Use the result of Exercise 27 of §1.4 to show that N×B = T and B×T = N. As a resul...

Use the result of Exercise 27 of §1.4 to show that

N×B = T and B×T = N.

As a result, we can arrange T, N, and B in a circle so that they correspond, respectively, to the vectors i, j, k appearing in Figure 1.54 and so that we may use a mnemonic for identifying cross products that is similar to the one described in Example 1 of §1.4. Let x be a path of class C3, parametrized by arclength s, with x’ ×x”≠ 0. We define the Darboux rotation vector (also called the angular velocity vector) by

1.4

1.4

Use the result of Exercise 27 of §1.4 to show that

N×B = T and B×T = N.

As a result, we can arrange T, N, and B in a circle so that they correspond, respectively, to the vectors i, j, k appearing in Figure 1.54 and so that we may use a mnemonic for identifying cross products that is similar to the one described in Example 1 of §1.4. Let x be a path of class C3, parametrized by arclength s, with x’ ×x”≠ 0. We define the Darboux rotation vector (also called the angular velocity vector) by

1.4

1.4