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82-497: James MacCullagh (1809 – 24 October 1847) was an Irish mathematician . MacCullagh was born in Landahaussy , near Plumbridge , County Tyrone , Ireland, but the family moved to Curly Hill, Strabane when James was about 10. He was the eldest of twelve children and demonstrated mathematical talent at an early age. He entered Trinity College Dublin as a student in 1824, winning a scholarship in 1827 and graduating in 1829. He became

164-522: A ) = F a b , {\displaystyle {\frac {\partial {\mathcal {L}}}{\partial (\partial _{b}A_{a})}}=F^{ab}\,,} obtains Maxwell's equations in vacuum. The source equations (Gauss' law for electricity and the Maxwell-Ampère law) are ∂ b F a b = μ 0 j a . {\displaystyle \partial _{b}F^{ab}=\mu _{0}j^{a}\,.} while

246-462: A . {\displaystyle F_{ab}=\partial _{a}A_{b}-\partial _{b}A_{a}.} To obtain the dynamics for this field, we try and construct a scalar from the field. In the vacuum, we have L = − 1 4 μ 0 F a b F a b . {\displaystyle {\mathcal {L}}=-{\frac {1}{4\mu _{0}}}F^{ab}F_{ab}\,.} We can use gauge field theory to get

328-413: A charge density ρ ( r , t ) and current density J ( r , t ), there will be both an electric and a magnetic field, and both will vary in time. They are determined by Maxwell's equations , a set of differential equations which directly relate E and B to the electric charge density (charge per unit volume) ρ and current density (electric current per unit area) J . Alternatively, one can describe

410-541: A continuous mass distribution ρ instead, the sum is replaced by an integral, g ( r ) = − G ∭ V ρ ( x ) d 3 x ( r − x ) | r − x | 3 , {\displaystyle \mathbf {g} (\mathbf {r} )=-G\iiint _{V}{\frac {\rho (\mathbf {x} )d^{3}\mathbf {x} (\mathbf {r} -\mathbf {x} )}{|\mathbf {r} -\mathbf {x} |^{3}}}\,,} Note that

492-510: A day over a country is described by assigning a vector to each point in space. Each vector represents the direction of the movement of air at that point, so the set of all wind vectors in an area at a given point in time constitutes a vector field . As the day progresses, the directions in which the vectors point change as the directions of the wind change. The first field theories, Newtonian gravitation and Maxwell's equations of electromagnetic fields were developed in classical physics before

574-413: A distribution of mass (or charge), the potential can be expanded in a series of spherical harmonics , and the n th term in the series can be viewed as a potential arising from the 2 -moments (see multipole expansion ). For many purposes only the monopole, dipole, and quadrupole terms are needed in calculations. Modern formulations of classical field theories generally require Lorentz covariance as this

656-644: A fellow of Trinity College Dublin in 1832 and was a contemporary there of William Rowan Hamilton . He became a member of the Royal Irish Academy in 1833. In 1835 he was appointed Erasmus Smith's Professor of Mathematics at Trinity College Dublin and in 1843 became Erasmus Smith's Professor of Natural and Experimental Philosophy . He was an inspiring teacher and taught notable scholars, including Samuel Haughton , Andrew Searle Hart , John Kells Ingram and George Salmon . He had been involved in repeated priority disputes with Hamilton. In 1832, Hamilton published

738-471: A financial economist might study the structural reasons why a company may have a certain share price , a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock ( see: Valuation of options ; Financial modeling ). According to the Dictionary of Occupational Titles occupations in mathematics include

820-576: A framework of telescoping rods, described in his paper On a Gyrostatic Adynamic Constitution for Ether (1890). MacCullagh was an idealistic nationalist, in the sense of the time. He unsuccessfully contested the election for the Dublin University constituency in 1847. Suffering from overwork and a bout of depression, he died in 1847 by cutting his throat in his rooms at Trinity College Dublin. After his death, Hamilton helped obtain pensions for his sisters. In May 2009, an Ulster History Circle plaque

