Shinban Liu

MAT 267

Brewer

Three Men Two Theorems

            George Green was born in 1793 in Britain.  He was a mathematician whose primary work was an essay called An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism.  His youth was spent in the city of Nottingham and he only went for about a year of formal school.  As he got older, Green worked on his father’s mill and when his father died he became proprietor of the job.  During this time, he started to study mathematics and eventually attended Cambridge University.  He graduated in 1837 and decided to stay as a faculty member.  While there, he wrote research papers on optics, acoustics, and hydrodynamics.  He also worked on the motion of waves and his work on light waves and the molecule ether became well known.  He died in 1841 from an illness after he returned to his birth town, Nottingham.

            George Stokes was born in 1819 in Ireland.  He was a mathematician and physicist who also went to Cambridge.  He attended schools in a number of cities in Ireland and after he graduated form Cambridge he was accepted as a fellow.  He was made a baronet in 1889 and served in the British Parliament.  He died in 1903 at Cambridge.  Stokes was the oldest one of the trio of natural philosophers.

            William Thomson, also known as Lord Kelvin, was born in 1824 in Ireland.  He was a mathematician, physicist, and engineer and did very important work in the analysis of electricity and thermodynamics.  He was the person who developed the Kelvin scale of temperature.  When Thomson was 24 years old he studied at the University of Glasgow and when he graduated he became a math teacher.  He attended Cambridge and was very active in school and sport activities.  While there, the specific subject of electricity greatly interested him and he even devised a hypothesis or electrical images.  He was also a telegraph engineer and inventor and made a machine to try and predict the tides.  Thomson died in December of 1907.

            Both George Green and George Stokes invented theorems that still exist today as functional properties for solving mathematical problems.  Green’s theorem states that:

Which results in the equation SScurl(F)∙k dA which applies to 2-D vector fields.
                                             D

Following a similar derivation, George Stokes was able to arrive at the function:

SScurl(F)∙k dA
 S

Where S is now a curved surface in 3 dimensions and k dA is equivalent to dS. With these modifications, Stokes’ theorem becomes:

     SS curl(F)∙<-z_x, -z_y, 1>dA
      D

Where D is now the projection of the curved surface S on the xy plane.

            While both of these equations deal with the flux through a surface, the similarities end when discussing the type of object being analyzed. As stated before, Green’s theorem deals primarily with surfaces that have only two dimensions such as a lamina that has negligible thickness. Stokes’ theorem on the other hand deals with the flux through an entire object in three dimensions. Using these functions, one can determine the flux of curl(F) across a surface or object, with the limitations that the function must be smooth and simple, positively oriented, and piecewise smooth.

            The applications of these two different yet similar theorems are still widely used today in the field of physics. The measurement of flux through a two dimensional or three dimensional object is commonly used in the field of electromagnetism when describing the flow of electrons, charge through a surface and also in hydrodynamics when calculating the flow of liquids through objects and other diverse areas in optics and acoustics.

            Although the authors of these two theorems seem very concrete, the process that the theorems underwent to gain the mathematical notoriety it holds today was quite complex. George Green, the man that the first theorem is named after, did not even get to see the full impact of his work during his lifetime.  While independently studying and running his father’s mill, he privately published An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism that only sold about 100 copies, with the majority of them sold to his friends who did not fully understand his work. Only a few weeks before Green’s death, William Thomson had been admitted to Cambridge where Green studied and in a paper by Robert Murphy, discovered a reference to Green’s publication. Thomson then located a copy of Green’s work several years later and in 1854, almost 26 years after Green published his pamphlets, Thomson republished the works that have exploded with popularity and are responsible to the mathematical theories of electricity underlying today’s technology. Stokes’ Theorem, the second theorem developed through the three men, was also due to George Green’s work. Stokes was a professor and Cambridge University and collaborated with Thomson through letters for over 50 years. In the 407 letters, Stokes learned of this theorem that is now accredited to him in 1850. This theorem acquired its name from Stokes’ habit of including it in the Cambridge prize examinations.

References:

D.M. Cannell, George Green, Mathematician and Physicist 1793-1841: The Background to His Life and Work (Philadelphia: Society for Industrial and Applied Mathematics, 2001).

Victor Katz, A History of Mathematics: An Introduction (New York: HarperColins, 1993), pp. 678-680.

Sylvanus P. Thompson, The Life of Lord Kelvin (New York: Chelsea, 1976).