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Great Oxygenation Event
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O2 build-up in the Earth's atmosphere. Red and green lines represent the range of the estimates while time is measured in billions of years ago (Ga).
Stage 1 (3.85–2.45 Ga): Practically no O2 in the atmosphere. The oceans were also largely anoxic with the possible exception of O2 gases in the shallow oceans.
Stage 2 (2.45–1.85 Ga): O2 produced, and rose to values of 0.02 and 0.04 atm, but absorbed in oceans and seabed rock.
Stage 3 (1.85–0.85 Ga): O2 starts to gas out of the oceans, but is absorbed by land surfaces. There was no significant change in terms of oxygen level.
Stages 4 and 5 (0.85–present): O2 sinks filled and the gas accumulates.[1]
The Great Oxygenation Event, the beginning of which is commonly known in scientific media as the Great Oxidation Event (GOE, also called the Oxygen Catastrophe, Oxygen Crisis, Oxygen Holocaust,[2] Oxygen Revolution, or Great Oxidation) was the biologically induced appearance of dioxygen (O2) in Earth's atmosphere.[3] Geological, isotopic, and chemical evidence suggest that this major environmental change happened around 2.45 billion years ago (2.45 Ga),[4] during the Siderian period, at the beginning of the Proterozoic eon. The causes of the event are not clear.[5] The current geochemical and biomarker evidence for the development of oxygenic photosynthesis before the Great Oxidation Event has been mostly inconclusive.[6]
Oceanic cyanobacteria, which evolved into coordinated (but not multicellular or even colonial) macroscopic forms more than 2.3 billion years ago (approximately 200 million years before the GOE),[7] are believed to have become the first microbes to produce oxygen by photosynthesis.[8] Before the GOE, any free oxygen they produced was chemically captured by dissolved iron or organic matter. The GOE started when these oxygen sinks became saturated, at which point oxygen produced by the cyanobacteria was free to escape into the atmosphere.
Cyanobacteria: Responsible for the buildup of oxygen in the earth's atmosphere
The increased production of oxygen set Earth's original atmosphere off balance.[9] Free oxygen is toxic to obligate anaerobic organisms, and the rising concentrations may have destroyed most such organisms at the time. Cyanobacteria were therefore responsible for one of the most significant mass extinctions in Earth's history. Besides marine cyanobacteria, there is also evidence of cyanobacteria on land.[citation needed]
A spike in chromium contained in ancient rock deposits formed underwater shows the accumulation had been washed off from the continental shelves. Chromium is not easily dissolved and its release from rocks would have required the presence of a powerful acid. One such acid, sulfuric acid (H2SO4), might have been created through bacterial reactions with pyrite.[10] Mats of oxygen-producing cyanobacteria can produce a thin layer, one or two millimeters thick, of oxygenated water in an otherwise anoxic environment even under thick ice, and before oxygen started accumulating in the atmosphere, these organisms would already be adapted to oxygen.[11] Additionally, the free oxygen would have reacted with atmospheric methane, a greenhouse gas, greatly reducing its concentration and triggering the Huronian glaciation, possibly the longest episode of glaciation in Earth's history and called snowball Earth.[12]
Eventually, the evolution of aerobic organisms that consumed oxygen established an equilibrium in its availability. Free oxygen has been an important constituent of the atmosphere ever since.[12]
Contents
[hide]
1 Timing
2 Time lag theory
2.1 Tectonic trigger
2.2 Nickel famine
2.3 Bistability
2.4 Hydrogen gas
3 Late evolution of oxy-photosynthesis theory
4 Role in mineral diversification
5 See also
6 References
7 External links
Timing[edit]
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Axis scale: millions of years ago.
Orange labels: ice ages.
Also see: Human timeline and Nature timeline
See also: Oxygen cycle
The most widely accepted chronology of the Great Oxygenation Event suggests that free oxygen was first produced by prokaryotic and then later eukaryotic organisms that carried out photosynthesis more efficiently, producing oxygen as a waste product. These organisms lived long before the GOE,[13] perhaps as early as 3,500 million years ago.
Initially, the oxygen they produced would have quickly been removed from the atmosphere by the chemical weathering of reducing (oxidizable) minerals, most notably iron. This 'mass rusting' led to the deposition of iron(III) oxide in the form of banded-iron formations such as the sediments in Minnesota and Pilbara, Western Australia. The saturation of these mineral sinks, and the resulting persistence of oxygen in the atmosphere, led within 50 million years to the start of the GOE.[14] Oxygen could have accumulated very rapidly: at today's rates of photosynthesis (much greater than those in the Precambrian without land plants), modern atmospheric O2 levels could be produced in only 2,000 years.[15]
