|
|
| Line 1: |
Line 1: |
| [[File:Tangent to a curve.svg|thumb|200px|width=150|length=150|The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line equals the derivative of the function at the marked point.]]
| | == but it is a white to see a handsome youth == |
| {{Calculus}}
| |
|
| |
|
| In [[mathematics]], '''differential calculus''' is a subfield of [[calculus]] concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being [[integral calculus]].
| | , Does not allow them to retreat, after all, they can not [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html カシオ 時計 電波] think, if Xiao Yan lost a few people, this one look like sand iron is not good, who will miss their hands 'fire energy.'<br><br>Since the escape, however, then fight it!<br><br>'how desperate counterattack intend??' Qiaode freshmen move sand iron brow of a challenge, took a little rough Senleng smile on the face: 'It seems that you still have to sort of [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ 腕時計 バンド] look forward to their strength, hehe , or, in this ghost forest we have not undergone nearly three days, and some are beginning to tickle the bones. '<br><br>'Wait!'<br><br>writhing in the sand when the iron fist of a sudden [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html casio 腕時計 メンズ] have a cry [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html カシオ腕時計 g-shock] sounded, the former frown, his eyes looked down the sound, but it is [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html casio 時計] a white to see a handsome youth, gaze swept the youth face, Some feel a little familiar sand iron.<br><br>'sand iron Brother, I do not know also |
| | 相关的主题文章： |
| | <ul> |
| | |
| | <li>[http://coinauction.org/cgi-bin/auction/logout.cgi http://coinauction.org/cgi-bin/auction/logout.cgi]</li> |
| | |
| | <li>[http://kamituki.3.pro.tok2.com/lds/epad/epad.cgi http://kamituki.3.pro.tok2.com/lds/epad/epad.cgi]</li> |
| | |
| | <li>[http://www.afzizhi.com/plus/feedback.php?aid=30 http://www.afzizhi.com/plus/feedback.php?aid=30]</li> |
| | |
| | </ul> |
|
| |
|
| The primary objects of study in differential calculus are the [[derivative]] of a [[Function (mathematics)|function]], related notions such as the [[Differential of a function|differential]], and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called '''differentiation'''. Geometrically, the derivative at a point is the [[slope]] of the [[tangent#Geometry|tangent line]] to the [[graph of a function|graph of the function]] at that point, provided that the derivative exists and is defined at that point. For a [[real-valued function]] of a single real variable, the derivative of a function at a point generally determines the best [[linear approximation]] to the function at that point.
| | == 'If you do not mind the words of the great elders == |
|
| |
|
| Differential calculus and integral calculus are connected by the [[fundamental theorem of calculus]], which states that differentiation is the reverse process to [[integration (mathematics)|integration]].
| | No better than two weak<br><br>'as partners with the tiger, the tiger was eventually [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html casio 腕時計 スタンダード] eat.'<br><br>Xiao Yan smiled, did not care to spend the evil demon king Senleng eyes, faint: 'Great elders, for day offerings were acting style, you should know better than anyone else, if the two together, it would be difficult to achieve real balance, sooner or later become masters of matter, time, spend days were probably have become another one of these offerings were a punishment. '<br><br>'Xiao [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html カシオの時計] Yan, Hu said you less!'<br><br>heard [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-10.html casio 腕時計 データバンク] this, but it is the face flower brocade 'color' for a change, thundered.<br><br>For this woman, Xiao Yan is reason to ignore, looked at the white-haired old woman, said: 'If you do not mind the words of the great elders, in the future I might be able to become [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html casio 腕時計 phys] the stars fell Court took were [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html カシオ 掛け時計] a reliable ally, presumably to family division reputation, there would be no doubt the great elders, right? '<br><br>'Oh,' medicine 'Venerable prominent throughout |
| | 相关的主题文章： |
| | <ul> |
| | |
| | <li>[http://hz-yuhua.cn/plus/view.php?aid=319133 http://hz-yuhua.cn/plus/view.php?aid=319133]</li> |
| | |
| | <li>[http://ausslife.com/mediawiki/index.php?title=User:Hijgovdbww#Xiao_Yan_is_no_hesitation http://ausslife.com/mediawiki/index.php?title=User:Hijgovdbww#Xiao_Yan_is_no_hesitation]</li> |
| | |
| | <li>[http://yl.0731zb.com/plus/feedback.php?aid=9 http://yl.0731zb.com/plus/feedback.php?aid=9]</li> |
| | |
| | </ul> |
|
| |
|
| Differentiation has applications to nearly all quantitative disciplines. For example, in [[physics]], the derivative of the [[displacement (vector)|displacement]] of a moving body with respect to [[time]] is the [[velocity]] of the body, and the derivative of velocity with respect to [[time]] is acceleration. [[Newton's laws of motion|Newton's second law of motion]] states that the derivative of the [[momentum]] of a body equals the force applied to the body. The [[reaction rate]] of a [[chemical reaction]] is a derivative. In [[operations research]], derivatives determine the most efficient ways to transport materials and design factories.
