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Runge's theorem

2014-11-25T07:15:55Z

67.164.164.45: Undid revision 633219571 by 192.68.254.220 (talk)

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Curve of constant width

2014-11-23T09:51:31Z

67.164.94.63: /* Examples */

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De Moivre–Laplace theorem

2013-12-13T06:47:32Z

67.164.92.20:

{{Calculus}}

The '''fundamental theorem of calculus''' is a theorem that links the concept of the [[derivative]] of a function with the concept of the [[integral]].

The first part of the theorem, sometimes called the '''first fundamental theorem of calculus''', is that an [[antiderivative|indefinite integration]]<ref>More exactly, the theorem deals with [[integral|definite integration]] with variable upper limit and arbitrarily selected lower limit. This particular kind of definite integration allows us to compute one of the infinitely many [[antiderivatives]] of a function (except for those that do not have a zero). Hence, it is almost equivalent to [[antiderivative|indefinite integration]], defined by most authors as an operation that yields any one of the possible antiderivatives of a function, including those without a zero.</ref> can be reversed by a differentiation. This part of the theorem is also important because it guarantees the existence of [[antiderivative]]s for [[continuous function]]s.<ref>{{Citation |last=Spivak|first=Michael|year=1980|title=Calculus|edition=2nd|publication-place=Houston, Texas|publisher=Publish or Perish Inc.}}</ref>

The second part, sometimes called the '''second fundamental theorem of calculus''', is that the [[definite integral]] of a function can be computed by using any one of its infinitely many [[antiderivative]]s. This part of the theorem has key practical applications because it markedly simplifies the computation of [[definite integral]]s.
{{TOC limit|3}}

==History==

{{Seealso|History of calculus}}

The fundamental theorem of calculus relates differentiation and integration, showing that these two operations are essentially inverses of one another. Before the discovery of this theorem, it was not recognized that these two operations are related. Ancient [[Greek mathematics | Greek mathematicians]] knew how to compute area via [[infinitesimals]], an operation that we would now call [[Integral | integration]]. The origins of [[Derivative | differentiation]] likewise predate the Fundamental Theorem of Calculus by hundreds of years; for example, in the fourteenth century the notions of ''continuity'' of functions and ''motion'' was studied by the [[Oxford Calculators]] and other scholars. The historical relevance of the Fundamental Theorem of Calculus is not the ability to calculate these operations, but the realization that the two seemingly distinct operations (calculation of geometric areas, and calculation of velocities) are actually closely related.

The first published statement and proof of a restricted version of the fundamental theorem was by [[James Gregory (astronomer and mathematician)|James Gregory]] (1638–1675).<ref>
See, e.g., Marlow Anderson, Victor J. Katz, Robin J. Wilson, ''Sherlock Holmes in Babylon and Other Tales of Mathematical History'', Mathematical Association of America, 2004, [http://books.google.com/books?vid=ISBN0883855461&id=BKRE5AjRM3AC&pg=PA114&lpg=PA114&ots=Z01TZKrQXY&dq=%22james+gregory%22+%22fundamental+theorem%22&sig=6xDqL0oNAhWw66IqPdI5fQX7euA p. 114].
</ref> [[Isaac Barrow]] (1630–1677) proved a more generalized version of the theorem<ref>http://www.archive.org/details/geometricallectu00barruoft</ref> while Barrow's student [[Isaac Newton]] (1643–1727) completed the development of the surrounding mathematical theory. [[Gottfried Leibniz]] (1646–1716) systematized the knowledge into a calculus for infinitesimal quantities and introduced the notation used today.

==Geometric meaning==
[[File:FTC geometric2.png|500px|thumb|right|The area shaded in red stripes can be estimated as ''h'' times ''f''(''x''). Alternatively, if the function ''A''(''x'') were known, it could be computed as {{nowrap|''A''(''x'' + ''h'') − ''A''(''x'').}} These two values are approximately equal, particularly for small ''h''.]]
For a continuous function {{nowrap|''y'' {{=}} ''f''(''x'')}} whose graph is plotted as a curve, each value of ''x'' has a corresponding area function ''A''(''x''), representing the area beneath the curve between 0 and ''x''. The function ''A''(''x'') may not be known, but it is given that it represents the area under the curve.

The area under the curve between ''x'' and {{nowrap|''x'' + ''h''}} could be computed by finding the area between 0 and {{nowrap|''x'' + ''h'',}} then subtracting the area between 0 and ''x''. In other words, the area of this “sliver” would be {{nowrap|''A''(''x'' + ''h'') − ''A''(''x'')}}.

There is another way to ''estimate'' the area of this same sliver. As shown in the accompanying figure, ''h'' is multiplied by ''f''(''x'') to find the area of a rectangle that is approximately the same size as this sliver. So:

:<math>A(x+h)-A(x) \approx f(x)h</math>

In fact, this estimate becomes a perfect equality if we add the red portion of the "excess" area shown in the diagram. So:

:<math>A(x+h)-A(x)=f(x)h+(Red Excess)</math>

Rearranging terms:

:<math>f(x) = \frac{A(x+h)-A(x)}{h} - \frac{(Red Excess)}{h}</math>.

As ''h'' approaches 0 in the [[limit of a function|limit]], the last fraction can be shown to go to zero.<ref>[[Lipman Bers|Bers, Lipman]]. ''Calculus'', pp. 180-181 (Holt, Rinehart and Winston (1976).</ref> This is true because the area of the red portion of excess region is less than the area of the tiny black-bordered rectangle; the area of that tiny rectangle, divided by ''h'', is simply the height of the tiny rectangle, which can be seen to go to zero as ''h'' goes to zero.

Removing the last fraction from our equation then, we have:

:<math>f(x) = \lim_{h\to 0}\frac{A(x+h)-A(x)}{h}</math>.

It can thus be shown that {{nowrap|''f''(''x'') {{=}} ''A''′(''x'')}}. That is, the derivative of the area function ''A''(''x'') is the original function ''f''(''x''); or, the area function is simply an [[antiderivative]] of the original function. Computing the derivative of a function and “finding the area” under its curve are "opposite" operations. This is the crux of the Fundamental Theorem of Calculus.

