆj  4u 3kˆ, find d 2 A d A du 2 d A  6uiˆ  2 ˆj  12u 2 kˆ d 2 A  6iˆ  24ukˆ du 2 d 2 dt 2 At  . In like manner higher If we wanted these derivatives at u=2, then we would have obtained d A and d 2 A du 2 u2  6iˆ  48kˆ Example 3.2 If A  iˆ  Int 2  1ˆj tan t kˆ , find d A dt , d 2 A dt 2 , d A dt , d 2 A dt 2 at t  0 d A dt 2t t t 0   0 ˆj kˆ u