| | == bone quiet last sentence just fallen == |
|
| |
|
| Derivatives are frequently used to find the [[maxima and minima]] of a function. Equations involving derivatives are called [[differential equations]] and are fundamental in describing [[Natural phenomenon|natural phenomena]]. Derivatives and their generalizations appear in many fields of mathematics, such as [[complex analysis]], [[functional analysis]], [[differential geometry]], [[measure theory]] and [[abstract algebra]].
| | But this is [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html 電波時計 casio] also skilled are bold bone quiet, semi-holy order other strength, so too he has enough [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-11.html 腕時計 メンズ casio] capital disdain for everyone here, under this unspeakable gap, even days of order fighting skills are Unable to play the role reversal of the universe, however, this bone is never quiet thought, before the soul of the family who play against strong and Xiao Yan, too, are holding this kind of mentality<br><br>'ancient tribe Kaoru children, Kamijina blood owner, ancient tribe known as the most [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ 電波ソーラー時計] perfect man of blood' bone quiet gaze, looking at not far from Kaoru children, surface touch dry smile on face, said: 'The potential indeed terrible, but now, you are still not old lady opponent! '<br><br>bone quiet last sentence [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html gps 腕時計 カシオ] just fallen, their stature, but it is suddenly strange disappearance in place, to see this scene, Kaoru children face slightly changed, Jiaoqu suddenly Putui, then filled with [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html カシオ腕時計 g-shock] gold 'color' flame of Yu Zhang , with a wave of terrible temperature, a space on the TV drama fiercely took over |
| | | 相关的主题文章： |
| == The derivative ==
| | <ul> |
| {{Main|Derivative}}
| | |
| | | <li>[http://www.jsjcpm.com.cn/plus/view.php?aid=67826 http://www.jsjcpm.com.cn/plus/view.php?aid=67826]</li> |
| Suppose that ''x'' and ''y'' are [[real number]]s and that ''y'' is a [[Function (mathematics)|function]] of ''x'', that is, for every value of ''x'', there is a corresponding value of ''y''. This relationship can be written as ''y'' = ''f''(''x''). If ''f''(''x'') is the equation for a straight line, then there are two real numbers ''m'' and ''b'' such that {{nobreak|''y'' {{=}} ''m'' ''x'' + ''b''}}. ''m'' is called the [[slope]] and can be determined from the formula:
| | |
| :<math>m=\frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x},</math>
| | <li>[http://www.lnxqd.com/bbs/forum.php?mod=viewthread&tid=2236 http://www.lnxqd.com/bbs/forum.php?mod=viewthread&tid=2236]</li> |
| where the symbol Δ (the uppercase form of the [[Greek alphabet|Greek]] letter [[Delta (letter)|Delta]]) is an abbreviation for "change in". It follows that {{nobreak|Δ''y'' {{=}} ''m'' Δ''x''}}.
| | |
| | | <li>[http://www.tokyotales.com/mt_admin/mt-search.cgi http://www.tokyotales.com/mt_admin/mt-search.cgi]</li> |
| A general function is not a line, so it does not have a slope. The '''derivative''' of ''f'' at the point ''x'' is the slope of the linear approximation to ''f'' at the point ''x''. It is usually denoted {{nowrap|''f'' ′(''x'')}} or ''dy''/''dx''. Together with the value of ''f'' at ''x'', the derivative of ''f'' determines the best linear approximation, or [[linearization]], of ''f'' near the point ''x''. This latter property is usually taken as the definition of the derivative.