==Physical intuition==
Intuitively, the theorem simply states that the sum of [[infinitesimal]] changes in a quantity over time (or over some other variable) adds up to the net change in the quantity.

Imagine for example using a stopwatch to mark-off tiny increments of time as a car travels down a highway. Imagine also looking at the car's speedometer as it travels, so that at every moment you know the velocity of the car. To understand the power of this theorem, imagine also that you are not allowed to look out the window of the car, so that you have no direct evidence of how far the car has traveled.

For any tiny interval of time in the car, you could calculate how far the car has traveled in that interval by multiplying the current speed of the car times the length of that tiny interval of time. (This is because ''distance'' = ''speed'' <math>\times</math> ''time''.)

Now imagine doing this instant after instant, so that for every tiny interval of time you know how far the car has traveled. In principle, you could then calculate the ''total'' distance traveled in the car (even though you've never looked out the window) by simply summing-up all those tiny distances.

:distance traveled = <math>\sum</math> the velocity at any instant <math>\times</math> a tiny interval of time

In other words,

:distance traveled = <math>\sum v(t) \times \Delta t</math>

On the right hand side of this equation, as <math>\Delta t</math> becomes infinitesimally small, the operation of "summing up" corresponds to [[Integral | integration]]. So what we've shown is that the integral of the velocity function can be used to compute how far the car has traveled.

Now remember that the velocity function is simply the derivative of the position function. So what we have really shown is that integrating the velocity simply recovers the original position function. This is the basic idea of the Theorem: that ''integration'' and ''differentiation'' are closely related operations, each essentially being the inverse of the other.

In other words, in terms of one's physical intuition, the theorem simply states that the sum of the changes in a quantity over time (such as ''position'', as calculated by multiplying ''velocity'' times ''time'') adds up to the total net change in the quantity. Or to put this more generally:
* Given a quantity <math>x</math> that changes over some variable <math>t</math>, and
* Given the velocity <math>v(t)</math> with which that quantity changes over that variable
then the idea that "distance equals speed times time" corresponds to the statement
:<math>dx = v(t) dt</math>
meaning that one can recover the original function <math>x(t)</math> by integrating its derivative, the velocity <math>v(t)</math>, over <math>t</math>.

==Formal statements==
There are two parts to the theorem. Loosely put, the first part deals with the derivative of an [[antiderivative]], while the second part deals with the relationship between antiderivatives and [[definite integral]]s.

===First part===
This part is sometimes referred to as the first fundamental theorem of calculus.<ref>{{harvnb|Apostol|1967|loc=§5.1}}</ref>

Let ''f'' be a continuous real-valued function defined on a [[Interval (mathematics)#Terminology|closed interval]] [''a'', ''b'']. Let ''F'' be the function defined, for all ''x'' in [''a'', ''b''], by
:<math>F(x) = \int_a^x\!f(t)\, dt.</math>
Then, ''F'' is continuous on [''a'', ''b''], differentiable on the open interval {{nowrap|(''a'', ''b''),}} and

:<math>F'(x) = f(x)\,</math>

for all ''x'' in (''a'', ''b'').

Alternatively, if ''f'' is merely [[Riemann integrable]], then ''F'' is continuous on [''a'', ''b''] (but not necessarily differentiable).

===Corollary===
The fundamental theorem is often employed to compute the definite integral of a function ''f'' for which an antiderivative ''F'' is known. Specifically, if ''f'' is a real-valued continuous function on {{nowrap|[''a'', ''b''],}} and ''F'' is an antiderivative of ''f'' in {{nowrap|[''a'', ''b''],}} then

:<math>\int_a^b f(t)\, dt = F(b)-F(a).</math>

The corollary assumes [[Continuous_function | continuity on the whole interval]]. This result is strengthened slightly in the following part of the theorem.

===Second part===
This part is sometimes referred to as the second fundamental theorem of calculus<ref>{{harvnb|Apostol|1967|loc=§5.3}}</ref> or the '''Newton–Leibniz axiom'''.

Let ''f'' and ''F'' be real-valued functions defined on a [[closed interval]] [''a'', ''b''] such that the derivative of ''F'' is ''f''. That is, ''f'' and ''F'' are functions such that for all ''x'' in {{nowrap|[''a'', ''b''],}}

:<math>F'(x) = f(x).\ </math>

If ''f'' is [[Riemann integrable]] on {{nowrap|[''a'', ''b'']}} then

:<math>\int_a^b f(x)\,dx = F(b) - F(a).</math>

The Second part is somewhat stronger than the Corollary because it does not assume that ''f'' is continuous.

When an antiderivative ''F'' exists, then there are infinitely many antiderivatives for ''f'', obtained by adding to ''F'' an arbitrary constant. Also, by the first part of the theorem, antiderivatives of ''f'' always exist when ''f'' is continuous.

==Proof of the first part==
For a given ''f''(''t''), define the function ''F''(''x'') as
:<math>F(x) = \int_a^x f(t) \,dt.</math>

For any two numbers ''x''1 and ''x''1 + Δ''x'' in [''a'', ''b''], we have
:<math>F(x_1) = \int_{a}^{x_1} f(t) \,dt</math>
and
:<math>F(x_1 + \Delta x) = \int_a^{x_1 + \Delta x} f(t) \,dt.</math>

Subtracting the two equalities gives
:<math>F(x_1 + \Delta x) - F(x_1) = \int_a^{x_1 + \Delta x} f(t) \,dt - \int_a^{x_1} f(t) \,dt. \qquad (1)</math>

It can be shown that
:<math>\int_{a}^{x_1} f(t) \,dt + \int_{x_1}^{x_1 + \Delta x} f(t) \,dt = \int_a^{x_1 + \Delta x} f(t) \,dt. </math>
:(The sum of the areas of two adjacent regions is equal to the area of both regions combined.)
Manipulating this equation gives
:<math>\int_{a}^{x_1 + \Delta x} f(t) \,dt - \int_{a}^{x_1} f(t) \,dt = \int_{x_1}^{x_1 + \Delta x} f(t) \,dt. </math>

Substituting the above into (1) results in
:<math>F(x_1 + \Delta x) - F(x_1) = \int_{x_1}^{x_1 + \Delta x} f(t) \,dt. \qquad (2)</math>

According to the [[mean value theorem]] for integration, there exists a real number <math>c(\Delta x)</math> in [''x''1, ''x''1 + Δ''x''] such that
:<math>\int_{x_1}^{x_1 + \Delta x} f(t) \,dt = f\left(c(\Delta x)\right) \Delta x.</math>

To keep the notation simple we will continue writing ''c'' instead of <math>c(\Delta x)</math> but one should keep in mind that ''c'' does depend on <math>\Delta x</math>.
Substituting the above into (2) we get
:<math>F(x_1 + \Delta x) - F(x_1) = f(c) \Delta x.</math>

Dividing both sides by Δ''x'' gives
:<math>\frac{F(x_1 + \Delta x) - F(x_1)}{\Delta x} = f(c).</math>
:The expression on the left side of the equation is Newton's [[difference quotient]] for ''F'' at ''x''1.