Another hypothesis is that oxygen producers did not evolve until a few million years before the major rise in atmospheric oxygen concentration.[16] This is based on a particular interpretation of a supposed oxygen indicator used in previous studies, the mass-independent fractionation of sulfur isotopes. This hypothesis would eliminate the need to explain a lag in time between the evolution of oxyphotosynthetic microbes and the rise in free oxygen.
In either case, oxygen did eventually accumulate in the atmosphere, with two major consequences.
Firstly, it oxidized atmospheric methane (a strong greenhouse gas) to carbon dioxide (a weaker one) and water. This decreased the greenhouse effect of the Earth's atmosphere, causing planetary cooling, and triggered the Huronian glaciation. Starting around 2.4 billion years ago, this lasted 300-400 million years, and may have been the longest ever snowball Earth episode.[16][17]
Secondly, the increased oxygen concentrations provided a new opportunity for biological diversification, as well as tremendous changes in the nature of chemical interactions between rocks, sand, clay, and other geological substrates and the Earth's air, oceans, and other surface waters. Despite the natural recycling of organic matter, life had remained energetically limited until the widespread availability of oxygen. This breakthrough in metabolic evolution greatly increased the free energy availabile to living organisms, with global environmental impacts. For example, mitochondria evolved after the GOE, giving organisms the energy to exploit new, more complex morphologies interacting in increasingly complex ecosystems.[18]
Timeline of glaciations, shown in blue.
Time lag theory[edit]
There may have been a gap of up to 900 million years between the start of photosynthetic oxygen production and the geologically rapid increase in atmospheric oxygen about 2.5–2.4 billion years ago. Several hypotheses propose to explain this time lag.
Tectonic trigger[edit]
2.1 billion year old rock showing banded iron formation
The oxygen increase had to await tectonically driven changes in the Earth, including the appearance of shelf seas, where reduced organic carbon could reach the sediments and be buried.[19] The newly produced oxygen was first consumed in various chemical reactions in the oceans, primarily with iron. Evidence is found in older rocks that contain massive banded iron formations apparently laid down as this iron and oxygen first combined; most present-day iron ore lies in these deposits. Evidence suggests oxygen levels spiked each time smaller land masses collided to form a super-continent. Tectonic pressure thrust up mountain chains, which eroded to release nutrients into the ocean to feed photosynthetic cyanobacteria.[20]
Nickel famine[edit]
Early chemosynthetic organisms likely produced methane, an important trap for molecular oxygen, since methane readily oxidizes to carbon dioxide (CO2) and water in the presence of UV radiation. Modern methanogens require nickel as an enzyme cofactor. As the Earth's crust cooled and the supply of volcanic nickel dwindled, oxygen-producing algae began to out-perform methane producers, and the oxygen percentage of the atmosphere steadily increased.[21] From 2.7 to 2.4 billion years ago, the rate of deposition of nickel declined steadily from a level 400 times today's.[22]
Bistability[edit]
Another hypothesis posits a model of the atmosphere that exhibits bistability: two steady states of oxygen concentration. The state of stable low oxygen conentration (0.02%) experiences a high rate of methane oxidation. If some event raises oxygen levels beyond a moderate threshold, the formation of an ozone layer shields UV rays and decreases methane oxidation, raising oxygen further to a stable state of 21% or more. The Great Oxygenation Event can then be understood as a transition from the lower to the upper steady states.[23]
Hydrogen gas[edit]
Another theory credits the appearance of cyanobacteria with suppressing hydrogen gas and increasing oxygen.
Some bacteria in the early oceans could separate water into hydrogen and oxygen. Under the Sun's rays, hydrogen molecules were incorporated into organic compounds, with oxygen as a by-product. If the hydrogen-heavy compounds were buried, it would have allowed oxygen to accumulate in the atmosphere.
However, in 2001 scientists realized that the hydrogen would instead escape into space through a process called methane photolysis, in which methane releases its hydrogen in a reaction with oxygen. This could explain why the early Earth stayed warm enough to sustain oxygen-producing lifeforms.[24]
Late evolution of oxy-photosynthesis theory[edit]
The oxygen indicator might have been misinterpreted. During the proposed lag era in the previous theory, there was a change in sediments from mass-independently fractionated (MIF) sulfur to mass-dependently fractionated (MDF) sulfur. This was assumed to show the appearance of oxygen in the atmosphere, since oxygen would have prevented the photolysis of sulfur dioxide, which causes MIF. However, the change from MIF to MDF of sulfur isotopes may instead have been caused by an increase in glacial weathering, or the homogenization of the marine sulfur pool as a result of an increased thermal gradient during the Huronian glaciation period (which in this interpretation was not caused by oxygenation).[16]
Role in mineral diversification[edit]
The Great Oxygenation Event triggered an explosive growth in the diversity of minerals, with many elements occuring in one or more oxidized forms near the Earth's surface.[25] It is estimated that the GOE was directly responsible for more than 2,500 of the total of about 4,500 minerals found on Earth today. Most of these new minerals were formed as hydrated and oxidized forms due to dynamic mantle and crust processes.[26]
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See also[edit]
Geological history of oxygen
Iodide
Medea hypothesis
Pasteur point
Rare Earth hypothesis
Stromatolite
References[edit]
Jump up ^ Holland, Heinrich D. "The oxygenation of the atmosphere and oceans". Philosophical Transactions of the Royal Society: Biological Sciences. Vol. 361. 2006. pp. 903–915.
Jump up ^ Margulis, Lynn; Sagan, Dorion (1986). "Chapter 6, "The Oxygen Holocaust"". Microcosmos: Four Billion Years of Microbial Evolution. California: University of California Press. p. 99. ISBN 9780520210646.
Jump up ^ Sosa Torres, Martha E.; Saucedo-Vázquez, Juan P.; Kroneck, Peter M.H. (2015). "Chapter 1, Section 2 "The rise of dioxygen in the atmosphere"". In Peter M.H. Kroneck and Martha E. Sosa Torres. Sustaining Life on Planet Earth: Metalloenzymes Mastering Dioxygen and Other Chewy Gases. Metal Ions in Life Sciences. 15. Springer. pp. 1–12. doi:10.1007/978-3-319-12415-5_1.
Jump up ^ Zimmer, Carl (3 October 2013). "Earth's Oxygen: A Mystery Easy to Take for Granted". The New York Times. Retrieved 3 October 2013.
Jump up ^ "University of Zurich. "Great Oxidation Event: More oxygen through multicellularity." ScienceDaily. ScienceDaily, 17 January 2013".
Jump up ^ Planavsky, Noah J.; et al. (24 January 2014). "Evidence for oxygenic photosynthesis half a billion years before the Great Oxidation Event". Nature. 7: 283–286. doi:10.1038/ngeo2122. Retrieved 14 March 2016.
Jump up ^ Flannery, D. T.; R.M. Walter (2012). "Archean tufted microbial mats and the Great Oxidation Event: new insights into an ancient problem". Australian Journal of Earth Sciences. 59 (1): 1–11. Bibcode:2012AuJES..59....1F. doi:10.1080/08120099.2011.607849.
Jump up ^ "The Rise of Oxygen - Astrobiology Magazine". Astrobiology Magazine. Retrieved 2016-04-06.
Jump up ^ "University of Zurich. "Great Oxidation Event: More oxygen through multicellularity." ScienceDaily. ScienceDaily, 17 January 2013".
Jump up ^ "Evidence of Earliest Oxygen-Breathing Life on Land Discovered". LiveScience.com. Retrieved 2016-04-06.
Jump up ^ Oxygen oasis in Antarctic lake reflects Earth in distant past
^ Jump up to: a b Frei, R.; Gaucher, C.; Poulton, S. W.; Canfield, D. E. (2009). "Fluctuations in Precambrian atmospheric oxygenation recorded by chromium isotopes". Nature. 461 (7261): 250–253. Bibcode:2009Natur.461..250F. doi:10.1038/nature08266. PMID 19741707. Lay summary.
Jump up ^ Dutkiewicz, A.; Volk, H.; George, S. C.; Ridley, J.; Buick, R. (2006). "Biomarkers from Huronian oil-bearing fluid inclusions: an uncontaminated record of life before the Great Oxidation Event". Geology. 34 (6): 437. Bibcode:2006Geo....34..437D. doi:10.1130/G22360.1.
Jump up ^ Anbar, A.; Duan, Y.; Lyons, T.; Arnold, G.; Kendall, B.; Creaser, R.; Kaufman, A.; Gordon, G.; Scott, C.; Garvin, J.; Buick, R. (2007). "A whiff of oxygen before the great oxidation event?". Science. 317 (5846): 1903–1906. Bibcode:2007Sci...317.1903A. doi:10.1126/science.1140325. PMID 17901330.
Jump up ^ Dole, M. (1965). "The Natural History of Oxygen". The Journal of General Physiology. 49 (1): Suppl:Supp5–27. doi:10.1085/jgp.49.1.5. PMC 2195461  . PMID 5859927.
^ Jump up to: a b c Robert E. Kopp; Joseph L. Kirschvink; Isaac A. Hilburn; Cody Z. Nash (2005). "The Paleoproterozoic snowball Earth: A climate disaster triggered by the evolution of oxygenic photosynthesis". Proceedings of the National Academy of Sciences of the United States of America. 102 (32): 11131–6. Bibcode:2005PNAS..10211131K. doi:10.1073/pnas.0504878102. PMC 1183582  . PMID 16061801.
Jump up ^ First breath: Earth's billion-year struggle for oxygen New Scientist, #2746, 5 February 2010 by Nick Lane.
Jump up ^ Sperling, Erik; Frieder, Christina; Raman, Akkur; Girguis, Peter; Levin, Lisa; Knoll, Andrew (Aug 2013). "Oxygen, ecology, and the Cambrian radiation of animals". Proceedings of the National Academy of Sciences of the United States of America. 110: 13446–13451. doi:10.1073/pnas.1312778110. PMC 3746845  . PMID 23898193. Retrieved 1 October 2014.
Jump up ^ Lenton, T. M.; H. J. Schellnhuber; E. Szathmáry (2004). "Climbing the co-evolution ladder". Nature. 431 (7011): 913. Bibcode:2004Natur.431..913L. doi:10.1038/431913a. PMID 15496901.
Jump up ^ American, Scientific. "Abundant Oxygen Indirectly Due to Tectonics". Scientific American. Retrieved 2016-04-06.
Jump up ^ American, Scientific. "Breathing Easy Thanks to the Great Oxidation Event". Scientific American. Retrieved 2016-04-06.
Jump up ^ Kurt O. Konhauser; et al. (2009). "Oceanic nickel depletion and a methanogen famine before the Great Oxidation Event". Nature. 458 (7239): 750–753. Bibcode:2009Natur.458..750K. doi:10.1038/nature07858. PMID 19360085.
Jump up ^ Goldblatt, C.; T.M. Lenton; A.J. Watson (2006). "The Great Oxidation at 2.4 Ga as a bistability in atmospheric oxygen due to UV shielding by ozone" (PDF). Geophysical Research Abstracts. 8: 00770.
Jump up ^ Franzen, Harald. "New Theory Explains How Earth's Early Atmosphere Became Oxygen-Rich". Scientific American. Retrieved 2016-04-06.
Jump up ^ Sverjensky, Dimitri A.; Lee, Namhey (2010-02-01). "The Great Oxidation Event and Mineral Diversification". Elements. 6 (1): 31–36. doi:10.2113/gselements.6.1.31. ISSN 1811-5209.
Jump up ^ "Evolution of Minerals", Scientific American, March 2010
External links[edit]
First breath: Earth's billion-year struggle for oxygen New Scientist, #2746, 5 February 2010 by Nick Lane. [1]
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Me and a Gun
From Wikipedia, the free encyclopedia
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"Me and a Gun"
Single by Tori Amos
from the album Little Earthquakes
A-side
"Silent All These Years"
B-side
"Upside Down"
"Thoughts"
Released
October 21, 1991
Format
7" single, 12" single and CD
Genre
A cappella
Label
Atlantic, EastWest
Songwriter(s)
Tori Amos
Producer(s)
Tori Amos, Eric Rosse
Tori Amos singles chronology
"Cool on Your Island"
(1988) "Me and a Gun"
(1991) "Silent All These Years"
(1991)
"Me and a Gun" is a song by American singer-songwriter and musician Tori Amos. It was released as the first single from her debut studio album Little Earthquakes. It was released on October 21, 1991 by Atlantic Records in North America and EastWest Records in the UK.
Contents
1 Background
2 Later appearances of the song
3 Covers and remixes
4 Track listing
5 References
Background[edit]
The song is three minutes and 44 seconds long. Amos wrote the song about being raped in Los Angeles when she was 21. After she performed at a bar, a patron asked her if he could have a ride home. She obliged, and he raped her at knifepoint. (Many journalists mistakenly state she was raped at gunpoint, possibly due to the song's title.) She escaped. Years later, in London, Amos saw the film Thelma and Louise and was stirred.[1][2]
Tori Amos explains the experience:
"I'll never talk about it at this level again but let me ask you. Why have I survived that kind of night, when other women didn't?
How am I alive to tell you this tale when he was ready to slice me up? In the song I say it was 'Me and a Gun' but it wasn't a gun. It was a knife he had. And the idea was to take me to his friends and cut me up, and he kept telling me that, for hours. And if he hadn't needed more drugs I would have been just one more news report, where you see the parents grieving for their daughter.
And I was singing hymns, as I say in the song, because he told me to. I sang to stay alive. Yet I survived that torture, which left me urinating all over myself and left me paralyzed for years. That's what that night was all about, mutilation, more than violation through sex.
I really do feel as though I was psychologically mutilated that night and that now I'm trying to put the pieces back together again. Through love, not hatred. And through my music. My strength has been to open again, to life, and my victory is the fact that, despite it all, I kept alive my vulnerability."[3]
This song did not take off very well as a single; in fact, the track was not the A-side to its own single. "Silent All These Years," another song from Little Earthquakes, was the first track on this single, with "Gun" appearing third (or as the B-side on the 7" release). "Silent" was a more accessible song, and radio stations began to play that instead. Ultimately the single was re-released with nearly identical packaging but retitled as Silent All These Years.
Amos made a habit of singing this song during live appearances. In 1994, the DC Rape Crisis Center awarded Amos a Visionary award for the song and the co-creation of RAINN. Amos stopped singing the song live in December 2001 and did not sing it live again until September 2007 (with the exception of one performance in Istanbul in August 2005).[4]
Later appearances of the song[edit]
Trance pioneers Salt Tank sampled the line "Do you know Carolina, where the biscuits are soft and sweet" on their 1996 release "Eugina."
A VHS video of music videos and live performances was released in 1992; it was also called Little Earthquakes. Amos sings "Me and a Gun" in an appearance on MTV Asia.
Amos performs "Me and a Gun" on the video Tori Amos: Live from New York, a recording of the 1997 RAINN benefit concert.
During 1997, a CD called Tori Amos the benefit for RAINN [sic] (Atlantic #PRCD 6995-2) was given out during several promotions in order to raise money for RAINN. An audio version of the above live concert version was released on this two-track CD. (The other track was the album version of "Putting the Damage On.")
In 1999, the song was featured on an MTV compilation called Fight for Your Rights: Take a Stand Against Violence.
"Me and a Gun" is also included on A Tori Amos Collection: Tales of a Librarian.
While performed for her stint on MTV Unplugged, "Me and a Gun" was cut from the final airing of the show. Clips of this performance surfaced on the MTV special Hips, Lips and Genderbenders: MTV's History of Sex, but it has otherwise remained unreleased.[5]
On November 6, 2007, during a concert in Chicago on her ADP tour, Amos performed "Me and a Gun" as Pip with her backing band, using a knife and gun as props.[6] This marked the first time she sang the song with instrumentation.
On November 12-14 and 17, 2011, during her Night of Hunters tour, Tori performed Me and a Gun for all four of her South Africa concerts. [7]
Covers and remixes[edit]