| | |
| | | </ul> |
| A closely related notion is the [[differential (calculus)|differential]] of a function.
| |
| | |
| [[Image:Tangent-calculus.svg|thumb|300px|The [[tangent line]] at (''x'',''f''(''x''))]]
| |
| When ''x'' and ''y'' are real variables, the derivative of ''f'' at ''x'' is the slope of the [[tangent line]] to the graph of ''f'' at ''x''. Because the source and target of ''f'' are one-dimensional, the derivative of ''f'' is a real number. If ''x'' and ''y'' are vectors, then the best linear approximation to the graph of ''f'' depends on how ''f'' changes in several directions at once. Taking the best linear approximation in a single direction determines a '''[[partial derivative]]''', which is usually denoted ∂''y''/∂''x''. The linearization of ''f'' in all directions at once is called the '''[[total derivative]]'''.
| |
| | |
| == History of differentiation ==
| |
| {{Main|History of calculus}}
| |
| | |
| The concept of a derivative in the sense of a [[tangent line]] is a very old one, familiar to [[Ancient Greece|Greek]] geometers such as
| |
| [[Euclid]] (c. 300 BC), [[Archimedes]] (c. 287–212 BC) and [[Apollonius of Perga]] (c. 262–190 BC).<ref>See [[Euclid's Elements]], The [[Archimedes Palimpsest]] and {{MacTutor Biography|id=Apollonius|title=Apollonius of Perga}}</ref> [[Archimedes]] also introduced the use of [[infinitesimal]]s, although these were primarily used to study areas and volumes rather than derivatives and tangents; see [[Archimedes' use of infinitesimals]].
| |
| | |
| The use of infinitesimals to study rates of change can be found in [[Indian mathematics]], perhaps as early as 500 AD, when the astronomer and mathematician [[Aryabhata]] (476–550) used infinitesimals to study the [[Orbit of the Moon|motion of the moon]].<ref>{{MacTutor Biography|id=Aryabhata_I|title=Aryabhata the Elder}}</ref> The use of infinitesimals to compute rates of change was developed significantly by [[Bhāskara II]] (1114–1185); indeed, it has been argued<ref>Ian G. Pearce. [http://turnbull.mcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch8_5.html Bhaskaracharya II.]</ref> that many of the key notions of differential calculus can be found in his work, such as "[[Rolle's theorem]]".<ref>{{Cite journal|first=T. A. A.|last=Broadbent|title=Reviewed work(s): ''The History of Ancient Indian Mathematics'' by C. N. Srinivasiengar|journal=The Mathematical Gazette|volume=52|issue=381|date=October 1968|pages=307–8|doi=10.2307/3614212|jstor=3614212|last2=Kline|first2=M.|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref> The [[Islamic mathematics|Persian mathematician]], [[Sharaf al-Dīn al-Tūsī]] (1135–1213), was the first to discover the [[derivative]] of [[Cubic function|cubic polynomials]], an important result in differential calculus;<ref>J. L. Berggren (1990). "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat", ''Journal of the American Oriental Society'' '''110''' (2), p. 304-309.</ref> his ''Treatise on Equations'' developed concepts related to differential calculus, such as the derivative [[Function (mathematics)|function]] and the [[maxima and minima]] of curves, in order to solve [[cubic equation]]s which may not have positive solutions.<ref name=Sharaf>{{MacTutor|id=Al-Tusi_Sharaf|title=Sharaf al-Din al-Muzaffar al-Tusi}}</ref>
| |
| | |
| The modern development of calculus is usually credited to [[Isaac Newton]] (1643–1727) and [[Gottfried Leibniz]] (1646–1716), who provided independent<ref>Newton began his work in 1666 and Leibniz began his in 1676. However, Leibniz published his first paper in 1684, predating Newton's publication in 1693. It is possible that Leibniz saw drafts of Newton's work in 1673 or 1676, or that Newton made use of Leibniz's work to refine his own. Both Newton and Leibniz claimed that the other plagiarized their respective works. This resulted in a bitter [[Newton v. Leibniz calculus controversy|controversy]] between the two men over who first invented calculus which shook the mathematical community in the early 18th century.