Take the limit as Δ''x'' → 0 on both sides of the equation.
:<math>\lim_{\Delta x \to 0} \frac{F(x_1 + \Delta x) - F(x_1)}{\Delta x} = \lim_{\Delta x \to 0} f(c). </math>

The expression on the left side of the equation is the definition of the derivative of ''F'' at ''x''1.
:<math>F'(x_1) = \lim_{\Delta x \to 0} f(c). \qquad (3) </math>

To find the other limit, we use the [[squeeze theorem]]. The number ''c'' is in the interval [''x''1, ''x''1 + Δ''x''], so ''x''1 ≤ ''c'' ≤ ''x''1 + Δ''x''.

Also, <math>\lim_{\Delta x \to 0} x_1 = x_1</math> and <math>\lim_{\Delta x \to 0} x_1 + \Delta x = x_1.\,</math>

Therefore, according to the squeeze theorem,
:<math>\lim_{\Delta x \to 0} c = x_1.</math>

Substituting into (3), we get
:<math>F'(x_1) = \lim_{c \to x_1} f(c).</math>

The function ''f'' is continuous at ''c'', so the limit can be taken inside the function. Therefore, we get
:<math>F'(x_1) = f(x_1).\ </math>
which completes the proof.

(Leithold ''et al.'', 1996) (a rigorous proof can be found http://www.imomath.com/index.php?options=438)

==Proof of the corollary==
Suppose ''F'' is an antiderivative of ''f'', with ''f'' continuous on {{nowrap|[''a'', ''b''].}} Let

: <math>G(x) = \int_a^x f(t)\, dt</math>.

By the ''first part'' of the theorem, we know ''G'' is also an antiderivative of ''f''. It follows by the mean value theorem that there is a number ''c'' such that {{nowrap|''G''(''x'') {{=}} ''F''(''x'') + ''c''}}, for all ''x'' in {{nowrap|[''a'', ''b''].}} Letting {{nowrap|''x'' {{=}} ''a''}}, we have

:<math>F(a) + c = G(a) = \int_a^a f(t)\, dt = 0,</math>

which means {{nowrap|''c'' {{=}} − ''F''(''a'').}} In other words {{nowrap|''G''(''x'') {{=}} ''F''(''x'') − ''F''(''a'')}}, and so

:<math>\int_a^b f(x)\, dx = G(b) = F(b) - F(a).</math>

==Proof of the second part==
This is a limit proof by [[Riemann integral|Riemann sums]].
Let ''f'' be (Riemann) integrable on the interval {{nowrap|[''a'', ''b''],}} and let ''f'' admit an antiderivative ''F'' on {{nowrap|[''a'', ''b''].}} Begin with the quantity {{nowrap|''F''(''b'') − ''F''(''a'')}}. Let there be numbers ''x''1, ..., ''x''''n''
such that

:<math>a = x_0 < x_1 < x_2 < \cdots < x_{n-1} < x_n = b. \, </math>

It follows that

:<math>F(b) - F(a) = F(x_n) - F(x_0). \, </math>

Now, we add each ''F''(''x''''i'') along with its additive inverse, so that the resulting quantity is equal:

:<math>\begin{align}
F(b) - F(a)
&= F(x_n) + [-F(x_{n-1}) + F(x_{n-1})] + \cdots + [-F(x_1) + F(x_1)] - F(x_0) \\
&= [F(x_n) - F(x_{n-1})] + [F(x_{n-1}) + \cdots - F(x_1)] + [F(x_1) - F(x_0)].
\end{align}</math>

The above quantity can be written as the following sum:

:<math>F(b) - F(a) = \sum_{i=1}^n \,[F(x_i) - F(x_{i-1})]. \qquad (1)</math>

Next, we employ the [[mean value theorem]]. Stated briefly,

Let ''F'' be continuous on the closed interval [''a'', ''b''] and differentiable on the open interval (''a'', ''b''). Then there exists some ''c'' in (''a'', ''b'') such that

:<math>F'(c) = \frac{F(b) - F(a)}{b - a}.</math>

It follows that

:<math>F'(c)(b - a) = F(b) - F(a). \,</math>

The function ''F'' is differentiable on the interval {{nowrap|[''a'', ''b''];}} therefore, it is also differentiable and continuous on each interval {{nowrap|[''x''''i''−1, ''x''''i'']}}. According to the mean value theorem (above),

:<math>F(x_i) - F(x_{i-1}) = F'(c_i)(x_i - x_{i-1}). \ </math>

Substituting the above into (1), we get

:<math>F(b) - F(a) = \sum_{i=1}^n \,[F'(c_i)(x_i - x_{i-1})].</math>

The assumption implies <math>F'(c_i) = f(c_i).</math> Also, <math>x_i - x_{i-1}</math> can be expressed as <math>\Delta x</math> of partition <math>i</math>.

:<math>F(b) - F(a) = \sum_{i=1}^n \,[f(c_i)(\Delta x_i)]. \qquad (2)</math>

[[File:Riemann integral irregular.gif|frame|right|A converging sequence of Riemann sums. The number in the upper left is the total area of the blue rectangles. They converge to the integral of the function.]]

We are describing the area of a rectangle, with the width times the height, and we are adding the areas together. Each rectangle, by virtue of the [[Mean Value Theorem]], describes an approximation of the curve section it is drawn over. Also <math>\Delta x_i</math> need not be the same for all values of ''i'', or in other words that the width of the rectangles can differ. What we have to do is approximate the curve with ''n'' rectangles. Now, as the size of the partitions get smaller and ''n'' increases, resulting in more partitions to cover the space, we get closer and closer to the actual area of the curve.

