George Green

How to use this course

George Green spent most of his life running a windmill. He had perhaps four terms of formal schooling, taught himself from the French analysts, and in 1828 printed a 70-page essay that almost nobody read. It contained the integral identity now called Green’s theorem, the first systematic use of the word “potential”, and the function that physicists and engineers now use to solve linear differential equations of every kind. This course tells the story of that unlikely career and works through the mathematics in modern notation.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Lessons 04 to 07 follow the 1828 Essay. Lessons 08 and 09 cover the eight papers Green wrote between 1832 and 1839. Lesson 10 follows his function into twentieth-century physics.

Timeline

A baker’s son in Sneinton, 1793–1807

George Green was baptised on 14 July 1793 in Sneinton, then a village on the eastern edge of Nottingham. His father, also George Green, was a baker; his mother was Sarah Butler. They had married in 1791, and a daughter, Ann, followed around 1795. Nottingham was a crowded, fast-growing town of lace and hosiery workers.

In 1801, aged about eight, Green was sent to the academy of Robert Goodacre in Nottingham, a school with a reputation for science teaching and one of the more expensive in town. He left in the summer of 1802 after four terms. That year and a half was the whole of his formal education until he went to Cambridge at forty. He then worked in his father’s bakery.

Biographers have looked for a mentor in these years. One candidate is John Toplis, a former headmaster of the Nottingham Free Grammar School who translated part of Pierre-Simon Laplace’s Mécanique céleste into English. There is no document linking the two. How Green learned the analysis of Laplace, Siméon Denis Poisson and Joseph-Louis Lagrange remains largely unknown.

The mill and the library, 1807–1828

On 26 February 1807 Green’s father bought a plot of land in Sneinton at auction and built a brick tower windmill, about 16 metres high, among the first brick mills in Nottinghamshire. The son worked in the mill and became, in practice, its manager. In 1817 the family moved into a house beside it. When the father died in 1829, Green inherited the business and a comfortable income.

Green lived with Jane Smith, daughter of William Smith, the mill’s manager. They had seven children, the first born in 1824 and the last in 1840, but they never married. Biographers suggest that his father disapproved of the match and that marriage might have cost Green his inheritance; later it would have cost him his Cambridge fellowship, which required celibacy.

In 1823 Green joined the Nottingham Subscription Library, housed in Bromley House in the town centre. The library gave him the Philosophical Transactions of the Royal Society and other journals, and its members were the educated middle class of the town. Its shelves did not hold all the French works he cited, and how he obtained Laplace and Poisson is not documented. What is clear is the result: by 1827 the miller had absorbed the most advanced mathematical physics of his time and had found something new in it.

The Essay of 1828

On 14 December 1827 Green advertised a forthcoming work, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, dedicated to the Duke of Newcastle. It appeared in March 1828, printed in Nottingham at 7s. 6d. a copy for about fifty subscribers, most of them members of the Subscription Library. Few could have followed it.

The preface is modest: Green apologised for the shortcomings of a writer whose education had been limited. The content was not modest. The Essay

• named the function \(V\) whose gradient gives the force, calling it the potential function;
• proved the integral identity relating volume and surface integrals of two functions that we now call Green’s theorem or Green’s second identity;
• used a special solution, vanishing on a boundary and singular at one point, to solve boundary value problems, the object now called the Green’s function;
• applied these tools to conductors, the Leyden jar, and the magnetism of bodies.

Earlier writers, among them Laplace, Lagrange and Poisson, had used potentials in gravitation and electricity. Green’s achievement was to make the potential the central object and to treat boundary value problems in general, not body by body. The Essay was not reviewed and nearly vanished. Lesson 12 tells how it was found again.

The potential function

For the general reader. Forces between charges, or between masses, point in different directions at every place. Green noticed that all this information is contained in one number at each point, the potential. The force is the slope of the potential, just as water flows down the slope of a hill. One number per point is much easier to handle than an arrow per point.

Definition

Let charge with density \(\rho\) occupy a bounded region of space. Green’s potential function at a point \(x\) is

A unit test charge at \(x\) feels the force \(F=-\nabla V\). Green called \(V\) the potential function and emphasised that one scalar function encodes all components of the force.

