Gauss's Applied Mathematics -- Physics and Statistics

One of Gauss's most important accomplishments is in the development of Gauss's Theorem (also known as the Divergence Theorem), which he used to make several important developments in the physical theory of electromagnetism. Qualitatively, Gauss's Theorem links the flux of a vector field through some closed surface to the divergence of the field inside the surface.Quantitatively, Gauss showed that the following was true:

 

Where V is any volume, F is any field, S is the surface enclosing V, and n is the vector normal to the surface. 

An intuitive way to think about this is that F describes something that is flowing (e.g., a gas or a fluid). The divergence of F would then describe the expansion or compression of the gas or fluid. Gauss's Theorem says that the total expansion (or compression) of the gas with some volume V is equal to how much "stuff" is entering (or leaving) the surface S.

Probably the most important application of Gauss's Theorem is in the physical theory of electromagnetism. Gauss was able to show that the electric flux is equal to the charge enclosed by the volume divided by a constant (ε₀). In other words,

Gauss did the same thing for the magnetic flux and found:

Physically, this shows that the electric field points radially away from positive charges (radially into negative charges) such that the total divergence is Q/ε₀. 

For magnetism, Gauss showed that the magnetic field has no divergence. The magnetic field can circulate, but cannot do anything else. In other words, all field lines that enter the Gaussian surface must also leave the surface. 

 

Gauss's Theorem provided the theoretical foundation for Gauss's laws of electricity and magnetism, which remain the backbone of electromagnetism to this day. 

Of equal importance was Gauss's development in probability theory of a way to handle random variables that each have unknown distributions. Today, this is known as a normal or Gaussian distribution. A fun way to visualize this is with a Galton board.

The Gaussian distribution is incredibly useful because of the central limit theorem, which says that the sum of independent random variables will always tend to a Gaussian distribution. This theorem makes the Gaussian distribution remarkably powerful and applicable, and it can be used to model useful data such as people's heights, blood pressure, IQ scores, and salaries. Because of their incredible applicability, Gaussian distributions are ubiquitous throughout all of branches of Applied Mathematics including physics, economics, and psychology.