Lecture 8, Friday, 9/12/08: Section 11.5: Tangent, normal, binormal vectors, tangential and normal components of acceleration

Topics

The TNB frame: Unit tangent vector T, principal unit normal vector N, binormal vector B
Tangential and normal components of acceleration, a_T, and a_N: Formulas in terms of s', s'' and curvature K (5.7); and relation between the two components given by Pythagoras theorem (5.8).

Read

Section 11.5. Optional: The part on Kepler's Law. This is an important application of the concepts developed in this chapter, but it is rather involved and would take us too far astray. Read it if you are interested.

Homework

Section 11.5: 1, 3, 9, 15, 17, 23, 29, 33, 34. Solutions.

Notes

2-dimensional versus 3-dimensional (i.e., space) curves: The definitions of the vectors T and N make perfect sense for 2-dimensional (as well as, of course, 3-dimensional) curves. However, the binormal vector B is only defined for 3-dimensional curves. This is because its definition of B involves a cross product, and cross products are only defined for 3-dimensional vectors. For similar reasons, the cross product formula for the curvature makes only sense in 3 dimensions, but the two other formulas (in terms of T) make sense in 2 dimensions.
Section 11.6. This section (like 10.6) is independent of the material of Chapters 10 and 11. I will not cover it at this point, and may get back to it at the end of the semester.

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Last modified Mon 22 Sep 2008 02:20:59 PM CDT