Derivation

Away from the charge, differentiate under the integral sign: for \(x\ne y\), a direct calculation gives \(\nabla_x^2\,|x-y|^{-1}=0\). Inside the charge, the integrand is singular. Surround \(x\) by a small ball \(B_\varepsilon\), and use the divergence theorem on the flux of \(\nabla V\) through its surface. The ball’s own charge contributes flux \(-4\pi\int_{B_\varepsilon}\rho\), because a point charge \(q\) gives flux \(-4\pi q\) through any surrounding sphere. Dividing by the volume and letting \(\varepsilon\to0\) gives \(\nabla^2V(x)=-4\pi\rho(x)\). Laplace had the first equation; Poisson added the second in 1813. Green used both.

Green’s theorem and reciprocity

For the general reader. Green found a way to trade what happens inside a region for what happens on its boundary. His identity says that a certain combination of two functions, added up over the volume, equals another combination added up over the surface. With it, you can learn everything about an electric field inside a box by measuring only at the walls.

Green wrote the identity with the normal pointing into the body, so his surface terms carry the opposite sign. He also allowed one function to be singular inside \(\Omega\), handling the singularity by cutting out a small sphere, which is how the Green’s function of Lesson 06 enters.

Proof

Apply the divergence theorem to the vector field \(U\nabla V\): since \(\nabla\cdot(U\nabla V)=\nabla U\cdot\nabla V+U\nabla^2V\),

This is Green’s first identity. Write it again with \(U\) and \(V\) exchanged and subtract; the symmetric term \(\nabla U\cdot\nabla V\) cancels.

Uniqueness

If \(V_1,V_2\) both solve \(\nabla^2V=-4\pi\rho\) in \(\Omega\) with the same boundary values, then \(W=V_1-V_2\) is harmonic and vanishes on \(\partial\Omega\). The first identity with \(U=V=W\) gives \(\int_\Omega|\nabla W|^2=0\), so \(W\) is constant, hence zero.

Reciprocity

Both sides equal \(\iint \rho(x)\rho'(y)/|x-y|\,d^3x\,d^3y\), which is symmetric in the two states. The same symmetry shows that the coefficients \(p_{ij}\) in \(V_i=\sum_j p_{ij}q_j\) satisfy \(p_{ij}=p_{ji}\).

The Green’s function

For the general reader. Put a single unit charge inside a closed metal box that is connected to the earth. The charges on the walls rearrange themselves to cancel its influence at the walls. The resulting potential is Green’s function for that box. Once you know it, you can find the potential for any arrangement of charge inside, and for any voltages on the walls, by adding up copies of it.

Definition

For a bounded domain \(\Omega\) and \(y\in\Omega\), the Dirichlet Green’s function is

where \(h_y\) is harmonic in \(\Omega\) and equals \(1/|x-y|\) for \(x\in\partial\Omega\). So \(\nabla_x^2G=-4\pi\delta_y\) and \(G=0\) on the boundary. Physically, \(-h_y\) is the potential of the charge induced on the grounded boundary by a unit charge at \(y\). Green argued for the existence of \(h_y\) on physical grounds: the induced charge must exist.

Proof sketch

Apply Green’s theorem to \(U=G(x,\cdot)\) and \(V\) on \(\Omega\) minus a small ball around \(x\). On the small sphere, \(G\approx1/r\), and the surface term \(-V\,\partial_n(1/r)\) integrates to \(4\pi V(x)\) as the radius shrinks; the other terms there vanish. Using \(\nabla^2V=-4\pi\rho\) and \(G=0\) on \(\partial\Omega\) gives the formula. For symmetry, apply the theorem to \(G(\cdot,x)\) and \(G(\cdot,y)\) with small balls removed around both poles: the two small-sphere terms give \(4\pi G(y,x)\) and \(4\pi G(x,y)\), which must be equal. This is reciprocity again: the potential at \(x\) due to a unit charge at \(y\) equals the potential at \(y\) due to a unit charge at \(x\).

