# The Inverse Square Law

In the early 1600's, Johann Kepler completed the Copernican revolution by establishing three laws describing planetary motion. Kepler's first law states that a planet has an elliptical orbit with the sun at one focus. Kepler's second law implies that a planet speeds up and slows down in its orbit, and his third law relates the semi-major axis of the ellipse to the amount of time required for one completion of the orbit.

A half a century later, a young mathematician named Isaac Newton showed that Kepler's laws followed from an inverse square law, which says that if M is the mass of the sun and m is the mass of the planet, then

Amount
of
gravity

= -G M m r2 where G  is the universal gravitational constant. Since then, it has been shown that the electromagnetic force also satisfies an inverse square law.

Thus, we conclude this textbook with an exploration of the inverse square law. In so doing, we will see nearly all of the topics from multivariable calculus come into play. Indeed, this chapter can be considered the culmination of all that calculus is about and what it is used for.

 C.1 Uniform Circular Motion C.5 Electric Fields and Gauss' Theorem C.2 Conservation Laws C.6 Magnetic Fields C.3 Kepler's Laws C.7 Maxwell's Equations C.4 Satellites and Planets C.8 Electromagnetic Radiation