Let u1, u2, u3, v1, v2, v3, w1, w2, and w3 be differentiablefunctions of t. Use Exercise

Chapter 12, Problem 55

(choose chapter or problem)

Let \(u_{1}, u_{2}, u_{3}, v_{1}, v_{2}, v_{3}, w_{1}, w_{2}\), and \(w_{3}\) be differentiable functions of \(t\) .

Use Exercise 54 to show that

\(\frac{d}{d t}\left|\begin{array}{ccc}u_{1} & u_{2} & u_{3} \\v_{1} & v_{2} & v_{3} \\w_{1} & w_{2} & w_{3}\end{array}\right|\)

\(=\left|\begin{array}{ccc}u^{\prime} 1 & u^{\prime} 2 & u^{\prime} 3 \\v_{1} & v_{2} & v_{3} \\w_{1} & w_{2} & w_{3}\end{array}\right|\)   \(+\left|\begin{array}{ccc}u_{1} & u_{2} & u_{3} \\v^{\prime}_{1} & v_{2}^{\prime} & v_{3}^{\prime} \\w_{1} & w_{2} & w_{3}\end{array}\right|\)    \(+\left|\begin{array}{ccc}u_{1} & u_{2} & u_{3} \\v_{1} & v_{2} & v_{3} \\w_{1}^{\prime} & w_{2}^{\prime} & w_{3}^{\prime}\end{array}\right|\)

Equation Transcription:

and

Text Transcription:

u_1,u_2,u_3,v_1,v_2,v_3,w_1,w_2,and w_3,

t

d/dt |_w_1 w_2 w_3 ^v_1 v_2 v_3 ^u_1 u_2 u_3|

= |_w_1 w_2 w_3 ^v_1 v_2 v_3 ^u’_1 u’_2 u’_3|

+|_w_1 w_2 w_3 ^v’_1 v’_2 v’_3 ^u_1 u_2 u_3|

+|_w’_1 w’_2 w’_3 ^v_1 v_2 v_3 ^u_1 u_2 u_3|