Explicit cases

Half-space \(\{x_3>0\}\): with \(\bar y=(y_1,y_2,-y_3)\), \(G(x,y)=|x-y|^{-1}-|x-\bar y|^{-1}\). Ball of radius \(a\) centred at 0: with \(y^*=a^2y/|y|^2\),

The second formula is Poisson’s integral for the ball. Green treated the sphere in the Essay. The “image” construction above was popularised later by William Thomson.

Conductors and Dirichlet’s principle

For the general reader. Electric charge on a metal body spreads out until the body is at a single voltage everywhere. Green turned this into mathematics. He also argued that among all possible ways of arranging the field, nature picks the one with the least energy, an idea later called Dirichlet’s principle. It took seventy years to make that argument rigorous.

Conductors

In equilibrium the field inside a conductor vanishes, so the potential is constant there and any charge sits on the surface. Outside, \(V\) is harmonic, equals the given constants on the conductors, and tends to 0 at infinity. The surface density is \(\sigma=-\frac{1}{4\pi}\partial V/\partial n\), with \(n\) pointing out of the conductor. Green proved that a closed grounded conducting shell screens its interior completely from external charges: inside, \(V\) is harmonic with zero boundary values, so it vanishes identically.

The capacity of a conductor is the charge it holds at unit potential. For a sphere of radius \(a\) it is \(a\). Green analysed the Leyden jar, the condenser of his day, as two conducting sheets separated by a thin layer of glass, and found that its charge grows as the glass gets thinner, in agreement with experience.

Dirichlet’s principle

Green, and independently Carl Friedrich Gauss (1839–40) and William Thomson (1847), argued that a minimiser must exist because \(E\ge0\). Bernhard Riemann, who learned the idea from Peter Gustav Lejeune Dirichlet’s lectures, named it Dirichlet’s principle and built his theory of Riemann surfaces on it. In 1870 Karl Weierstrass showed that a non-negative functional need not attain its infimum. David Hilbert rescued the principle in 1900, and the modern proof uses the Hilbert space \(H^1\) and the direct method of the calculus of variations.

Pendulums, canals and Green’s law

For the general reader. After the Essay, Green turned to fluids. He calculated how much a surrounding liquid slows a swinging pendulum, and how a wave grows as it travels up a canal that becomes narrower and shallower. His canal result explains why tsunamis tower over the shore after crossing the ocean almost unnoticed.

Pendulums in fluid media

In a paper read to the Royal Society of Edinburgh on 16 December 1833, Green computed the motion of an ellipsoid oscillating in an ideal incompressible fluid. The fluid flows with a velocity potential \(\phi\), harmonic outside the body, and its kinetic energy is proportional to the square of the body’s speed. The body therefore behaves as though its mass were increased by an added mass. For a sphere, the added mass is half the mass of the fluid it displaces.

Waves in a variable canal

In a paper read to the Cambridge Philosophical Society on 15 May 1837, Green considered long waves in a canal whose breadth \(b(x)\) and depth \(h(x)\) change slowly. The linear shallow-water equation for the surface elevation \(\eta\) is

Derivation

Seek \(\eta=A(x)\,e^{i(\omega t-\int k\,dx)}\) with \(A\) slowly varying. The leading order gives \(\omega^2=ghk^2\), so \(k=\omega/\sqrt{gh}\). The next order gives \((b\,h\,k\,A^2)'=0\), so \(A^2\propto1/(bhk)\propto1/(b\sqrt h)\). The same law follows from conservation of energy flux: energy per unit length \(\propto bA^2\), carried at speed \(\sqrt{gh}\). Green’s method of expanding about a slowly varying phase anticipated by almost a century the approximation of Wentzel, Kramers and Brillouin in quantum mechanics. In an 1839 note he compared his result with John Scott Russell’s observations of canal waves.

Elastic solids, sound and light

For the general reader. In the 1830s light was thought to be a vibration of an invisible elastic solid, the ether. To describe how light bounces off and bends into glass, Green needed a theory of elastic solids. Rather than guess at forces between molecules, he started from energy. The approach gave the right count of elastic constants and remains the foundation of elasticity.

The strain energy function

Let \(u(x)\) be a small displacement and \(e_{ij}=\tfrac12(\partial_iu_j+\partial_ju_i)\) the strain. In his paper on light read on 11 December 1837, Green assumed only that the elastic forces derive from a potential energy \(\phi\), and that \(\phi\) is a quadratic function of the strain for small deformations. He then applied Lagrange’s method to obtain the equations of motion.