902-400: A manner which will help ensure that the plans are maintained on a sound financial basis. As another example, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while

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984-420: A mechanical interpretation of the luminiferous aether because he readily admits that no known physical medium could have such a potential function resisting only the rotation of its elements. "Concerning the peculiar constitution of the ether, we know nothing and shall suppose nothing, except what is involved in the foregoing assumptions [rectilinear vibrations in a medium of constant density]... Having arrived at

1066-427: A path ℓ will exert a force on nearby charged particles that is quantitatively different from the electric field force described above. The force exerted by I on a nearby charge q with velocity v is F ( r ) = q v × B ( r ) , {\displaystyle \mathbf {F} (\mathbf {r} )=q\mathbf {v} \times \mathbf {B} (\mathbf {r} ),} where B ( r )

1148-766: A political dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked and scraping off her skin with clamshells (some say roofing tiles). Science and mathematics in the Islamic world during the Middle Ages followed various models and modes of funding varied based primarily on scholars. It was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. Funding for translation of scientific texts in other languages

1230-411: A potential function for a dynamical theory for the transmission of light. MacCullagh found that a conventional potential function proportional to the squared norm of the displacement field was incompatible with known properties of light waves. In order to support only transverse waves , he found that the potential function must be proportional to the squared norm of the curl of the displacement field. It

1312-450: A prediction of conical refraction . In 1833, MacCullagh claimed that it is a special case of a theorem he published in 1830 that he did not explicate since it was not relevant to that particular paper. In 1842, Hamilton speculated on a model of ether, to which MacCullagh claimed that he had speculated on the same model. Although he worked mostly on optics , he is also remembered for his work on geometry ; his most significant work in optics

1394-739: A scalar potential, V ( r ) E ( r ) = − ∇ V ( r ) . {\displaystyle \mathbf {E} (\mathbf {r} )=-\nabla V(\mathbf {r} )\,.} Gauss's law for electricity is in integral form ∬ E ⋅ d S = Q ε 0 {\displaystyle \iint \mathbf {E} \cdot d\mathbf {S} ={\frac {Q}{\varepsilon _{0}}}} while in differential form ∇ ⋅ E = ρ e ε 0 . {\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho _{e}}{\varepsilon _{0}}}\,.} A steady current I flowing along

1476-450: A small test mass m located at r , and then dividing by m : g ( r ) = F ( r ) m . {\displaystyle \mathbf {g} (\mathbf {r} )={\frac {\mathbf {F} (\mathbf {r} )}{m}}.} Stipulating that m is much smaller than M ensures that the presence of m has a negligible influence on the behavior of M . According to Newton's law of universal gravitation , F ( r )

1558-572: Is ∇ 2 ϕ = σ {\displaystyle \nabla ^{2}\phi =\sigma } where σ is a source function (as a density, a quantity per unit volume) and ø the scalar potential to solve for. In Newtonian gravitation, masses are the sources of the field so that field lines terminate at objects that have mass. Similarly, charges are the sources and sinks of electrostatic fields: positive charges emanate electric field lines, and field lines terminate at negative charges. These field concepts are also illustrated in

1640-506: Is ∇ ⋅ g = − 4 π G ρ m {\displaystyle \nabla \cdot \mathbf {g} =-4\pi G\rho _{m}} Therefore, the gravitational field g can be written in terms of the gradient of a gravitational potential φ ( r ) : g ( r ) = − ∇ ϕ ( r ) . {\displaystyle \mathbf {g} (\mathbf {r} )=-\nabla \phi (\mathbf {r} ).} This

1722-402: Is ∬ B ⋅ d S = 0 , {\displaystyle \iint \mathbf {B} \cdot d\mathbf {S} =0,} while in differential form it is ∇ ⋅ B = 0. {\displaystyle \nabla \cdot \mathbf {B} =0.} The physical interpretation is that there are no magnetic monopoles . In general, in the presence of both