</ref> and unified approaches to differentiation and derivatives. The key insight, however, that earned them this credit, was the [[fundamental theorem of calculus]] relating differentiation and integration: this rendered obsolete most previous methods for computing areas and volumes,<ref>This was a monumental achievement, even though a restricted version had been proven previously by [[James Gregory (astronomer and mathematician)|James Gregory]] (1638–1675), and some key examples can be found in the work of [[Pierre de Fermat]] (1601–1665).</ref> which had not been significantly extended since the time of [[Ibn al-Haytham]] (Alhazen).<ref name=Katz>Victor J. Katz (1995), "Ideas of Calculus in Islam and India", ''Mathematics Magazine'' '''68''' (3): 163-174 [165-9 & 173-4]</ref> For their ideas on derivatives, both Newton and Leibniz built on significant earlier work by mathematicians such as [[Isaac Barrow]] (1630–1677), [[René Descartes]] (1596–1650), [[Christiaan Huygens]] (1629–1695), [[Blaise Pascal]] (1623–1662) and [[John Wallis]] (1616–1703). Isaac Barrow is generally given credit for the early development of the derivative.<ref>Eves, H. (1990).</ref> Nevertheless, Newton and Leibniz remain key figures in the history of differentiation, not least because Newton was the first to apply differentiation to [[theoretical physics]], while Leibniz systematically developed much of the notation still used today.
| |
| | |
| Since the 17th century many mathematicians have contributed to the theory of differentiation. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as [[Augustin Louis Cauchy]] (1789–1857), [[Bernhard Riemann]] (1826–1866), and [[Karl Weierstrass]] (1815–1897). It was also during this period that the differentiation was generalized to [[Euclidean space]] and the [[complex plane]].
| |
| | |
| == Applications of derivatives ==
| |
| === Optimization ===
| |
| If ''f'' is a [[differentiable function]] on '''R''' (or an [[open interval]]) and ''x'' is a [[local maximum]] or a [[local minimum]] of ''f'', then the derivative of ''f'' at ''x'' is zero; points where {{nobreak|''f<nowiki>'</nowiki>''(''x'') {{=}} 0}} are called ''[[critical point (mathematics)|critical points]]'' or ''[[stationary point]]s'' (and the value of ''f'' at ''x'' is called a ''[[critical value]]''). (The definition of a critical point is sometimes extended to include points where the derivative does not exist.) Conversely, a critical point ''x'' of ''f'' can be analysed by considering the [[second derivative]] of ''f'' at ''x'':
| |
| * if it is positive, ''x'' is a local minimum;
| |
| * if it is negative, ''x'' is a local maximum;
| |
| * if it is zero, then ''x'' could be a local minimum, a local maximum, or neither. (For example, {{nobreak|''f''(''x'') {{=}} ''x''<sup>3</sup>}} has a critical point at {{nobreak|''x'' {{=}} 0}}, but it has neither a maximum nor a minimum there, whereas {{nobreak|''f''(''x'') {{=}} ±''x''<sup>4</sup>}} has a critical point at {{nobreak|''x'' {{=}} 0}} and a minimum and a maximum, respectively, there.)
| |
| This is called the [[second derivative test]]. An alternative approach, called the [[first derivative test]], involves considering the sign of the ''f<nowiki>'</nowiki>'' on each side of the critical point.
| |
| | |
| Taking derivatives and solving for critical points is therefore often a simple way to find local minima or maxima, which can be useful in [[Optimization (mathematics)|optimization]]. By the [[extreme value theorem]], a continuous function on a [[closed interval]] must attain its minimum and maximum values at least once. If the function is differentiable, the minima and maxima can only occur at critical points or endpoints.
| |
| | |
| This also has applications in graph sketching: once the local minima and maxima of a differentiable function have been found, a rough plot of the graph can be obtained from the observation that it will be either increasing or decreasing between critical points.
| |
| | |
| In higher dimensions, a critical point of a scalar valued function is a point at which the gradient is zero. The second derivative test can still be used to analyse critical points by considering the [[eigenvalue]]s of the [[Hessian matrix]] of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is a [[saddle point]], and if none of these cases hold (i.e., some of the eigenvalues are zero) then the test is inconclusive.
| |
| | |
| ==== Calculus of variations ====
| |
| {{Main|Calculus of variations}}
| |
| | |
| One example of an optimization problem is: Find the shortest curve between two points on a surface, assuming that the curve must also lie on the surface. If the surface is a plane, then the shortest curve is a line. But if the surface is, for example, egg-shaped, then the [[Shortest path problem|shortest path]] is not immediately clear. These paths are called [[geodesic]]s, and one of the simplest problems in the calculus of variations is finding geodesics. Another example is: Find the smallest area surface filling in a closed curve in space. This surface is called a [[minimal surface]] and it, too, can be found using the calculus of variations.
| |
| | |
| === Physics ===
| |
| Calculus is of vital importance in physics: many physical processes are described by equations involving derivatives, called [[differential equation]]s. Physics is particularly concerned with the way quantities change and evolve over time, and the concept of the "'''[[time derivative]]'''" — the rate of change over time — is essential for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in [[Newtonian physics]]:
| |
| | |
| * [[velocity]] is the derivative (with respect to time) of an object's displacement (distance from the original position)
| |
| * [[acceleration]] is the derivative (with respect to time) of an object's velocity, that is, the second derivative (with respect to time) of an object's position.
| |
| | |
| For example, if an object's position on a line is given by
| |
| | |
| : <math>x(t) = -16t^2 + 16t + 32 , \,\!</math>
| |
| | |
| then the object's velocity is
| |
| | |
| : <math>\dot x(t) = x'(t) = -32t + 16, \,\!</math>
| |
| | |
| and the object's acceleration is
| |
| | |
| : <math>\ddot x(t) = x''(t) = -32, \,\!</math>
| |
| | |
| which is constant.
| |
| | |
| === Differential equations ===
| |
| {{Main|Differential equation}}
| |
| | |
| A '''differential equation''' is a relation between a collection of functions and their derivatives. An '''ordinary differential equation''' is a differential equation that relates functions of one variable to their derivatives with respect to that variable. A '''partial differential equation''' is a differential equation that relates functions of more than one variable to their partial derivatives. Differential equations arise naturally in the physical sciences, in mathematical modelling, and within mathematics itself. For example, [[Newton's second law]], which describes the relationship between acceleration and force, can be stated as the ordinary differential equation
| |
| :<math>F(t) = m\frac{d^2x}{dt^2}.</math>
| |
| The [[heat equation]] in one space variable, which describes how heat diffuses through a straight rod, is the partial differential equation
| |
| :<math>\frac{\partial u}{\partial t} = \alpha\frac{\partial^2 u}{\partial x^2}.</math>
| |
| Here ''u''(''x'',''t'') is the temperature of the rod at position ''x'' and time ''t'' and α is a constant that depends on how fast heat diffuses through the rod.
| |
| | |
| === Mean value theorem === | |
| {{Main|Mean value theorem}}
| |
| | |
| The mean value theorem gives a relationship between values of the derivative and values of the original function. If ''f''(''x'') is a real-valued function and ''a'' and ''b'' are numbers with {{nobreak|''a'' < ''b''}}, then the mean value theorem says that under mild hypotheses, the slope between the two points (''a'',''f''(''a'')) and (''b'',''f''(''b'')) is equal to the slope of the tangent line to ''f'' at some point ''c'' between ''a'' and ''b''. In other words,
| |
| :<math>f'(c) = \frac{f(b) - f(a)}{b - a}.</math> | |
| In practice, what the mean value theorem does is control a function in terms of its derivative. For instance, suppose that ''f'' has derivative equal to zero at each point. This means that its tangent line is horizontal at every point, so the function should also be horizontal. The mean value theorem proves that this must be true: The slope between any two points on the graph of ''f'' must equal the slope of one of the tangent lines of ''f''. All of those slopes are zero, so any line from one point on the graph to another point will also have slope zero. But that says that the function does not move up or down, so it must be a horizontal line. More complicated conditions on the derivative lead to less precise but still highly useful information about the original function.
