The Line

Introduction:

The line is one of the most important curves in all mathematics, because it is simple enough to be used as an approximation to other curves. Calculus at all levels is based on solving a problem by chopping it up into tiny linear pieces and then taking a limit as those pieces become smaller and smaller. We choose to cut things into linear pieces because it is easy to find the length of a line, or its slope, or the area of a box.

Definition:

,

where (u, v) is a vector parallel to the line and (x, y) is a point on the line.

Properties:

Velocity and Acceleration:

The Tangent Vector:

Arc Length:

Curvature: