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"Joseph Fourier was mathematician, known also as an Egyptologist and \ administrator, who exerted strong influence on mathematical physics through \ his ", StyleBox["Th\[EAcute]orie analytique de la chaleur", FontSlant->"Italic"], " (1822; The Analytical Theory of Heat). He showed how the conduction of \ heat in solid bodies may be analyzed in terms of infinite mathematical series \ now called by his name, the ", StyleBox["Fourier series", FontWeight->"Bold"], ". Far transcending the particular subject of heat conduction, his work \ stimulated research in mathematical physics, which has since been often \ identified with the solution of boundary-value problems, encompassing many \ natural occurrences such as sunspots, tides, and the weather. " }], "Text"], Cell["\<\ After working as a teacher in mathematics at the military school of his youth \ and later as a Professor of mathematics at the \[CapitalEAcute]cole \ Polytechnique, Fourier accompanied Napoleon on his expedition to Egypt. Until \ 1801 he was engaged in extensive research on Egyptian antiquities, gave \ advice on engineering and diplomatic undertakings, and served for three years \ as the secretary of the Institut d'\[CapitalEAcute]gypte, which Napoleon \ establishedin Cairo in 1798.\ \>", "Text"], Cell[TextData[{ "After his return to France, Fourier was charged with the publication of the \ enormous mass of Egyptian materials. This became the ", StyleBox["Description de l'\[CapitalEAcute]gypte", FontSlant->"Italic"], ", to which he also wrote a lengthy historical preface on the ancient \ civilization of Egypt. He was also appointed prefect (administrator for the \ national government and d\[EAcute]partement) of the Is\[EGrave]re \ d\[EAcute]partement, a position he held from 1802 to 1814, with his \ headquarters at Grenoble. He showed great administrative ability, as in \ directing the drainage of swamps, while continuing his Egyptological and \ mathematical work. In 1809 Napoleon made him a baron. Following Napoleon's \ fall from power in 1815, Fourier was appointed director of the Statistical \ Bureau of the Seine, allowing him a period of quiet academic life in Paris. \ In 1817 he was elected to the Acad\[EAcute]mie des Sciences, of which, in \ 1822, he became perpetual secretary. Because of his work in Egyptology he was \ elected in 1826 to the Acad\[EAcute]mie Fran\[CCedilla]aise and the Acad\ \[EAcute]mie de M\[EAcute]decine." }], "Text"], Cell[TextData[{ "Fourier began his work on the", StyleBox[" Th\[EAcute]orie analytique de la chaleur", FontSlant->"Italic"], " in Grenoble in 1807 and completed it in Paris in 1822. His work enabled \ him to express the conduction of heat in two-dimensional objects (", StyleBox["i.e", FontSlant->"Italic"], "., very thin sheets of material) in terms of the differential equation\n\t\ ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[PartialD]", "t"], " ", "T"}], TraditionalForm]]], " = ", StyleBox["k", FontSlant->"Italic"], " (", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[PartialD]", RowBox[{"x", ",", "x"}]], "T"}], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[PartialD]", RowBox[{"y", ",", "y"}]], "T"}], TraditionalForm]]], ")\nthat describes the flow of heat from hot to cold on a uniform plane. The \ constant ", StyleBox["k", FontSlant->"Italic"], " is the thermal conductivity." }], "Text"], Cell[TextData[{ "Fourier introduced a series expansion for ", StyleBox["T", FontSlant->"Italic"], "(", StyleBox["x, y", FontSlant->"Italic"], ") in terms of ", StyleBox["Cos[...]", FontWeight->"Bold"], " and ", StyleBox["Sin[...] ", FontWeight->"Bold"], "functions. Although this series expansion was", " occasionally used by Leonhard Euler and other 18th-century mathematicians, \ Fourier elevated this technique to an important position in modern \ mathematics. He also extended this concept into the so-called Fourier \ integral. " }], "Text"], Cell["\<\ Today, Fourier, or spectral, techniques play an essential role in the \ solution of partial differential equations. 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