| |
| | |
| === Taylor polynomials and Taylor series ===
| |
| {{Main|Taylor polynomial|Taylor series}}
| |
| | |
| The derivative gives the best possible linear approximation, but this can be very different from the original function. One way of improving the approximation is to take a quadratic approximation. That is to say, the linearization of a real-valued function ''f''(''x'') at the point ''x''<sub>0</sub> is a linear polynomial {{nobreak|''a'' + ''b''(''x'' − ''x''<sub>0</sub>)}}, and it may be possible to get a better approximation by considering a quadratic polynomial {{nobreak|''a'' + ''b''(''x'' − ''x''<sub>0</sub>) + ''c''(''x'' − ''x''<sub>0</sub>)<sup>2</sup>}}. Still better might be a cubic polynomial {{nobreak|''a'' + ''b''(''x'' − ''x''<sub>0</sub>) + ''c''(''x'' − ''x''<sub>0</sub>)<sup>2</sup> + ''d''(''x'' − ''x''<sub>0</sub>)<sup>3</sup>}}, and this idea can be extended to arbitrarily high degree polynomials. For each one of these polynomials, there should be a best possible choice of coefficients ''a'', ''b'', ''c'', and ''d'' that makes the approximation as good as possible.
| |
| | |
| In the [[Neighbourhood (mathematics)|neighbourhood]] of ''x''<sub>0</sub>, for ''a'' the best possible choice is always ''f''(''x''<sub>0</sub>), and for ''b'' the best possible choice is always ''f<nowiki>'</nowiki>''(''x''<sub>0</sub>). For ''c'', ''d'', and higher-degree coefficients, these coefficients are determined by higher derivatives of ''f''. ''c'' should always be ''f<nowiki>''</nowiki>''(''x''<sub>0</sub>)/2, and ''d'' should always be ''f<nowiki>'''</nowiki>''(''x''<sub>0</sub>)/3[[factorial|!]]. Using these coefficients gives the '''Taylor polynomial''' of ''f''. The Taylor polynomial of degree ''d'' is the polynomial of degree ''d'' which best approximates ''f'', and its coefficients can be found by a generalization of the above formulas. [[Taylor's theorem]] gives a precise bound on how good the approximation is. If ''f'' is a polynomial of degree less than or equal to ''d'', then the Taylor polynomial of degree ''d'' equals ''f''.
| |
| | |
| The limit of the Taylor polynomials is an infinite series called the '''Taylor series'''. The Taylor series is frequently a very good approximation to the original function. Functions which are equal to their Taylor series are called [[analytic function]]s. It is impossible for functions with discontinuities or sharp corners to be analytic, but there are [[smooth function]]s which are not analytic.
| |
| | |
| === Implicit function theorem === | |
| {{Main|Implicit function theorem}}
| |
| | |
| Some natural geometric shapes, such as [[circle]]s, cannot be drawn as the [[graph of a function]]. For instance, if {{nobreak|''f''(''x'',''y'') {{=}} ''x''<sup>2</sup> + ''y''<sup>2</sup> − 1}}, then the circle is the set of all pairs (''x'',''y'') such that {{nobreak|''f''(''x'',''y'') {{=}} 0}}. This set is called the zero set of ''f''. It is not the same as the graph of ''f'', which is a [[cone (geometry)|cone]]. The implicit function theorem converts relations such as {{nobreak|''f''(''x'',''y'') {{=}} 0}} into functions. It states that if ''f'' is [[continuously differentiable]], then around most points, the zero set of ''f'' looks like graphs of functions pasted together. The points where this is not true are determined by a condition on the derivative of ''f''. The circle, for instance, can be pasted together from the graphs of the two functions <math>\pm\sqrt{1-x^2}</math>. In a neighborhood of every point on the circle except (−1,0) and (1,0), one of these two functions has a graph that looks like the circle. (These two functions also happen to meet (−1,0) and (1,0), but this is not guaranteed by the implicit function theorem.)
