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| {{for|the interpretation of this theorem in terms of symmetry of second derivatives of a mapping <math>f \colon \mathbb{R}^n \to \mathbb{R}</math> |Symmetry of second derivatives}}
| | It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br><br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>Should you cherished this information and also you would want to get more details about [http://www.youtube.com/watch?v=90z1mmiwNS8 Washington DC Dentist] i implore you to pay a visit to our own web page. |
| [[File:Elipsoid zplostely.png|thumb |200px |Figure 1: An ellipsoid]]
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| [[File:Gnuplot ellipsoid.svg|thumb|200px|Figure 2: Wireframe rendering of an ellipsoid (oblate spheroid)]]
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| '''Clairaut's theorem''', published in 1743 by [[Alexis Clairaut|Alexis Claude Clairaut]] in his ''Théorie de la figure de la terre, tirée des principes de l'hydrostatique'',<ref name=RoyalSoc>[http://books.google.com/books?id=3owAAAAAYAAJ&pg=PA134&lpg=PA134&dq=%22Th%C3%A9orie+de+la+figure+de+la+terre%22&source=web&ots=an0JW-H3C8&sig=BMkuXfZEsK3p0tzrZ1Jvfcy7hmw&hl=en&sa=X&oi=book_result&resnum=10&ct=result From the catalogue of the scientific books in the library of the Royal Society.]</ref> synthesized physical and geodetic evidence that the Earth is an oblate rotational [[ellipsoid]].<ref name= Torge>{{cite book |title=Geodesy: An Introduction |edition=3rd |author=Wolfgang Torge |page=10 |url=http://books.google.com/books?id=pFO6VB_czRYC&pg=PA109&dq=%22Clairaut%27s+theorem%22&lr=&as_brr=0&sig=ACfU3U34GaPhl4tA9duUMLQpm77hiKb-RQ#PPA10,M1 |isbn=3-11-017072-8 |year=2001 |publisher=Walter de Gruyter }}</ref><ref name=Routh>{{cite book |author=Edward John Routh |title=A Treatise on Analytical Statics with Numerous Examples |page=154 |year=2001|isbn=1-4021-7320-2 |publisher=Adamant Media Corporation |volume=Vol. 2 |url=http://books.google.com/books?id=yKmdk4LZxhMC&pg=RA1-PA40&dq=isbn=1-4021-7320-2&sig=ACfU3U2uhAKDJtIYZEY-Jf-1e5wf7UgG1w#PPA154,M1 }} A reprint of the original work published in 1908 by Cambridge University Press.</ref> It is a general mathematical law applying to spheroids of revolution. It was initially used to relate the gravity at any point on the Earth's surface to the position of that point, allowing the [[ellipticity]] of the Earth to be calculated from measurements of gravity at different latitudes.
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| ==Formula==
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| Clairaut's formula for the acceleration due to gravity ''g'' on the surface of a spheroid at latitude φ, was:<ref name=Ball>[http://www.maths.tcd.ie/pub/HistMath/People/Clairaut/RouseBall/RB_Clairaut.html W. W. Rouse Ball ''A Short Account of the History of Mathematics'' (4th edition, 1908)]</ref><ref name=Rouse2>{{cite book |title=A short account of the history of mathematics |author= Walter William Rouse Ball |page=384 |url=http://books.google.com/books?id=O-UGAAAAYAAJ&dq=A+Short+Account+of+the+History+of+Mathematics'+(4th+edition,+1908)+by+W.+W.+Rouse+Ball.&pg=PP1&ots=327JhZ192M&sig=w-HWPhOnc6JAlzlMoralry7rIL4&hl=en&sa=X&oi=book_result&resnum=1&ct=result#PPA384,M1
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| |year=1901 |publisher=Macmillan |edition=3rd }}</ref>
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| :<math> g = G \left[ 1 + \left(\frac{5}{2} m - f\right) \sin^2 \phi \right] \ , </math>
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| where ''G'' is the value of the acceleration of gravity at the equator, ''m'' the ratio of the centrifugal force to gravity at the equator, and ''f'' the [[flattening]] of a [[meridian (geography)|meridian]] section of the earth, defined as:
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| :<math>f = \frac {a-b}{a} \ , </math>
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| (where ''a'' = semimajor axis, ''b''=semiminor axis ). | |
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| Clairaut derived the formula under the assumption that the body was composed of concentric coaxial spheroidal layers of constant density.<ref>{{cite book
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| | last = Poynting
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| | first = John Henry
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| | authorlink =
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| | coauthors = Joseph John Thompson
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| | title = A Textbook of Physics, 4th Ed.