The molecular theories of Claude-Louis Navier, Poisson and Augustin-Louis Cauchy, built from central forces between point particles, impose six further relations (the Cauchy relations) and give at most 15 constants; in the isotropic case they force \(\lambda=\mu\). Followers of the two views were later called multi-constant and rari-constant theorists, and experiment eventually sided with Green.

From \(\phi\) the isotropic equation of motion is \(\rho\,\ddot u=(\lambda+\mu)\nabla(\nabla\cdot u)+\mu\nabla^2u\). It carries longitudinal waves at speed \(\sqrt{(\lambda+2\mu)/\rho}\) and transverse waves at speed \(\sqrt{\mu/\rho}\). Light is transverse, so Green assumed the ether’s longitudinal speed to be very large, making longitudinal waves negligible.

Reflection and refraction

Green’s companion paper on sound, read the same day, treated two fluids meeting at a plane. He used continuity of displacement and pressure at the interface and checked his results against conservation of energy, one of the earliest uses of energy flux to test reflection formulas. In these papers he also treated incidence beyond the critical angle, where the transmitted disturbance decays exponentially away from the interface, what we now call an evanescent wave. In the light paper he recovered Fresnel’s sine law for one polarisation. His 1839 paper on crystallized media used the energy method to simplify Cauchy’s theory of light in crystals.

Green’s functions after Green

For the general reader. Green’s idea works for any linear equation: find the response to a single sharp kick, then build any other response by adding kicks. Engineers call it the impulse response. Physicists call it the propagator. Quantum field theory is in large part a theory of Green’s functions.

Examples

• Heat equation on \(\mathbb R^n\): \(G(x,t)=(4\pi t)^{-n/2}e^{-|x|^2/4t}\) for \(t>0\).
• Helmholtz equation \((\nabla^2+k^2)G=-4\pi\delta\) in \(\mathbb R^3\): outgoing solution \(G=e^{ik|x|}/|x|\), the basis of scattering theory.
• Wave equation: the retarded Green’s function carries the effect of a source at the speed of the wave, and gives the retarded potentials of electrodynamics.
• Quantum theory: the Feynman propagator is a Green’s function of the Klein–Gordon or Dirac operator with a particular boundary condition in time. Julian Schwinger, who had used Green’s functions in wartime radar work, made them central to his formulation of quantum electrodynamics.

Bromhead, Cambridge and the last years, 1828–1841

One subscriber to the Essay was Sir Edward ffrench Bromhead, a Lincolnshire baronet, a Cambridge graduate and a founder of the Analytical Society. He wrote to Green offering to forward any future papers to the Royal Society of London, the Royal Society of Edinburgh or the Cambridge Philosophical Society. Green replied, after a long delay, in January 1830. Over the next three years the two met at Bromhead’s home, Thurlby Hall near Lincoln, and Bromhead communicated three papers: on the equilibrium of fluids analogous to the electric fluid (read 12 November 1832), on the attractions of ellipsoids of variable density (read 6 May 1833), and on pendulums in fluid media (read 16 December 1833). The ellipsoid paper worked in an arbitrary number of dimensions, unusual for its time.

Bromhead suggested that Green should study at Cambridge. In October 1833, about forty years old, Green entered Gonville and Caius College, Bromhead’s own college. He graduated in January 1837 as fourth wrangler; James Joseph Sylvester was second that year. Over the next two years he wrote his papers on canal waves, on sound, on light and on crystallized media, all published by the Cambridge Philosophical Society. On 31 October 1839 he was elected to a Perse Fellowship at Caius.

He held the fellowship only briefly. By May 1840 he was back in Nottingham, ill. He made his will in July 1840 and died at Sneinton on 31 May 1841, aged 47, at the house of Jane Smith and their children. Influenza is the cause usually given. He was buried in the churchyard of St Stephen’s, Sneinton.