By taking the limit of the expression as the norm of the partitions approaches zero, we arrive at the [[Riemann integral]]. We know that this limit exists because ''f'' was assumed to be integrable. That is, we take the limit as the largest of the partitions approaches zero in size, so that all other partitions are smaller and the number of partitions approaches infinity.

So, we take the limit on both sides of (2). This gives us

:<math>\lim_{\| \Delta x_i \| \to 0} F(b) - F(a) = \lim_{\| \Delta x_i \| \to 0} \sum_{i=1}^n \,[f(c_i)(\Delta x_i)].</math>

Neither ''F''(''b'') nor ''F''(''a'') is dependent on <math>||\Delta x_i\|</math>, so the limit on the left side remains {{nowrap|''F''(''b'') − ''F''(''a'').}}

:<math>F(b) - F(a) = \lim_{\| \Delta x_i \| \to 0} \sum_{i=1}^n \,[f(c_i)(\Delta x_i)].</math>

The expression on the right side of the equation defines the integral over ''f'' from ''a'' to ''b''. Therefore, we obtain

:<math>F(b) - F(a) = \int_a^b f(x)\,dx,</math>

which completes the proof.

It almost looks like the first part of the theorem follows directly from the second. That is, suppose ''G'' is an antiderivative of ''f''. Then by the second theorem, <math>G(x) - G(a) = \int_a^x f(t) \, dt</math>. Now, suppose <math>F(x) = \int_a^x f(t)\, dt\ = G(x) - G(a)</math>. Then ''F'' has the same derivative as ''G'', and therefore {{nowrap|''F''′ {{=}} ''f''}}. This argument only works, however, if we already know that ''f'' has an antiderivative, and the only way we know that all continuous functions have antiderivatives is by the first part of the Fundamental Theorem.<ref>{{Citation |last=Spivak|first=Michael|year=1980|title=Calculus|edition=2nd|publication-place=Houston, Texas|publisher=Publish or Perish Inc.}}</ref>
For example if {{nowrap|''f''(''x'') {{=}} e−''x''2,}} then ''f'' has an antiderivative, namely

:<math>G(x) = \int_0^x f(t) \, dt\,</math>

and there is no simpler expression for this function. It is therefore important not to interpret the second part of the theorem as the definition of the integral. Indeed, there are many functions that are integrable but lack antiderivatives that can be written as an [[elementary function]]. Conversely, many functions that have antiderivatives are not Riemann integrable (see [[Volterra's function]]).

==Examples==
As an example, suppose the following is to be calculated:

:<math>\int_2^5 x^2\, dx. </math>

Here, <math>f(x) = x^2 \,</math> and we can use <math>F(x) = \frac{x^3}{3} </math> as the antiderivative. Therefore:

:<math>\int_2^5 x^2\, dx = F(5) - F(2) = \frac{5^3}{3} - \frac{2^3}{3} = \frac{125}{3} - \frac{8}{3} = \frac{117}{3} = 39.</math>

Or, more generally, suppose that

:<math>\frac{d}{dx} \int_0^x t^3\, dt </math>
is to be calculated. Here, <math>f(t) = t^3 \,</math> and <math>F(t) = \frac{t^4}{4} </math> can be used as the antiderivative. Therefore:

:<math>\frac{d}{dx} \int_0^x t^3\, dt = \frac{d}{dx} F(x) - \frac{d}{dx} F(0) = \frac{d}{dx} \frac{x^4}{4} = x^3.</math>

Or, equivalently,

:<math>\frac{d}{dx} \int_0^x t^3\, dt = f(x) \frac{dx}{dx} - f(0) \frac{d0}{dx} = x^3.</math>

==Generalizations==
We don't need to assume continuity of ''f'' on the whole interval. Part I of the theorem then says: if ''f'' is any [[Lebesgue integration|Lebesgue integrable]] function on {{nowrap|[''a'', ''b'']}} and ''x''0 is a number in {{nowrap|[''a'', ''b'']}} such that ''f'' is continuous at ''x''0, then

:<math>F(x) = \int_a^x f(t)\, dt</math>

is differentiable for {{nowrap|''x'' {{=}} ''x''0}} with {{nowrap|''F''′(''x''0) {{=}} ''f''(''x''0).}} We can relax the conditions on ''f'' still further and suppose that it is merely locally integrable. In that case, we can conclude that the function ''F'' is differentiable [[almost everywhere]] and {{nowrap|''F''′(''x'') {{=}} ''f''(''x'')}} almost everywhere. On the real line this statement is equivalent to [[Lebesgue differentiation theorem|Lebesgue's differentiation theorem]]. These results remain true for the Henstock–Kurzweil integral, which allows a larger class of integrable functions {{harv|Bartle|2001|loc=Thm. 4.11}}.

In higher dimensions Lebesgue's differentiation theorem generalizes the Fundamental theorem of calculus by stating that for almost every ''x'', the average value of a function ''f'' over a ball of radius ''r'' centered at ''x'' tends to ''f''(''x'') as ''r'' tends to 0.

Part II of the theorem is true for any Lebesgue integrable function ''f'', which has an antiderivative ''F'' (not all integrable functions do, though). In other words, if a real function ''F'' on {{nowrap|[''a'', ''b'']}} admits a derivative ''f''(''x'') at ''every'' point ''x'' of {{nowrap|[''a'', ''b'']}} and if this derivative ''f'' is Lebesgue integrable on {{nowrap|[''a'', ''b''],}} then

:<math>F(b) - F(a) = \int_a^b f(t) \, dt.</math><ref>{{harvnb|Rudin|1987|loc=th. 7.21}}</ref>

This result may fail for continuous functions ''F'' that admit a derivative ''f''(''x'') at almost every point ''x'', as the example of the [[Cantor function]] shows. However, if ''F'' is [[Absolute continuity|absolutely continuous]], it admits a derivative ''F′''(''x'') at almost every point ''x'', and moreover ''F′'' is integrable, with {{nowrap|''F''(''b'') − ''F''(''a'')}} equal to the integral of ''F′'' on {{nowrap|[''a'', ''b''].}} Conversely, if ''f'' is any integrable function, then ''F'' as given in the first formula will be absolutely continuous with ''F′'' = ''f'' a.e.