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1804-420: Is mathematics that studies entirely abstract concepts . From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics , and at variance with the trend towards meeting the needs of navigation , astronomy , physics , economics , engineering , and other applications. Another insightful view put forth is that pure mathematics

1886-534: Is Newton's gravitational constant . Therefore, the gravitational field of M is g ( r ) = F ( r ) m = − G M r 2 r ^ . {\displaystyle \mathbf {g} (\mathbf {r} )={\frac {\mathbf {F} (\mathbf {r} )}{m}}=-{\frac {GM}{r^{2}}}{\hat {\mathbf {r} }}.} The experimental observation that inertial mass and gravitational mass are equal to unprecedented levels of accuracy leads to

1968-451: Is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on problems, often concrete but sometimes abstract. As professionals focused on problem solving, applied mathematicians look into the formulation, study, and use of mathematical models in science , engineering , business , and other areas of mathematical practice. Pure mathematics

2050-496: Is a consequence of the gravitational force F being conservative . A charged test particle with charge q experiences a force F based solely on its charge. We can similarly describe the electric field E generated by the source charge Q so that F = q E : E ( r ) = F ( r ) q . {\displaystyle \mathbf {E} (\mathbf {r} )={\frac {\mathbf {F} (\mathbf {r} )}{q}}.} Using this and Coulomb's law

2132-484: Is a continuity equation, representing the conservation of mass ∂ ρ ∂ t + ∇ ⋅ ( ρ u ) = 0 {\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf {u} )=0} and the Navier–Stokes equations represent the conservation of momentum in the fluid, found from Newton's laws applied to

2214-419: Is given by F ( r ) = − G M m r 2 r ^ , {\displaystyle \mathbf {F} (\mathbf {r} )=-{\frac {GMm}{r^{2}}}{\hat {\mathbf {r} }},} where r ^ {\displaystyle {\hat {\mathbf {r} }}} is a unit vector pointing along the line from M to m , and G

2296-410: Is not conservative in general, and hence cannot usually be written in terms of a scalar potential. However, it can be written in terms of a vector potential , A ( r ): B ( r ) = ∇ × A ( r ) {\displaystyle \mathbf {B} (\mathbf {r} )=\nabla \times \mathbf {A} (\mathbf {r} )} Gauss's law for magnetism in integral form

2378-400: Is not necessarily applied mathematics : it is possible to study abstract entities with respect to their intrinsic nature, and not be concerned with how they manifest in the real world. Even though the pure and applied viewpoints are distinct philosophical positions, in practice there is much overlap in the activity of pure and applied mathematicians. To develop accurate models for describing

2460-411: Is now recognised as a fundamental aspect of nature. A field theory tends to be expressed mathematically by using Lagrangians . This is a function that, when subjected to an action principle , gives rise to the field equations and a conservation law for the theory. The action is a Lorentz scalar, from which the field equations and symmetries can be readily derived. Throughout we use units such that

2542-425: Is produced. Newtonian gravitation is now superseded by Einstein's theory of general relativity , in which gravitation is thought of as being due to a curved spacetime , caused by masses. The Einstein field equations, G a b = κ T a b {\displaystyle G_{ab}=\kappa T_{ab}} describe how this curvature is produced by matter and radiation, where G ab

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2624-538: Is the Einstein tensor , G a b = R a b − 1 2 R g a b {\displaystyle G_{ab}\,=R_{ab}-{\frac {1}{2}}Rg_{ab}} written in terms of the Ricci tensor R ab and Ricci scalar R = R ab g , T ab is the stress–energy tensor and κ = 8 πG / c is a constant. In the absence of matter and radiation (including sources)