Rediscovery and legacy

In 1845 the young William Thomson, about to graduate from Cambridge, saw the Essay cited in a paper by Robert Murphy and could not find a copy. His coach, William Hopkins, gave him copies in January 1845, just after the examinations. Thomson took them to Paris that summer and showed them to Joseph Liouville and Charles Sturm. Sturm recognised results he had worked to obtain himself. Thomson then arranged for the Essay to be reprinted, with an introduction, in Crelle’s journal in 1850, 1852 and 1854. Green’s ideas entered the work of Thomson, George Gabriel Stokes and James Clerk Maxwell, whose Treatise on Electricity and Magnetism uses Green’s theorem throughout. In 1871 Norman Macleod Ferrers collected Green’s papers in one volume.

Legacy

• Mathematics: Green’s identities, Green’s functions and the Dirichlet problem are the foundation of potential theory and of the theory of elliptic partial differential equations. The planar identity \(\oint P\,dx+Q\,dy=\iint(\partial_xQ-\partial_yP)\,dA\), taught as “Green’s theorem” in calculus courses, is a close relative named in his honour.
• Physics: propagators in quantum field theory, response functions in condensed matter physics, Green’s tensor in elasticity and seismology, and the reciprocity theorems of electromagnetism and acoustics.
• Commemoration: Green’s Mill in Sneinton was restored to working order in 1985 and houses a science centre. In July 1993, the bicentenary of his birth, a memorial stone was unveiled in the nave of Westminster Abbey near the graves of Isaac Newton and Lord Kelvin. For the bicentenary Schwinger wrote a tribute, “The Greening of Quantum Field Theory: George and I.” The University of Nottingham’s science library and a lunar crater bear his name.

Connections

People linked to George Green across this collection. A name without a link has no page here yet. See George Green in the network of all 337 lives →

Influenced by

• Augustin-Louis Cauchy
• Claude-Louis Navier
• Joseph-Louis Lagrange
• Pierre-Simon Laplace
• Siméon Denis Poisson

Influenced

• Bernhard Riemann
• George Gabriel Stokes
• James Clerk Maxwell
• William Thomson, Lord Kelvin 1845

Colleagues and collaborators

• Edward Witten

Friends

• Edward ffrench Bromhead 1828-1841

Also mentioned in this life

• Apollonius of Perga
• Charles-Augustin de Coulomb
• Paul Dirac
• Peter Gustav Lejeune Dirichlet
• Augustin-Jean Fresnel
• Carl Friedrich Gauss
• Hermann von Helmholtz
• David Hilbert
• Joseph Liouville
• Isaac Newton
• Bertrand Russell
• John Wallis
• Karl Weierstrass

Fields

• potential theory 9 lives
• mathematical physics 45 lives
• electrostatics 4 lives
• hydrodynamics 4 lives
• elasticity 11 lives

On CoursesHub

• Fluid Mechanics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, George Green, MacTutor History of Mathematics.
2. George Green (mathematician), Wikipedia.
3. D. M. Cannell, George Green, Mathematician and Physicist 1793–1841: The Background to His Life and Work, Athlone Press, 1993; 2nd ed., SIAM, 2001.
4. L. Challis and F. Sheard, The Green of Green functions, Physics Today 56(12), December 2003.
5. Westminster Abbey, George Green (memorial).
6. P. J. Wallis, Green, George, Dictionary of Scientific Biography.

Mathematics and physics

1. G. Green, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, Nottingham, 1828. Transcription on arXiv:0807.0088.
2. N. M. Ferrers (ed.), Mathematical Papers of the late George Green, Macmillan, 1871.
3. G. Green, On the motion of waves in a variable canal of small depth and width, Trans. Cambridge Phil. Soc. 6 (1838).
4. G. Green, On the laws of the reflexion and refraction of light at the common surface of two non-crystallized media, Trans. Cambridge Phil. Soc. 7.
5. J. Schwinger, The Greening of quantum field theory: George and I, arXiv:hep-ph/9310283.
6. O. D. Kellogg, Foundations of Potential Theory, Springer, 1929.
7. F. W. Hehl and Y. Itin, The Cauchy relations in linear elasticity theory, arXiv:cond-mat/0206175.
8. I. Todhunter and K. Pearson, A History of the Theory of Elasticity and of the Strength of Materials, vol. 1, Cambridge, 1886.