The conditions of this theorem may again be relaxed by considering the integrals involved as [[Henstock–Kurzweil integral]]s. Specifically, if a continuous function ''F''(''x'') admits a derivative ''f''(''x'') at all but countably many points, then ''f''(''x'') is Henstock–Kurzweil integrable and {{nowrap|''F''(''b'') − ''F''(''a'')}} is equal to the integral of ''f'' on {{nowrap|[''a'', ''b''].}} The difference here is that the integrability of ''f'' does not need to be assumed. {{harv|Bartle|2001|loc=Thm. 4.7}}

The version of [[Taylor's theorem]], which expresses the error term as an integral, can be seen as a generalization of the Fundamental Theorem.

There is a version of the theorem for [[complex number|complex]] functions: suppose ''U'' is an open set in '''C''' and {{nowrap|''f'' : ''U'' → '''C'''}} is a function that has a [[holomorphic function|holomorphic]] antiderivative ''F'' on ''U''. Then for every curve {{nowrap|γ : [''a'', ''b''] → ''U'',}} the [[curve integral]] can be computed as

:<math>\int_\gamma f(z) \,dz = F(\gamma(b)) - F(\gamma(a)).</math>

The fundamental theorem can be generalized to curve and surface integrals in higher dimensions and on [[manifold]]s. One such generalization offered by the [[calculus of moving surfaces]] is the [[time evolution of integrals]]. The most familiar extensions of the Fundamental theorem of calculus in higher dimensions are the [[Divergence theorem]] and the [[Gradient theorem]].

One of the most powerful statements in this direction is [[Stokes' theorem]]: Let ''M'' be an oriented [[piecewise]] smooth [[manifold]] of [[dimension]] ''n'' and let <math>\omega</math> be an ''n''−1 form that is a [[compactly supported]] [[differential form]] on ''M'' of class C1. If ∂''M'' denotes the [[manifold|boundary]] of ''M'' with its induced [[Orientation (mathematics)|orientation]], then

:<math>\int_M d\omega = \oint_{\partial M} \omega.</math>

Here ''d'' is the [[exterior derivative]], which is defined using the manifold structure only.

The theorem is often used in situations where ''M'' is an embedded oriented submanifold of some bigger manifold on which the form <math>\omega</math> is defined.

==See also==
{{Portal|Mathematics}}
* [[Differentiation under the integral sign]]

==Notes==
<references/>

==References==
* {{Citation | last1=Apostol | first1=Tom M. | author1-link=Tom M. Apostol | title=Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra | publisher=[[John Wiley & Sons]] | location=New York | edition=2nd | isbn=978-0-471-00005-1 | year=1967}}.
* {{Citation | last1=Bartle | first1=Robert | title=A Modern Theory of Integration | publisher=AMS | isbn=0-8218-0845-1 | year=2001}}.
* {{citation|last1=Larson|first1=Ron|first2=Bruce H.|last2=Edwards|first3=David E.|last3=Heyd|title=Calculus of a single variable|edition=7th| isbn=978-0-618-14916-2 | publication-place=Boston|publisher=Houghton Mifflin Company|year=2002}}.
* {{citation|last=Leithold|first=L.|year=1996|title=The calculus of a single variable|edition=6th|publication-place=New York|publisher=HarperCollins College Publishers}}.
* Malet, A, ''Studies on James Gregorie (1638-1675)'' (PhD Thesis, Princeton, 1989).
* {{citation|last=Rudin|first=Walter|year=1987|title=Real and Complex Analysis|edition=third|publication-place=New York|publisher=McGraw-Hill Book Co.|isbn=0-07-054234-1 }}
* {{citation|last=Stewart|first=J.|year=2003|contribution=Fundamental Theorem of Calculus|title=Calculus: early transcendentals|publication-place=Belmont, California|publisher=Thomson/Brooks/Cole}}.
* {{citation|editor=Turnbull, H. W.|title=The James Gregory Tercentenary Memorial Volume|publication-place=London|year=1939}}.
* {{Citation |last=Spivak|first=Michael|author1-link=Michael_Spivak|year=1980|title=Calculus|edition=2nd|publication-place=Houston, Texas|publisher=Publish or Perish Inc.}}.
* {{citation|last1=Courant|first1=Richard|last2=John|first2=Fritz|title=Introduction to Calculus and Analysis|publisher=Springer|year=1965}}.

==External links==
*[http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=388&bodyId=343 James Gregory's Euclidean Proof of the Fundamental Theorem of Calculus] at [http://mathdl.maa.org/convergence/1/ Convergence]
*[http://school.maths.uwa.edu.au/~schultz/L18Barrow.html Isaac Barrow's proof of the Fundamental Theorem of Calculus]
*[http://www.imomath.com/index.php?options=438 Fundamental Theorem of Calculus at imomath.com]
* [http://www.encyclopediaofmath.org/index.php/Newton-Leibniz_Formula Fundamental Theorem of Calculus] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

{{Fundamental theorems}}

{{DEFAULTSORT:Fundamental Theorem Of Calculus}}
[[Category:Articles containing proofs]]
[[Category:Fundamental theorems|Calculus]]
[[Category:Theorems in calculus]]
[[Category:Theorems in real analysis]]

{{Link GA|de}}

Virasoro algebra

2013-12-04T04:54:14Z

67.164.35.70: /* Definition */

The '''van der Pauw Method''' is a technique commonly used to measure the [[resistivity]] and the [[Hall coefficient]] of a sample. Its power lies in its ability to accurately measure the properties of a sample of any arbitrary shape, so long as the sample is approximately two-dimensional (i.e. it is much thinner than it is wide), solid (no holes), and the [[electrode]]s are placed on its [[perimeter]].

From the measurements made, the following properties of the material can be calculated:
* The [[resistivity]] of the material
* The [[Doping (semiconductor)|doping]] type (i.e. whether it is a [[P-type semiconductor|P-type]] or [[N-type semiconductor|N-type]] material)
* The sheet carrier density of the [[charge carrier|majority carrier]] (the number of majority carriers per unit area). From this the charge density and doping level can be found
* The [[Electron mobility|mobility]] of the majority carrier
The method was first propounded by Leo J. van der Pauw in 1958 .<ref>{{cite journal | last = Van der Pauw | first = L.J. | title = A method of measuring specific resistivity and Hall effect of discs of arbitrary shape | journal = Philips Research Reports | volume = 13 | pages = 1–9 | year = 1958 | url = http://astro1.panet.utoledo.edu/~relling2/teach/6180-7180/Hall_effect_van%20der%20Pauw_1958.pdf | format = [[PDF]]}})</ref>

== Conditions ==
There are five conditions that must be satisfied to use this technique:<ref>{{cite book|last=Webster|first=John G|title=The measurement, instrumentation, and sensors handbook|year=1999|publisher=CRC Press LLC|location=New York|isbn=3-540-64830-5|pages=43-1}}</ref> 
1. The sample must have a flat shape of uniform thickness 
2. The sample must not have any isolated holes 
3. The sample must be [[homogeneous]] and [[isotropic]] 
4. All four contacts must be located at the edges of the sample 
5. The area of contact of any individual contact should be at least an [[order of magnitude]] smaller than the area of the entire sample.

== Sample preparation ==
In order to use the van der Pauw method, the sample thickness must be much less than the width and length of the sample. In order to reduce errors in the calculations, it is preferable that the sample is symmetrical. There must also be no isolated holes within the sample.

[[Image:VanderPauwContactPlacement.jpg|thumb|right|Some possible contact placements]]

The measurements require that four [[ohmic contact]]s be placed on the sample. Certain conditions for their placement need to be met:
* They must be on the boundary of the sample (or as close to it as possible).
* They must be infinitely small. Practically, they must be as small as possible; any errors given by their non-zero size will be of the order ''D/L'', where ''D'' is the average diameter of the contact and ''L'' is the distance between the contacts.

In addition to this, any leads from the contacts should be constructed from the same batch of wire to minimise [[thermoelectric]] effects. For the same reason, all four contacts should be of the same material.

== Measurement definitions ==
* The contacts are numbered from 1 to 4 in a counter-clockwise order, beginning at the top-left contact.
* The [[Electric current|current]] ''I''12 is a positive DC current injected into contact ''1'' and taken out of contact ''2'', and is measured in [[ampere]]s (A).
* The [[voltage]] ''V''34 is a DC voltage measured between contacts ''3'' and ''4'' with no externally applied magnetic field, measured in [[volt]]s (V).
* The [[resistivity]] ''ρ'' is measured in [[ohm (unit)|ohms]]⋅[[metre]]s (Ω⋅m).
* The thickness of the sample ''t'' is measured in [[metre]]s (m).
* The [[sheet resistance]] ''RS'' is measured in [[ohm (unit)|ohms]] (Ω).

== Resistivity measurements ==
The average resistivity of a sample is given by ''ρ = RS⋅t'', where the sheet resistance ''RS'' is determined as follows. For an anisotropic material, the individual resistivity components, e.g. ''ρx'' or ''ρy'', can be calculated using the [[Montgomery method]].

=== Basic measurements ===
To make a measurement, a current is caused to flow along one edge of the sample (for instance, ''I12'') and the voltage across the opposite edge (in this case, ''V''34) is measured. From these two values, a resistance (for this example, <math>R_{12,34}</math>) can be found using [[Ohm's law]]:

:<math>R_{12,34} = \frac{V_{34}}{I_{12}}</math>

In his paper, van der Pauw showed that the sheet resistance of samples with arbitrary shapes can be determined from two of these resistances - one measured along a vertical edge, such as <math>R_{12,34}</math>, and a corresponding one measured along a horizontal edge, such as <math>R_{23,41}</math>. The actual sheet resistance is related to these resistances by the van der Pauw formula

:<math>e^{-\pi R_{12,34}/R_s}+e^{-\pi R_{23,41}/R_s}=1</math>

=== Reciprocal measurements ===

The [[Reciprocity (electromagnetism)|reciprocity]] theorem [http://www.du.edu/~jcalvert/tech/reciproc.htm] tells us that

:<math>R_{AB,CD} = R_{CD,AB}</math>

Therefore, it is possible to obtain a more precise value for the resistances <math>R_{12,34}</math> and <math>R_{23,41}</math> by making two additional measurements of their reciprocal values <math>R_{34,12}</math> and <math>R_{41,23}</math> and averaging the results.

We define

:<math>R_{\text{vertical}} = \frac{R_{12,34} + R_{34,12}}{2}</math>

and

:<math>R_{\text{horizontal}} = \frac{R_{23,41} + R_{41,23}}{2}</math>

Then, the van der Pauw formula becomes

:<math>e^{-\pi R_{\text{vertical}}/R_S}+e^{-\pi R_{\text{horizontal}}/R_S}=1</math>

=== Reversed polarity measurements ===
A further improvement in the accuracy of the resistance values can be obtained by repeating the resistance measurements after switching polarities of both the current source and the voltage meter. Since this is still measuring the same portion of the sample, just in the opposite direction, the values of ''Rvertical'' and ''Rhorizontal'' can still be calculated as the averages of the standard and reversed polarity measurements. The benefit of doing this is that any offset voltages, such as thermoelectric potentials due to the [[Seebeck effect]], will be cancelled out.

Combining these methods with the reciprocal measurements from above leads to the formulas for the resistances being

:<math>R_{\text{vertical}} = \frac{R_{12,34} + R_{34,12} + R_{21,43} + R_{43,21}}{4}</math>

and

:<math>R_{\text{horizontal}} = \frac{R_{23,41} + R_{41,23} + R_{32,14} + R_{14,32}}{4}</math>

The van der Pauw formula takes the same form as in the previous section.

=== Measurement accuracy ===

Both of the above procedures check the repeatability of the measurements. If any of the reversed polarity measurements don't agree to a sufficient degree of accuracy (usually within 3%) with the corresponding standard polarity measurement, then there is probably a source of error somewhere in the setup, which should be investigated before continuing. The same principle applies to the reciprocal measurements—they should agree to a sufficient degree before they are used in any calculations.

=== Calculating sheet resistance ===

In general, the van der Pauw formula cannot be rearranged to give the sheet resistance ''RS'' in terms of known functions. The most notable exception to this is when ''Rvertical = R = Rhorizontal''; in this scenario the sheet resistance is given by

:<math>R_s = \frac{\pi R}{\ln 2}</math>

In most other scenarios, an [[iterative method]] is used to solve the van der Pauw formula numerically for RS. Unfortunately, the formula doesn't fulfill the preconditions for the [[Banach fixed point theorem]], thus methods based on it don't work. Instead, [[nested intervals]] converge slowly but steadily.

== Hall measurements ==

=== Background ===
{{main|Hall effect}}

When a charged particle—such as an electron—is placed in a [[magnetic field]], it experiences a [[Lorentz force]] proportional to the strength of the field and the velocity at which it is traveling through it. This force is strongest when the direction of motion is perpendicular to the direction of the magnetic field; in this case the force
:<math>F_L = qvB\,\!</math>
where <math>q</math> is the charge on the particle in [[coulomb]]s, <math>v</math> the velocity it is traveling at (centimeters per [[second]]), and <math>B</math> the strength of the magnetic field ([[Weber (unit)|Wb]]/cm²). Note that centimeters are often used to measure length in the semiconductor industry, which is why they are used here instead of the [[International System of Units|SI units]] of meters.

[[Image:Van der Pauw Method - Hall Effect.png|thumb|300px|right|The Hall effect as it is used for the van der Pauw method. '''(a)''' - a current flowing through a piece of semiconductor material '''(b)''' - the electrons flowing due to the current '''(c)''' - the electrons accumulating at one edge due to the magnetic field '''(d)''' - the resulting electric field and Hall voltage <math>V_H</math>]]

When a current is applied to a piece of semiconducting material, this results in a steady flow of electrons through the material (as shown in parts '''(a)''' and '''(b)''' of the accompanying figure). The velocity the electrons are traveling at is (see [[Electric current#The drift speed of electric charges|electric current]]):
:<math>v = \frac{I}{nAq}</math>
where <math>n</math> is the electron density, <math>A</math> is the cross-sectional area of the material and <math>q</math> the [[elementary charge]] (1.602×10−19 [[coulomb]]s).

If an external magnetic field is then applied perpendicular to the direction of current flow, then the resulting Lorentz force will cause the electrons to accumulate at one edge of the sample (see part '''(c)''' of the figure). Combining the above two equations, and noting that <math>q</math> is the charge on an electron, results in a formula for the Lorentz force experienced by the electrons:
:<math>F_L = \frac{IB}{nA}</math>

This accumulation will create an [[electric field]] across the material due to the uneven distribution of charge, as shown in part '''(d)''' of the figure. This in turn leads to a [[potential difference]] across the material, known as the Hall voltage <math>V_H</math>. The current, however, continues to only flow along the material, which indicates that the force on the electrons due to the electric field balances the Lorentz force. Since the force on an electron from an electric field <math>\epsilon</math> is <math>q\epsilon</math>, we can say that the strength of the electric field is therefore
:<math>\epsilon = \frac{IB}{qnA}</math>

Finally, the magnitude of the Hall voltage is simply the strength of the electric field multiplied by the width of the material; that is,
:<math>\begin{align}
V_H &= w\epsilon \\
&= \frac{wIB}{qnA} \\
&= \frac{IB}{qnd}
\end{align}</math>
where <math>d</math> is the depth of the material. Since the sheet density <math>n_s</math> is defined as the density of electrons multiplied by the depth of the material, we can define the Hall voltage in terms of the sheet density:
:<math>V_H = \frac{IB}{qn_s}</math>

=== Making the measurements ===

Two sets of measurements need to be made: one with a magnetic field in the positive ''z''-direction as shown above, and one with it in the negative ''z''-direction. From here on in, the voltages recorded with a positive field will have a subscript P (for example, ''V13, P'') and those recorded with a negative field will have a subscript N (such as ''V13, N''). For all of the measurements, the magnitude of the injected current should be kept the same; the magnitude of the magnetic field needs to be the same in both directions also.

First of all with a positive magnetic field, the current ''I24'' is applied to the sample and the voltage ''V13, P'' is recorded; note that the voltages can be positive or negative. This is then repeated for ''I13'' and ''V42, P''.

As before, we can take advantage of the reciprocity theorem to provide a check on the accuracy of these measurements. If we reverse the direction of the currents (i.e. apply the current ''I42'' and measure ''V31, P'', and repeat for ''I31'' and ''V24, P''), then ''V13, P'' should be the same as ''V31, P'' to within a suitably small degree of error. Similarly, ''V42, P'' and ''V24, P'' should agree.

Having completed the measurements, a negative magnetic field is applied in place of the positive one, and the above procedure is repeated to obtain the voltage measurements ''V13, N'', ''V42, N'', ''V31, N'' and ''V24, N''.

=== Calculations ===

First of all, the difference of the voltages for positive and negative magnetic fields needs to be worked out:

''V''13 = ''V''13, P − ''V''13, N'' 
''V''24 = ''V''24, P − ''V''24, N'' 
''V''31 = ''V''31, P − ''V''31, N'' 
''V''42 = ''V''42, P − ''V''42, N''

The overall Hall voltage is then
:<math>V_H = \frac{V_{13} + V_{24} + V_{31} + V_{42}}{8}</math>.

The polarity of this Hall voltage indicates the type of material the sample is made of; if it is positive, the material is P-type, and if it is negative, the material is N-type.

The formula given in the background can then be rearranged to show that the sheet density
:<math>n_s = \frac{IB}{q|V_H|}</math>

Note that the strength of the magnetic field ''B'' needs to be in units of Wb/cm² if ns is in cm-2. For instance, if the strength is given in the commonly used units of [[Tesla (unit)|teslas]], it can be converted by multiplying it by 10-4.

== Other calculations ==

=== Mobility ===
The resistivity of a semiconductor material can be shown to be<ref>{{cite book | last = Sze | first = S.M. | authorlink = Simon Sze | title = Semiconductor Devices: Physics and Technology | publisher = Wiley | year = 2001 | location = New York | pages = 53 | isbn = 0-471-33372-7}}</ref>
:<math>\rho = \frac{1}{q(n\mu_n + p\mu_p)}</math>
where ''n'' and ''p'' are the concentration of electrons and holes in the material respectively, and ''μn'' and ''μp'' are the mobility of the electrons and holes respectively.