Fan remixes of this track exist; however, no official club or house remix exists. It was remastered for the album Tales of a Librarian.
Rock band Stabbing Westward covered the song in 1996 during a live concert.[8]
Amanda Palmer covered the song live on September 12, 2009.[9]
Track listing[edit]
CD single and 12" single
"Silent All These Years" – 4:11
"Upside Down" – 4:22
"Me and a Gun" – 3:42
"Thoughts" – 2:36
Tracks 1 & 3 are from the album. Track 2 also appeared on the single for "Winter". Track 4 was otherwise unavailable until it was remastered and released on the deluxe reissue of Little Earthquakes in 2015.
7" single
"Silent All These Years" – 4:11
"Me and a Gun" – 3:42
References[edit]
Jump up ^ "Chasing Away the Demons". 20/20. February 15, 1999. Archived from the original on June 25, 2001.
Jump up ^ "Tori Amos on her new album and the "muses" that inspire her songs". CBS This Morning. November 11, 2017.
Jump up ^ Editorial about the definition of rape, with Tori Amos quotes
Jump up ^ Tori Amos Setlist Database on Undented
Jump up ^ Me and a Gun "Unplugged" footage
Jump up ^ Tori Amos - Me and A Gun on YouTube
Jump up ^ "Tori Amos Night of Hunters Tour Dates". Archived from the original on 2013-12-25.
Jump up ^ Fan report on Stabbing Westward's cover
Jump up ^ Amanda Palmer - Me And A Gun (Tori Amos cover) on YouTube
hide
v
t
e
Tori Amos
Discography
Studio albums
Little Earthquakes
Under the Pink
Boys for Pele
From the Choirgirl Hotel
To Venus and Back
Strange Little Girls
Scarlet's Walk
The Beekeeper
American Doll Posse
Abnormally Attracted to Sin
Midwinter Graces
Night of Hunters
Gold Dust
Unrepentant Geraldines
Native Invader
Compilations
Tales of a Librarian
A Piano: The Collection
Live albums
To Venus and Back
The Original Bootlegs
Legs and Boots
Live at Montreux 1991/1992
From Russia with Love
EPs
Crucify
Hey Jupiter
Scarlet's Hidden Treasures
Exclusive Session
Flavor (Peter Rauhofer Mixes)
Singles
"Me and a Gun"
"Silent All These Years"
"China"
"Winter"
"Crucify"
"Cornflake Girl"
"God"
"Pretty Good Year"
"Past the Mission"
"Caught a Lite Sneeze"
"Talula"
"Professional Widow"
"Hey Jupiter"
"In the Springtime of His Voodoo"
"Spark"
"Jackie's Strength"
"Cruel"/"Raspberry Swirl"
"Bliss"
"1000 Oceans"
"Glory of the 80's"
"Concertina"
"Strange Little Girl"
"A Sorta Fairytale"
"Taxi Ride"
"Don't Make Me Come to Vegas"
"Sleeps with Butterflies"
"Sweet the Sting"
"Big Wheel"
"Bouncing off Clouds"
"Welcome to England"
"Maybe California"
"A Silent Night with You"
"Carry"
"Flavor"
"Trouble's Lament"
Videos
Little Earthquakes
Live from New York
Complete Videos: 1991–1998
A Sorta Fairytale
Welcome to Sunny Florida
Fade to Red
Live at Montreux 1991/1992
Tours
5 ½ Weeks Tour
To Dallas and Back
Strange Little Tour
On Scarlet's Walk Tour
Original Sinsuality Tour
American Doll Posse World Tour
Sinful Attraction Tour
Night of Hunters Tour
Gold Dust Orchestral Tour
Unrepentant Geraldines Tour
Related articles
Piece by Piece
Comic Book Tattoo
Delirium
Y Kant Tori Read
"Blue Skies"
RAINN
"Baltimore"
The Light Princess
Original Cast Recording
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Total derivative
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In the mathematical field of differential calculus, a total derivative or full derivative of a function f {\displaystyle f} of several variables, e.g., t {\displaystyle t} , x {\displaystyle x} , y {\displaystyle y} , etc., with respect to an exogenous argument, e.g., t {\displaystyle t} , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function.
The total derivative of a function is different from its corresponding partial derivative ( ∂ {\displaystyle \partial } ). Calculation of the total derivative of f {\displaystyle f} with respect to t {\displaystyle t} does not assume that the other arguments are constant while t {\displaystyle t} varies; instead, it assumes that the other arguments too depend on t {\displaystyle t} . The total derivative includes these indirect dependencies to find the overall dependency of f {\displaystyle f} on t . {\displaystyle t.} [1]:198–203 For example, the total derivative of f ( t , x ( t ) , y ( t ) ) {\displaystyle f(t,x(t),y(t))} with respect to t {\displaystyle t} is
d  f d  t = ∂ f ∂ t d  t d  t + ∂ f ∂ x d  x d  t + ∂ f ∂ y d  y d  t , {\displaystyle {\frac {\operatorname {d} f}{\operatorname {d} t}}={\frac {\partial f}{\partial t}}{\frac {\operatorname {d} t}{\operatorname {d} t}}+{\frac {\partial f}{\partial x}}{\frac {\operatorname {d} x}{\operatorname {d} t}}+{\frac {\partial f}{\partial y}}{\frac {\operatorname {d} y}{\operatorname {d} t}},}
which simplifies to
d  f d  t = ∂ f ∂ t + ∂ f ∂ x d  x d  t + ∂ f ∂ y d  y d  t . {\displaystyle {\frac {\operatorname {d} f}{\operatorname {d} t}}={\frac {\partial f}{\partial t}}+{\frac {\partial f}{\partial x}}{\frac {\operatorname {d} x}{\operatorname {d} t}}+{\frac {\partial f}{\partial y}}{\frac {\operatorname {d} y}{\operatorname {d} t}}.}
Consider multiplying both sides of the equation by the differential d  t {\displaystyle \operatorname {d} t} :
d  f = ∂ f ∂ t d  t + ∂ f ∂ x d  x + ∂ f ∂ y d  y . {\displaystyle \operatorname {d} f={\frac {\partial f}{\partial t}}\operatorname {d} t+{\frac {\partial f}{\partial x}}\operatorname {d} x+{\frac {\partial f}{\partial y}}\operatorname {d} y.}