2706-539: Is the magnetic field , which is determined from I by the Biot–Savart law : B ( r ) = μ 0 I 4 π ∫ d ℓ × d r ^ r 2 . {\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}I}{4\pi }}\int {\frac {d{\boldsymbol {\ell }}\times d{\hat {\mathbf {r} }}}{r^{2}}}.} The magnetic field

2788-493: Is the mass density , ρ e the charge density , G the gravitational constant and k e = 1/4πε 0 the electric force constant. Incidentally, this similarity arises from the similarity between Newton's law of gravitation and Coulomb's law . In the case where there is no source term (e.g. vacuum, or paired charges), these potentials obey Laplace's equation : ∇ 2 ϕ = 0. {\displaystyle \nabla ^{2}\phi =0.} For

2870-1913: Is the volume form in curved spacetime. ( g ≡ det ( g μ ν ) ) {\displaystyle (g\equiv \det(g_{\mu \nu }))} Therefore, the Lagrangian itself is equal to the integral of the Lagrangian density over all space. Then by enforcing the action principle , the Euler–Lagrange equations are obtained δ S δ ϕ = ∂ L ∂ ϕ − ∂ μ ( ∂ L ∂ ( ∂ μ ϕ ) ) + ⋯ + ( − 1 ) m ∂ μ 1 ∂ μ 2 ⋯ ∂ μ m − 1 ∂ μ m ( ∂ L ∂ ( ∂ μ 1 ∂ μ 2 ⋯ ∂ μ m − 1 ∂ μ m ϕ ) ) = 0. {\displaystyle {\frac {\delta {\mathcal {S}}}{\delta \phi }}={\frac {\partial {\mathcal {L}}}{\partial \phi }}-\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi )}}\right)+\cdots +(-1)^{m}\partial _{\mu _{1}}\partial _{\mu _{2}}\cdots \partial _{\mu _{m-1}}\partial _{\mu _{m}}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu _{1}}\partial _{\mu _{2}}\cdots \partial _{\mu _{m-1}}\partial _{\mu _{m}}\phi )}}\right)=0.} Two of

2952-408: Is used. The electromagnetic four-potential is defined to be A a = (− φ , A ) , and the electromagnetic four-current j a = (− ρ , j ) . The electromagnetic field at any point in spacetime is described by the antisymmetric (0,2)-rank electromagnetic field tensor F a b = ∂ a A b − ∂ b A

3034-634: The Pythagorean school , whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypatia of Alexandria ( c.  AD 350 – 415). She succeeded her father as librarian at the Great Library and wrote many works on applied mathematics. Because of

3116-656: The Schock Prize , and the Nevanlinna Prize . The American Mathematical Society , Association for Women in Mathematics , and other mathematical societies offer several prizes aimed at increasing the representation of women and minorities in the future of mathematics. Several well known mathematicians have written autobiographies in part to explain to a general audience what it is about mathematics that has made them want to devote their lives to its study. These provide some of

3198-478: The graduate level . In some universities, a qualifying exam serves to test both the breadth and depth of a student's understanding of mathematics; the students who pass are permitted to work on a doctoral dissertation . Mathematicians involved with solving problems with applications in real life are called applied mathematicians . Applied mathematicians are mathematical scientists who, with their specialized knowledge and professional methodology, approach many of

3280-474: The ' vacuum field equations , G a b = 0 {\displaystyle G_{ab}=0} can be derived by varying the Einstein–Hilbert action , S = ∫ R − g d 4 x {\displaystyle S=\int R{\sqrt {-g}}\,d^{4}x} with respect to the metric, where g is the determinant of the metric tensor g . Solutions of

3362-700: The 4-potential A , and it's this potential which enters the Euler-Lagrange equations. The EM field F is not varied in the EL equations. Therefore, ∂ b ( ∂ L ∂ ( ∂ b A a ) ) = ∂ L ∂ A a . {\displaystyle \partial _{b}\left({\frac {\partial {\mathcal {L}}}{\partial \left(\partial _{b}A_{a}\right)}}\right)={\frac {\partial {\mathcal {L}}}{\partial A_{a}}}\,.} Evaluating