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| | publisher = Charles Griffin & Co.
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| | year = 1907
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| | location = London
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| | pages = 22–23
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| | url = http://books.google.com/books?id=TL4KAAAAIAAJ&pg=PA22
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| | doi =
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| | id =
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| | isbn = }}</ref>
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| This work was subsequently pursued by [[Pierre-Simon Laplace|Laplace]], who relaxed the initial assumption that surfaces of equal density were spheroids.<ref name=Todhunter>{{cite book |author=Isaac Todhunter |title=A History of the Mathematical Theories of Attraction and the Figure of the Earth from the Time of Newton to that of Laplace |volume=Vol. 2 |publisher=Elibron Classics |isbn=1-4021-1717-5 |url=http://books.google.com/books?id=blZ_Tar9IRMC&pg=RA1-PA500&dq=%22Clairaut%27s+theorem%22&lr=&as_brr=0&sig=ACfU3U0BK0IMg4DPZTFon_yf_DyT4wOlcQ#PPA62,M1 }} Reprint of the original edition of 1873 published by Macmillan and Co.</ref> | |
| [[Sir George Stokes, 1st Baronet|Stokes]] showed in 1849 that the theorem applied to any law of density so long as the external surface is a spheroid of equilibrium.<ref name=Fisher>{{cite book |title=Physics of the Earth's Crust |author=Osmond Fisher |page=27 |url=http://books.google.com/books?id=o8oPAAAAIAAJ&pg=PA27&dq=%22Clairaut%27s+theorem%22&lr=&as_brr=0
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| |year=1889 |publisher=Macmillan and Co. }}</ref><ref name= Poynting>{{cite book |title=A Textbook of Physics |author= John Henry Poynting & Joseph John Thomson |url=http://books.google.com/books?id=TL4KAAAAIAAJ&pg=PA23&dq=%22Clairaut%27s+theorem%22&lr=&as_brr=0#PPA22,M1 |page=22 |year=1907 |publisher=C. Griffin }}</ref> A history of the subject, and more detailed equations for ''g'' can be found in Khan.<ref name=Khan>[http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19690003446_1969003446.pdf NASA case file ''On the equilibrium figure of the earth'' by Mohammad A. Khan (1968)]</ref>
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| The above expression for ''g'' has been supplanted by the Somigliana equation:
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| :<math>g = G \left[ \frac{1+k\sin^2 \phi}{\sqrt{1-e^2 \sin^2 \phi }} \right] \ , </math>
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| where, for the Earth, G =9.7803267714 ms<sup>−2</sup>; k =0.00193185138639 ; e<sup>2</sup> =0.00669437999013.<ref name=Somigliana>[http://ocw.mit.edu/NR/rdonlyres/Earth--Atmospheric--and-Planetary-Sciences/12-201Fall-2004/E7A9DF78-ADC6-49A7-8812-1D8244939398/0/ch2.pdf Eq. 2.57 in MIT Earth Atmospheric and Planetary Sciences OpenCourseWare notes]</ref>
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| ==Clairaut's relation==
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| {{Main|Clairaut's relation}}
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| A formal mathematical statement of the (unrelated) Clairaut's theorem is:<ref name=Pressley>{{cite book |author=Andrew Pressley |title=Elementary Differential Geometry |page=183 |url=http://books.google.com/books?id=UXPyquQaO6EC&pg=PA185&dq=%22Clairaut%27s+theorem%22&lr=&as_brr=0&sig=ACfU3U214J0zkWQcRXLTohVjdHUD3Fuk2A#PPA183,M1
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| |isbn=1-85233-152-6 |publisher=Springer |year=2001 }}</ref>
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| {{quotation|Let γ be a [[geodesic]] on a [[surface of revolution]] ''S'', let ρ be the distance of a point of ''S'' from the [[axis of rotation]], and let ψ be the angle between γ and the [[Meridian (geography)|meridians]] of ''S''. Then ρ sin ψ is constant along γ. Conversely, if ρ sin ψ is constant along some curve γ in the surface, and if no part of γ is part of some parallel of ''S'', then γ is a geodesic.|Andrew Pressley: ''Elementary Differential Geometry'', p. 183}}
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| Pressley (p. 185) explains this theorem as an expression of conservation of angular momentum about the axis of revolution when a particle slides along a geodesic under no forces other than those that keep it on the surface.
