4100 AWL/Thomas_ch16p1143-1228

4100 AWL/Thomas_ch15p1067-1142

4100 AWL/Thomas_ch13p906-964

As a particle moves along a smooth curve in the plane, turns as the curve bends. Since T is a unit vector, its length remains constant and only its direction changes as the particle moves along the curve. The rate at which T turns per unit of length along the curve is called the curvature (Figure 13.19). The traditional symbol for the curvature function is the Greek letter (“kappa”). k T = dr>d...

4100 AWL/Thomas_ch12p848-905

In the calculus of functions of a single variable, we used our knowledge of lines to study curves in the plane. We investigated tangents and found that, when highly magnified, differentiable curves were effectively linear. To study the calculus of functions of more than one variable in the next chapter, we start with planes and use our knowledge of planes to study the surfaces that are the grap...

4100 AWL/Thomas_ch13p906-964

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 unit tangent vector T), the direction in which your path is turning (the unit normal vector N), and the tendency of your motion to “twist” ...

4100 AWL/Thomas_ch12p848-905

In studying lines in the plane, when we needed to describe how a line was tilting, we used the notions of slope and angle of inclination. In space, we want a way to describe how a plane is tilting. We accomplish this by multiplying two vectors in the plane together to get a third vector perpendicular to the plane. The direction of this third vector tells us the “inclination” of the plane. The p...