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3444-547: The Advancement of Science . He corresponded with many notable scientists, including John Herschel and Charles Babbage . In Passages from the Life of a Philosopher , Charles Babbage wrote that MacCullagh was "an excellent friend of mine" and discussed the benefits and drawbacks of the analytical engine with him. MacCullagh's most important paper on optics, An essay towards a dynamical theory of crystalline reflection and refraction ,

3526-578: The Italian and German universities, but as they already enjoyed substantial freedoms and autonomy the changes there had begun with the Age of Enlightenment , the same influences that inspired Humboldt. The Universities of Oxford and Cambridge emphasized the importance of research , arguably more authentically implementing Humboldt's idea of a university than even German universities, which were subject to state authority. Overall, science (including mathematics) became

3608-412: The action functional can be constructed by integrating over spacetime, S = ∫ L − g d 4 x . {\displaystyle {\mathcal {S}}=\int {{\mathcal {L}}{\sqrt {-g}}\,\mathrm {d} ^{4}x}.} Where − g d 4 x {\displaystyle {\sqrt {-g}}\,\mathrm {d} ^{4}x}

3690-457: The advent of relativity theory in 1905, and had to be revised to be consistent with that theory. Consequently, classical field theories are usually categorized as non-relativistic and relativistic . Modern field theories are usually expressed using the mathematics of tensor calculus . A more recent alternative mathematical formalism describes classical fields as sections of mathematical objects called fiber bundles . Michael Faraday coined

3772-422: The best glimpses into what it means to be a mathematician. The following list contains some works that are not autobiographies, but rather essays on mathematics and mathematicians with strong autobiographical elements. Field equations A physical field can be thought of as the assignment of a physical quantity at each point of space and time . For example, in a weather forecast, the wind velocity during

3854-470: The comma indicates a partial derivative . After Newtonian gravitation was found to be inconsistent with special relativity , Albert Einstein formulated a new theory of gravitation called general relativity . This treats gravitation as a geometric phenomenon ('curved spacetime ') caused by masses and represents the gravitational field mathematically by a tensor field called the metric tensor . The Einstein field equations describe how this curvature

3936-413: The density ρ , pressure p , deviatoric stress tensor τ of the fluid, as well as external body forces b , are all given. The velocity field u is the vector field to solve for. In 1839, James MacCullagh presented field equations to describe reflection and refraction in "An essay toward a dynamical theory of crystalline reflection and refraction". The term " potential theory " arises from

4018-449: The derivative of the Lagrangian density with respect to the field components ∂ L ∂ A a = μ 0 j a , {\displaystyle {\frac {\partial {\mathcal {L}}}{\partial A_{a}}}=\mu _{0}j^{a}\,,} and the derivatives of the field components ∂ L ∂ ( ∂ b A

4100-440: The direction of the field points from the position r to the position of the masses r i ; this is ensured by the minus sign. In a nutshell, this means all masses attract. In the integral form Gauss's law for gravity is ∬ g ⋅ d S = − 4 π G M {\displaystyle \iint \mathbf {g} \cdot d\mathbf {S} =-4\pi GM} while in differential form it

4182-500: The earliest known mathematicians was Thales of Miletus ( c.  624  – c.  546 BC ); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry , by deriving four corollaries to Thales's theorem . The number of known mathematicians grew when Pythagoras of Samos ( c.  582  – c.  507 BC ) established

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4264-407: The electric field due to a single charged particle is E = 1 4 π ε 0 Q r 2 r ^ . {\displaystyle \mathbf {E} ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r^{2}}}{\hat {\mathbf {r} }}\,.} The electric field is conservative , and hence is given by the gradient of