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| ==Geodesy==
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| The spheroidal shape of the Earth is the result of the interplay between [[gravity]] and [[centrifugal force]] caused by the Earth's rotation about its axis.<ref name=Vinti>{{cite book |title=Orbital and Celestial Mechanics |series=Progress in astronautics and aeronautics, v. 177 |author=John P. Vinti, Gim J. Der, Nino L. Bonavito |page=171 |url=http://books.google.com/books?id=-dXzdYHvPgMC&pg=PA172&dq=Earth+spheroid+centrifugal+date:1990-2008&lr=&as_brr=0&sig=ACfU3U0YCa9N8606CejuHyuolKmh56JOtw#PPA171,M1 |isbn=1-56347-256-2 |year=1998 |publisher=American Institute of Aeronautics and Astronautics}}</ref><ref name=Webster>{{cite book |title=The Dynamics of Particles and of Rigid, Elastic, and Fluid Bodies: being lectures on mathematical physics |author=Arthur Gordon Webster |year=1904 |publisher=B.G. Teubner |url=http://books.google.com/books?id=2kMNAAAAYAAJ&printsec=titlepage#PPA468,M1 |page=468 }}</ref> In his ''Principia'', [[Isaac Newton|Newton]] proposed the equilibrium shape of a homogeneous rotating Earth was a rotational ellipsoid with a flattening ''f'' given by 1/230.<ref name=Newton>Isaac Newton: ''Principia'' Book III Proposition XIX Problem III, p. 407 in Andrew Motte translation.</ref><ref name=Principia>See the ''Principia'' on line at [http://ia310114.us.archive.org/2/items/newtonspmathema00newtrich/newtonspmathema00newtrich.pdf Andrew Motte Translation]</ref> As a result gravity increases from the equator to the poles. By applying Clairaut's theorem, [[Pierre-Simon Laplace|Laplace]] was able to deduce from 15 gravity values that ''f'' = 1/330. A modern estimate is 1/298.25642.<ref>[ftp://tai.bipm.org/iers/convupdt/chapter1/icc1.pdf Table 1.1 IERS Numerical Standards (2003)])</ref> See [[Figure of the Earth]] for more detail. | |
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| For a detailed account of the construction of the [[Reference ellipsoid|reference Earth model]] of geodesy, see Chatfield.<ref name=Chatfield>{{cite book |title= Fundamentals of High Accuracy Inertial Navigation |url=http://books.google.com/books?id=2hJTDpT2U1UC&pg=PA1&dq=frame+coordinate+%22state+of+motion%22&lr=&as_brr=0&sig=ACfU3U2NOYvih-VaDyv1CxAkTc7L1AaRXQ#PPA7,M1
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| |isbn=1-56347-243-0 |year=1997 |author=Averil B. Chatfield |publisher=American Institute of Aeronautics and Astronautics |series=Volume 174 in ''Progress in Astronautics and Aeronautics'' |nopp= true |pages= Chapter 1, Part VIII p. 7 }}</ref>
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| ==References==
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| <references/>
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| [[Category:Geodesy]]
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| [[Category:Global Positioning System]]
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| [[Category:Navigation]]
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| [[Category:Surveying]]
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| [[Category:Physics theorems]]
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| [[Category:Gravimetry]]
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It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.
Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.
At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.
Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.
Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.
Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.
Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.
Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.
In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.
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