The result is the differential change d  f {\displaystyle \operatorname {d} f} in, or total differential of, the function f {\displaystyle f} . Because f {\displaystyle f} depends on t {\displaystyle t} , some of that change will be due to the partial derivative of f {\displaystyle f} with respect to t {\displaystyle t} . However, some of that change will also be due to the partial derivatives of f {\displaystyle f} with respect to the variables x {\displaystyle x} and y {\displaystyle y} . So, the differential d  t {\displaystyle \operatorname {d} t} is applied to the total derivatives of x {\displaystyle x} and y {\displaystyle y} to find differentials d  x {\displaystyle \operatorname {d} x} and d  y {\displaystyle \operatorname {d} y} , which can then be used to find the contribution to d  f {\displaystyle \operatorname {d} f} .
"Total derivative" is sometimes also used as a synonym for the material derivative, D D  t {\displaystyle {\frac {\operatorname {D} }{\operatorname {D} t}}} , in fluid mechanics.
Contents
[hide]
1 Differentiation with indirect dependencies
2 The total derivative via differentials
3 The total derivative as a linear map
4 Total differential equation
5 Application to equation systems
6 References
7 External links
Differentiation with indirect dependencies[edit]
Suppose that f is a function of two variables, x and y. Normally these variables are assumed to be independent. However, in some situations they may be dependent on each other. For example y could be a function of x, constraining the domain of f to a curve in R 2 {\displaystyle \mathbb {R} ^{2}} . In this case the partial derivative of f with respect to x does not give the true rate of change of f with respect to changing x because changing x necessarily changes y. The total derivative takes such dependencies into account.
For example, suppose
f ( x , y ) = x y {\displaystyle f(x,y)=xy} .
The rate of change of f with respect to x is usually the partial derivative of f with respect to x; in this case,
∂ f ∂ x = y {\displaystyle {\frac {\partial f}{\partial x}}=y} .
However, if y depends on x, the partial derivative does not give the true rate of change of f as x changes because it holds y fixed.
Suppose we are constrained to the line
y = x ; {\displaystyle y=x;}
then
f ( x , y ) = f ( x , x ) = x 2 {\displaystyle f(x,y)=f(x,x)=x^{2}} .
In that case, the total derivative of f with respect to x is
d f d x = 2 x {\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} x}}=2x} .
Instead of immediately substituting for y in terms of x, this can be found equivalently using the chain rule:
d f d x = ∂ f ∂ x + ∂ f ∂ y d y d x = y + x ⋅ 1 = x + y = 2 x . {\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} x}}={\frac {\partial f}{\partial x}}+{\frac {\partial f}{\partial y}}{\frac {\mathrm {d} y}{\mathrm {d} x}}=y+x\cdot 1=x+y=2x.}
Notice that this is not equal to the partial derivative:
d f d x = 2 x ≠ ∂ f ∂ x = y = x {\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} x}}=2x\neq {\frac {\partial f}{\partial x}}=y=x} .
While one can often perform substitutions to eliminate indirect dependencies, the chain rule provides for a more efficient and general technique. Suppose M(t, p1, ..., pn) is a function of time t and n variables p i {\displaystyle p_{i}} which themselves depend on time. Then, the total time derivative of M is
d  M d  t = d d  t M ( t , p 1 ( t ) , … , p n ( t ) ) . {\displaystyle {\operatorname {d} M \over \operatorname {d} t}={\frac {\operatorname {d} }{\operatorname {d} t}}M{\bigl (}t,p_{1}(t),\ldots ,p_{n}(t){\bigr )}.}
The chain rule for differentiating a function of several variables implies that
d  M d  t = ∂ M ∂ t + ∑ i = 1 n ∂ M ∂ p i d  p i d  t = ( ∂ ∂ t + ∑ i = 1 n d  p i d  t ∂ ∂ p i ) ( M ) . {\displaystyle {\operatorname {d} M \over \operatorname {d} t}={\frac {\partial M}{\partial t}}+\sum _{i=1}^{n}{\frac {\partial M}{\partial p_{i}}}{\frac {\operatorname {d} p_{i}}{\operatorname {d} t}}={\biggl (}{\frac {\partial }{\partial t}}+\sum _{i=1}^{n}{\frac {\operatorname {d} p_{i}}{\operatorname {d} t}}{\frac {\partial }{\partial p_{i}}}{\biggr )}(M).}
This expression is often used in physics for a gauge transformation of the Lagrangian, as two Lagrangians that differ only by the total time derivative of a function of time and the n generalized coordinates lead to the same equations of motion. An interesting example concerns the resolution of causality concerning the Wheeler–Feynman time-symmetric theory. The operator in brackets (in the final expression) is also called the total derivative operator (with respect to t).
For example, the total derivative of f(x(t), y(t)) is
d  f d  t = ∂ f ∂ x d  x d  t + ∂ f ∂ y d  y d  t . {\displaystyle {\frac {\operatorname {d} f}{\operatorname {d} t}}={\partial f \over \partial x}{\operatorname {d} x \over \operatorname {d} t}+{\partial f \over \partial y}{\operatorname {d} y \over \operatorname {d} t}.}
Here there is no ∂f/∂t term since f itself does not depend on the independent variable t directly.
The total derivative via differentials[edit]
Differentials provide a simple way to understand the total derivative. For instance, suppose M ( t , p 1 , … , p n ) {\displaystyle M(t,p_{1},\dots ,p_{n})} is a function of time t and n variables p i {\displaystyle p_{i}} as in the previous section. Then, the differential of M is