4346-486: The fact that, in 19th century physics, the fundamental forces of nature were believed to be derived from scalar potentials which satisfied Laplace's equation . Poisson addressed the question of the stability of the planetary orbits , which had already been settled by Lagrange to the first degree of approximation from the perturbation forces, and derived the Poisson's equation , named after him. The general form of this equation

4428-476: The fluid, ∂ ∂ t ( ρ u ) + ∇ ⋅ ( ρ u ⊗ u + p I ) = ∇ ⋅ τ + ρ b {\displaystyle {\frac {\partial }{\partial t}}(\rho \mathbf {u} )+\nabla \cdot (\rho \mathbf {u} \otimes \mathbf {u} +p\mathbf {I} )=\nabla \cdot {\boldsymbol {\tau }}+\rho \mathbf {b} } if

4510-494: The focus of universities in the 19th and 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. The German university system fostered professional, bureaucratically regulated scientific research performed in well-equipped laboratories, instead of

4592-992: The following. There is no Nobel Prize in mathematics, though sometimes mathematicians have won the Nobel Prize in a different field, such as economics or physics. Prominent prizes in mathematics include the Abel Prize , the Chern Medal , the Fields Medal , the Gauss Prize , the Nemmers Prize , the Balzan Prize , the Crafoord Prize , the Shaw Prize , the Steele Prize , the Wolf Prize ,

4674-1002: The general divergence theorem , specifically Gauss's law's for gravity and electricity. For the cases of time-independent gravity and electromagnetism, the fields are gradients of corresponding potentials g = − ∇ ϕ g , E = − ∇ ϕ e {\displaystyle \mathbf {g} =-\nabla \phi _{g}\,,\quad \mathbf {E} =-\nabla \phi _{e}} so substituting these into Gauss' law for each case obtains ∇ 2 ϕ g = 4 π G ρ g , ∇ 2 ϕ e = 4 π k e ρ e = − ρ e ε 0 {\displaystyle \nabla ^{2}\phi _{g}=4\pi G\rho _{g}\,,\quad \nabla ^{2}\phi _{e}=4\pi k_{e}\rho _{e}=-{\rho _{e} \over \varepsilon _{0}}} where ρ g

4756-773: The identification of the gravitational field strength as identical to the acceleration experienced by a particle. This is the starting point of the equivalence principle , which leads to general relativity . For a discrete collection of masses, M i , located at points, r i , the gravitational field at a point r due to the masses is g ( r ) = − G ∑ i M i ( r − r i ) | r − r i | 3 , {\displaystyle \mathbf {g} (\mathbf {r} )=-G\sum _{i}{\frac {M_{i}(\mathbf {r} -\mathbf {r_{i}} )}{|\mathbf {r} -\mathbf {r} _{i}|^{3}}}\,,} If we have

4838-629: The imposing problems presented in related scientific fields. With professional focus on a wide variety of problems, theoretical systems, and localized constructs, applied mathematicians work regularly in the study and formulation of mathematical models . Mathematicians and applied mathematicians are considered to be two of the STEM (science, technology, engineering, and mathematics) careers. The discipline of applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry; thus, "applied mathematics"

4920-411: The interaction of charged matter with the electromagnetic field. The first formulation of this field theory used vector fields to describe the electric and magnetic fields. With the advent of special relativity, a more complete formulation using tensor fields was found. Instead of using two vector fields describing the electric and magnetic fields, a tensor field representing these two fields together

5002-460: The interaction term, and this gives us L = − 1 4 μ 0 F a b F a b − j a A a . {\displaystyle {\mathcal {L}}=-{\frac {1}{4\mu _{0}}}F^{ab}F_{ab}-j^{a}A_{a}\,.} To obtain the field equations, the electromagnetic tensor in the Lagrangian density needs to be replaced by its definition in terms of

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5084-569: The kind of research done by private and individual scholars in Great Britain and France. In fact, Rüegg asserts that the German system is responsible for the development of the modern research university because it focused on the idea of "freedom of scientific research, teaching and study." Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education , and then proceed to specialize in topics of their own choice at

