Differentiation of Polar Equations**

  • In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by:

    • a distance from a reference point

    • an angle from a reference direction.

  • The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis.

  • The radial coordinate is often denoted by r, and the angular coordinate by θ.

    • Angles in polar notation are generally expressed in either degrees or radians.

      • For this course, we will always be in radians.

  • The initial motivation for the introduction of the polar system was the study of circular and orbital motion.

  • Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point.

    • The mathematical function that describes a spiral can be expressed using rectangular (or Cartesian) coordinates.

    • However, if we change our coordinate system to something that works a bit better with circular patterns, the function becomes much simpler to describe.

    • The polar coordinate system is well suited for describing curves of this type.

  • Many physical systems—such as those concerned with bodies moving around a central point or with phenomena originating from a central point—are simpler and more intuitive to model using polar coordinates.

Essential Questions

How are derivatives of polar equations found?