| |
| | |
| The implicit function theorem is closely related to the [[inverse function theorem]], which states when a function looks like graphs of [[invertible function]]s pasted together.
| |
| | |
| == See also ==
| |
| {{sisterlinks|Differential calculus}}
| |
| * [[Differential (calculus)]]
| |
| * [[Differential geometry]]
| |
| * [[Numerical differentiation]]
| |
| * [[Techniques for differentiation]]
| |
| * [[List of calculus topics]]
| |
| | |
| == References ==
| |
| {{Reflist}}
| |
| | |
| *{{cite book | author=J. Edwards | title=Differential Calculus
| |
| | publisher= MacMillan and Co.| location=London | year=1892
| |
| |url=http://books.google.com/books?id=unltAAAAMAAJ&pg=PA1#v=onepage&q&f=false}}
| |
| | |
| [[Category:Differential calculus| ]]
| |
| | |
| {{Link FA|de}}
| |
| {{Link FA|lmo}}
| |
| {{Link FA|mk}}
| |
but it is a white to see a handsome youth
, Does not allow them to retreat, after all, they can not カシオ 時計 電波 think, if Xiao Yan lost a few people, this one look like sand iron is not good, who will miss their hands 'fire energy.'
Since the escape, however, then fight it!
'how desperate counterattack intend??' Qiaode freshmen move sand iron brow of a challenge, took a little rough Senleng smile on the face: 'It seems that you still have to sort of カシオ 腕時計 バンド look forward to their strength, hehe , or, in this ghost forest we have not undergone nearly three days, and some are beginning to tickle the bones. '
'Wait!'
writhing in the sand when the iron fist of a sudden casio 腕時計 メンズ have a cry カシオ腕時計 g-shock sounded, the former frown, his eyes looked down the sound, but it is casio 時計 a white to see a handsome youth, gaze swept the youth face, Some feel a little familiar sand iron.
'sand iron Brother, I do not know also
相关的主题文章：
'If you do not mind the words of the great elders
No better than two weak
'as partners with the tiger, the tiger was eventually casio 腕時計 スタンダード eat.'
Xiao Yan smiled, did not care to spend the evil demon king Senleng eyes, faint: 'Great elders, for day offerings were acting style, you should know better than anyone else, if the two together, it would be difficult to achieve real balance, sooner or later become masters of matter, time, spend days were probably have become another one of these offerings were a punishment. '
'Xiao カシオの時計 Yan, Hu said you less!'
heard casio 腕時計 データバンク this, but it is the face flower brocade 'color' for a change, thundered.
For this woman, Xiao Yan is reason to ignore, looked at the white-haired old woman, said: 'If you do not mind the words of the great elders, in the future I might be able to become casio 腕時計 phys the stars fell Court took were カシオ 掛け時計 a reliable ally, presumably to family division reputation, there would be no doubt the great elders, right? '
'Oh,' medicine 'Venerable prominent throughout
相关的主题文章：
bone quiet last sentence just fallen
But this is 電波時計 casio also skilled are bold bone quiet, semi-holy order other strength, so too he has enough 腕時計 メンズ casio capital disdain for everyone here, under this unspeakable gap, even days of order fighting skills are Unable to play the role reversal of the universe, however, this bone is never quiet thought, before the soul of the family who play against strong and Xiao Yan, too, are holding this kind of mentality
'ancient tribe Kaoru children, Kamijina blood owner, ancient tribe known as the most カシオ 電波ソーラー時計 perfect man of blood' bone quiet gaze, looking at not far from Kaoru children, surface touch dry smile on face, said: 'The potential indeed terrible, but now, you are still not old lady opponent! '
bone quiet last sentence gps 腕時計 カシオ just fallen, their stature, but it is suddenly strange disappearance in place, to see this scene, Kaoru children face slightly changed, Jiaoqu suddenly Putui, then filled with カシオ腕時計 g-shock gold 'color' flame of Yu Zhang , with a wave of terrible temperature, a space on the TV drama fiercely took over
相关的主题文章：