5166-470: The king of Prussia , Fredrick William III , to build a university in Berlin based on Friedrich Schleiermacher 's liberal ideas; the goal was to demonstrate the process of the discovery of knowledge and to teach students to "take account of fundamental laws of science in all their thinking." Thus, seminars and laboratories started to evolve. British universities of this period adopted some approaches familiar to

5248-678: The merits of one of his papers on light ... on the occasion of presenting him with a Gold Medal in 1838 ... followed his coffin on foot from the College through the streets of Dublin; cooperated in procuring a pension for his sisters; and subscribed to the MacCullogh Testimonial. Mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems . Mathematicians are concerned with numbers , data , quantity , structure , space , models , and change . One of

5330-404: The most well-known Lorentz-covariant classical field theories are now described. Historically, the first (classical) field theories were those describing the electric and magnetic fields (separately). After numerous experiments, it was found that these two fields were related, or, in fact, two aspects of the same field: the electromagnetic field . Maxwell 's theory of electromagnetism describes

5412-537: The other two (Gauss' law for magnetism and Faraday's law) are obtained from the fact that F is the 4-curl of A , or, in other words, from the fact that the Bianchi identity holds for the electromagnetic field tensor. 6 F [ a b , c ] = F a b , c + F c a , b + F b c , a = 0. {\displaystyle 6F_{[ab,c]}\,=F_{ab,c}+F_{ca,b}+F_{bc,a}=0.} where

5494-531: The probability and likely cost of the occurrence of an event such as death, sickness, injury, disability, or loss of property. Actuaries also address financial questions, including those involving the level of pension contributions required to produce a certain retirement income and the way in which a company should invest resources to maximize its return on investments in light of potential risk. Using their broad knowledge, actuaries help design and price insurance policies, pension plans, and other financial strategies in

5576-484: The real world, many applied mathematicians draw on tools and techniques that are often considered to be "pure" mathematics. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research. Many professional mathematicians also engage in the teaching of mathematics. Duties may include: Many careers in mathematics outside of universities involve consulting. For instance, actuaries assemble and analyze data to estimate

5658-403: The seventeenth century at Oxford with the scientists Robert Hooke and Robert Boyle , and at Cambridge where Isaac Newton was Lucasian Professor of Mathematics & Physics . Moving into the 19th century, the objective of universities all across Europe evolved from teaching the "regurgitation of knowledge" to "encourag[ing] productive thinking." In 1810, Alexander von Humboldt convinced

5740-595: The speed of light in vacuum is 1, i.e. c = 1. Given a field tensor ϕ {\displaystyle \phi } , a scalar called the Lagrangian density L ( ϕ , ∂ ϕ , ∂ ∂ ϕ , … , x ) {\displaystyle {\mathcal {L}}(\phi ,\partial \phi ,\partial \partial \phi ,\ldots ,x)} can be constructed from ϕ {\displaystyle \phi } and its derivatives. From this density,

5822-773: The system in terms of its scalar and vector potentials V and A . A set of integral equations known as retarded potentials allow one to calculate V and A from ρ and J , and from there the electric and magnetic fields are determined via the relations E = − ∇ V − ∂ A ∂ t {\displaystyle \mathbf {E} =-\nabla V-{\frac {\partial \mathbf {A} }{\partial t}}} B = ∇ × A . {\displaystyle \mathbf {B} =\nabla \times \mathbf {A} .} Fluid dynamics has fields of pressure, density, and flow rate that are connected by conservation laws for energy and momentum. The mass continuity equation

5904-461: The term "field" and lines of forces to explain electric and magnetic phenomena. Lord Kelvin in 1851 formalized the concept of field in different areas of physics. Some of the simplest physical fields are vector force fields. Historically, the first time that fields were taken seriously was with Faraday's lines of force when describing the electric field . The gravitational field was then similarly described. The first field theory of gravity