d  M = ∂ M ∂ t d  t + ∑ i = 1 n ∂ M ∂ p i d  p i . {\displaystyle \operatorname {d} M={\frac {\partial M}{\partial t}}\operatorname {d} t+\sum _{i=1}^{n}{\frac {\partial M}{\partial p_{i}}}\operatorname {d} p_{i}.}
This expression is often interpreted heuristically as a relation between infinitesimals. However, if the variables t and p i {\displaystyle p_{i}} are interpreted as functions, and M ( t , p 1 , … , p n ) {\displaystyle M(t,p_{1},\dots ,p_{n})} is interpreted to mean the composite of M with these functions, then the above expression makes perfect sense as an equality of differential 1-forms, and is immediate from the chain rule for the exterior derivative. The advantage of this point of view is that it takes into account arbitrary dependencies between the variables. For example, if p 1 2 = p 2 p 3 {\displaystyle p_{1}^{2}=p_{2}p_{3}} then 2 p 1 d  p 1 = p 3 d  p 2 + p 2 d  p 3 {\displaystyle 2p_{1}\operatorname {d} p_{1}=p_{3}\operatorname {d} p_{2}+p_{2}\operatorname {d} p_{3}} . In particular, if the variables p i {\displaystyle p_{i}} are all functions of t, as in the previous section, then
d  M = ∂ M ∂ t d  t + ∑ i = 1 n ∂ M ∂ p i ∂ p i ∂ t d  t . {\displaystyle \operatorname {d} M={\frac {\partial M}{\partial t}}\operatorname {d} t+\sum _{i=1}^{n}{\frac {\partial M}{\partial p_{i}}}{\frac {\partial p_{i}}{\partial t}}\,\operatorname {d} t.}
Dividing through by dt gives the total derivative dM/dt.
The total derivative as a linear map[edit]
Let U ⊆ R n {\displaystyle U\subseteq \mathbb {R} ^{n}} be an open subset. Then a function f : U → R m {\displaystyle f:U\rightarrow \mathbb {R} ^{m}} is said to be (totally) differentiable at a point u ∈ U {\displaystyle u\in U} , if there exists a linear map d  f u : R n → R m {\displaystyle \operatorname {d} f_{u}:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m}} (also denoted Duf or Df(u)) such that
lim x → u ∥ f ( x ) − f ( u ) − d  f u ( x − u ) ∥ ∥ x − u ∥ = 0. {\displaystyle \lim _{x\rightarrow u}{\frac {\|f(x)-f(u)-\operatorname {d} f_{u}(x-u)\|}{\|x-u\|}}=0.}
The linear map d  f u {\displaystyle \operatorname {d} f_{u}} is called the (total) derivative or (total) differential of f {\displaystyle f} at u {\displaystyle u} . A function is (totally) differentiable if its total derivative exists at every point in its domain.
Note that f is differentiable if and only if each of its components f i : U → R {\displaystyle f_{i}:U\rightarrow \mathbb {R} } is differentiable. For this it is necessary, but not sufficient, that the partial derivatives of each function fi exist. However, if these partial derivatives exist and are continuous, then f is differentiable and its differential at any point is the linear map determined by the Jacobian matrix of partial derivatives at that point.[2]
Total differential equation[edit]
Main article: Total differential equation
A total differential equation is a differential equation expressed in terms of total derivatives. Since the exterior derivative is a natural operator, in a sense that can be given a technical meaning, such equations are intrinsic and geometric.
Application to equation systems[edit]
In economics, it is common for the total derivative to arise in the context of a system of equations.[1]:pp. 217–220 For example, a simple supply-demand system might specify the quantity q of a product demanded as a function D of its price p and consumers' income I, the latter being an exogenous variable, and might specify the quantity supplied by producers as a function S of its price and two exogenous resource cost variables r and w. The resulting system of equations,
q = D ( p , I ) , {\displaystyle q=D(p,I),}
q = S ( p , r , w ) , {\displaystyle q=S(p,r,w),}
determines the market equilibrium values of the variables p and q. The total derivative of, for example, p with respect to r, d  p d  r , {\displaystyle {\frac {\operatorname {d} p}{\operatorname {d} r}},} gives the sign and magnitude of the reaction of the market price to the exogenous variable r. In the indicated system, there are a total of six possible total derivatives, also known in this context as comparative static derivatives: dp/dr, dp/dw, dp/dI, dq/dr, dq/dw, and dq/dI. The total derivatives are found by totally differentiating the system of equations, dividing through by, say dr, treating dq/dr and dp/dr as the unknowns, setting dI=dw=0, and solving the two totally differentiated equations simultaneously, typically by using Cramer's rule.
References[edit]
^ Jump up to: a b Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics (Third ed.). McGraw-Hill. ISBN 0-07-010813-7.
Jump up ^ Abraham, Ralph; Marsden, J. E.; Ratiu, Tudor (2012). Manifolds, Tensor Analysis, and Applications. Springer Science & Business Media. p. 78.
A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2
From thesaurus.maths.org total derivative
External links[edit]
Weisstein, Eric W. "Total Derivative". MathWorld.
http://www.sv.vt.edu/classes/ESM4714/methods/df2D.html
Retrieved from "https://en.wikipedia.org/w/index.php?title=Total_derivative&oldid=805495624"
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values: \frac{\operatorname df}{\operatorname dt}=\frac{\partial f}{\partial t} + \frac{\partial f}{\partial x} \frac{\operatorname dx}{\operatorname dt} + \frac{\partial f}{\partial y} \frac{\operatorname dy}{\operatorname dt}.

