Week 4: Curvature and the unit normal vector N, Torsion and the unit binormal vector B, Planetary motion and satellites

Thomas Calculus ,11th edition, (pages: 936 - 959)

Sec 13.4 Curvature and The unit Normal Vector

In this section we study how a curve turns or bends. We look first at curves in the coordinate plane, and then at curves in space.

Sec 13.5 Torsion and the Unit Binormal Vector

If you are traveling along a space curve, the Cartesian i, j, and k coordinate system for representing the vectors describing your motion are not truly relevant to you. What is meaningful instead are the vectors representative of your forward direction (the unittangent vector T), the direction in which your path is turning (the unit normal vector N), and the tendency of your motion to “twist” out of the plane created by these vectors in the direction perpendicular to this plane (defined by the unit binormal vector ). Expressing the acceleration vector along the curve as a linear combination of this TNB frame of mutually orthogonal unit vectors traveling with the motion (Figure 13.25) is particularly revealing of the nature of the path and motion along it.

Sec 13.6 Planetary Motion and Satellites

In this section, we derive Kepler’s laws of planetary motion from Newton’s laws of motion and gravitation and discuss the orbits of Earth satellites. The derivation of Kepler’s laws from Newton’s is one of the triumphs of calculus. It draws on almost everything we have studied so far, including the algebra and geometry of vectors in space, the calculus of vector functions, the solutions of differential equations and initial value problems, and the polar coordinate description of conic sections.