5986-424: The value of [the potential function], we may now take it for the starting point of our theory, and dismiss the assumptions by which we were conducted to it." Despite the success of the theory, physicists and mathematicians were not receptive to the idea of reducing physics to a set of abstract field equations divorced from a mechanical model. The notion of the ether as a compressible fluid or similar physical entity

6068-938: Was Al-Khawarizmi . A notable feature of many scholars working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics , maths and astronomy of Ibn al-Haytham . The Renaissance brought an increased emphasis on mathematics and science to Europe. During this period of transition from a mainly feudal and ecclesiastical culture to a predominantly secular one, many notable mathematicians had other occupations: Luca Pacioli (founder of accounting ); Niccolò Fontana Tartaglia (notable engineer and bookkeeper); Gerolamo Cardano (earliest founder of probability and binomial expansion); Robert Recorde (physician) and François Viète (lawyer). As time passed, many mathematicians gravitated towards universities. An emphasis on free thinking and experimentation had begun in Britain's oldest universities beginning in

6150-462: Was Newton's theory of gravitation in which the mutual interaction between two masses obeys an inverse square law . This was very useful for predicting the motion of planets around the Sun. Any massive body M has a gravitational field g which describes its influence on other massive bodies. The gravitational field of M at a point r in space is found by determining the force F that M exerts on

6232-431: Was accepted that his radical choice ruled out any hope for a mechanical model for the ethereal medium. Nevertheless, the field equations stemming from this purely gyrostatic medium were shown to be in accord with all known laws, including those of Snell and Augustin-Jean Fresnel . At several points, MacCullagh addresses the physical nature of an ethereal medium having such properties. Not surprisingly, he argues against

6314-512: Was difficult to live with him; and I am thankful that I escaped, so well as I did, from a quarrel, partly because I do not live in College, nor in Dublin. I fear that all this must seem a little unkind; but you will understand me. I was on excellent terms with MacCullogh; ... spoke of those early papers of his, in 1832, to the British Association, when it first met at Oxford; took pains to exhibit

6396-431: Was ongoing throughout the reign of certain caliphs, and it turned out that certain scholars became experts in the works they translated, and in turn received further support for continuing to develop certain sciences. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences. An example of a translator and mathematician who benefited from this type of support

6478-546: Was presented to the Royal Irish Academy in December 1839. The paper begins by defining what was then a new concept, the curl of a vector field . (The term 'curl' was first used by James Clerk Maxwell in 1870.) MacCullagh first showed that the curl is a covariant vector in the sense that its components are transformed in the appropriate manner under coordinate rotation. Taking his cue from George Green , he set out to develop

6560-578: Was published in the mid-to-late 1830s; his most significant work on geometry On surfaces of the second order was published in 1843. He was awarded the Cunningham Medal of the Royal Irish Academy in 1838 for his paper on On the laws of crystalline reflexion and refraction . He won the Copley medal for his work on the nature of light in 1842. MacCullagh was involved with the British Association for

6642-533: Was too deeply ingrained in nineteenth-century physical thinking, even for decades after the publication of Maxwell's electromagnetic theory in 1864. MacCullagh's ideas were largely abandoned and forgotten until 1880, when George Francis FitzGerald re-discovered and re-interpreted his findings in the light of Maxwell's work. William Thomson, 1st Baron Kelvin succeeded in developing a physically realizable model of MacCullagh's rotationally elastic but translationally insensitive ether, consisting of gyrostats mounted on

6724-569: Was unveiled at his family tomb at St Patrick's Church in Upper Badoney. The plaque was part of events organised by the Glenelly Historical Society to mark his life. ... my intercourse with poor MacCullogh, who was constantly fancying that people were plundering his stores, which certainly were worth the robbing. This was no doubt a sort of premonitory symptom of that insanity that produced his awful end. He could inspire love and yet it

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