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values: \frac{\mathrm{d}f}{\mathrm{d}x} = 2 x \neq \frac{\partial f}{\partial x} = y = x

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Exogenous variable

values: vertical-align: -0.838ex; width:12.018ex; height:2.843ex;

values: vertical-align: -0.838ex; width:14.188ex; height:2.843ex;

values: vertical-align: -2.005ex; width:4.363ex; height:5.509ex;

values:
reflist mw-references-wrap
list-style-type: decimal;

values: Category:Articles lacking in-text citations from July 2013

values: Category:All articles lacking in-text citations

values: Discussion about edits from this IP address [ctrl-alt-n]

values: A list of edits made from this IP address [ctrl-alt-y]

values: You are encouraged to create an account and log in; however, it is not mandatory

values: You're encouraged to log in; however, it's not mandatory. [ctrl-alt-o]

values:
/wiki/Total_derivative
View the content page [ctrl-alt-c]
c

values:
mw-wiki-logo
/wiki/Main_Page
Visit the main page

values:
/wiki/Main_Page
Visit the main page [ctrl-alt-z]
z

values: Featured content – the best of Wikipedia

values: Find background information on current events

values:
/wiki/Special:Random
Load a random article [ctrl-alt-x]
x

values: portal navigation p-navigation p-navigation-label

values: Guidance on how to use and edit Wikipedia

values: About the project, what you can do, where to find things

values: A list of recent changes in the wiki [ctrl-alt-r]

values: portal navigation p-interaction p-interaction-label

values:
uls-settings-trigger
Language settings
0
button
true

values: تفاضل كامل – Arabic

values: مشتق کل – Persian

values: Полная производная функции – Russian

values: Повна похідна – Ukrainian

values: /static/images/wikimedia-button-1.5x.png 1.5x, /static/images/wikimedia-button-2x.png 2x

values: /static/images/poweredby_mediawiki_132x47.png 1.5x, /static/images/poweredby_mediawiki_176x62.png 2x

values: mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Total_derivative rootpage-Total_derivative vector-experimental-print-styles vector-nav-directionality skin-vector action-view