Generally, the material is sufficiently doped so that there is many orders-of-magnitude difference between the two concentrations, and so this equation can be simplified to
:<math>\rho = \frac{1}{qn_m\mu_m}</math>
where ''nm'' and ''μm'' are the doping level and mobility of the majority carrier respectively.

If we then note that the sheet resistance RS is the resistivity divided by the thickness of the sample, and that the sheet density nS is the doping level multiplied by the thickness, we can divide the equation through by the thickness to get
:<math>R_s = \frac{1}{qn_s\mu_m}</math>

This can then be rearranged to give the majority carrier mobility in terms of the previously calculated sheet resistance and sheet density:
:<math>\mu_m = \frac{1}{qn_sR_s}</math>

== Footnotes ==

{{reflist}}

== References ==
*{{cite journal | last = van der Pauw | first = L.J. | title = A method of measuring specific resistivity and Hall effect of discs of arbitrary shape | journal = Philips Research Reports | volume = 13 | pages = 1–9 | year = 1958 | url = http://astro1.panet.utoledo.edu/~relling2/teach/6180-7180/Hall_effect_van%20der%20Pauw_1958.pdf | format = [[PDF]]}}
*{{cite journal | last = van der Pauw | first = L.J. | title = A method of measuring the resistivity and Hall coefficient on lamellae of arbitrary shape | journal = Philips Technical Review | volume = 20 | pages = 220–224 | year = 1958 | url = http://electron.mit.edu/~gsteele/vanderpauw/vanderpauw.pdf | format = [[PDF]]}}
*{{cite web | title = Hall Effect Measurements | publisher = National Institute of Standards and Technology | url = http://www.eeel.nist.gov/812/hall.html }}

{{DEFAULTSORT:Van Der Pauw Method}}
[[Category:Electrical engineering]]

Mix network

2013-11-13T02:43:25Z

67.164.92.75: /* Return Addresses */ some more: R -> R0, Kx -> K_x

{{Unreferenced stub|auto=yes|date=December 2009}}
Temperature is a statistical quantity. The formal definition is T = dU/dS, the change in internal energy with respect to entropy holding volume and particle number constant. A practical definition comes from the fact that the atoms, molecules, or whatever particles in your system have average kinetic energy. The average specifically means to average over the kinetic energy of all the particles in your system.

If the [[velocity|velocities]] of a group of [[electron]]s, e.g., in a [[plasma (physics)|plasma]], follow a [[Maxwell-Boltzmann distribution#Distribution of the velocity vector|Maxwell-Boltzmann distribution]], then the '''electron temperature''' is well-defined as the [[temperature]] of that distribution. For other distributions, two-thirds of the average energy is often referred to as the temperature, since for a Maxwell-Boltzmann distribution with three [[Degrees of freedom (physics and chemistry)|degrees of freedom]], <math>\langle E \rangle = (3/2) \langle k_BT \rangle</math>.

The [[International System of Units|SI]] unit of temperature is the [[kelvin]] (K), but using the above relation the electron temperature is often expressed in terms of the energy unit [[electronvolt]] (eV). Each kelvin (1 K) corresponds to 8.6173324(78)×10−5 eV; this factor is the ratio of the [[Boltzmann constant]] to the [[elementary charge]].

The electron temperature of a plasma can be several orders of magnitude higher than the temperature of the neutral species or of the [[ion]]s. This is a result of two facts. Firstly, many [[plasma source]]s heat the electrons more strongly than the ions. Secondly, atoms and ions are much heavier than electrons, and energy transfer in a two-body [[collision]] is much more efficient if the masses are similar.

{{DEFAULTSORT:Electron Temperature}}

[[Category:Plasma physics]]

{{Physics-stub}}

Hedetniemi's conjecture

2013-09-17T05:27:35Z

67.164.92.75: /* References */ restoring alphabetization

{{Technical|date=January 2013}}

'''Musean hypernumbers''' are an algebraic concept envisioned by [[Charles Musès|Charles A. Musès]] (1919–2000) to form a complete, integrated, connected, and natural number system.<ref name="Muses1972">
{{cite journal
| last = Musès
| first = Charles A.
| title = Hypernumbers and their Spaces: a Summary of New Findings
| journal = J. Study. Consciousness
| volume = 5
| pages = 251–256
| year = 1972 }}
</ref><ref name="Muses1977exp">
{{cite journal
| last = Musès
| first = Charles A.
| title = Explorations in mathematics
| journal = Impact of science on society
| volume = 27
| pages = 67–85
| year = 1977 }}
</ref><ref name="Muses1978">
{{cite journal
| last = Musès
| first = Charles A.
| title = Hypernumbers—II. further concepts and computational applications
| journal = Appl. Math. Comput.
| volume = 4
| pages = 45–66
| year = 1978
| doi = 10.1016/0096-3003(78)90026-7 }}
</ref><ref name="Muses1979bio">
{{cite journal
| last = Musès
| first = Charles A.
| title = Computing in the bio-sciences with hypernumbers: a survey
| journal = Intl. J. Bio-Med. Comput.
| volume = 10
| pages = 519–525
| year = 1979
| doi = 10.1016/0020-7101(79)90032-1
| issue = 6 }}
</ref><ref name="Muses1983">
{{cite journal
| last = Musès
| first = Charles A.
| title = Hypernumbers and time operators
| journal = Appl. Math. Comput.
| volume = 12
| pages = 139–167
| year = 1983
| doi = 10.1016/0096-3003(83)90004-8
| issue = 2–3 }}
</ref> Musès sketched certain fundamental types of hypernumbers and arranged them in ten "levels", each with its own associated [[arithmetic]] and [[geometry]].

Mostly criticized for lack of mathematical rigor and unclear defining relations, Musean hypernumbers are often perceived as an unfounded mathematical speculation. This impression was not helped by Musès' outspoken confidence in applicability to fields far beyond what one might expect from a number system, including consciousness, religion, and metaphysics.

The term "M-algebra" was used by Musès for investigation into a subset of his hypernumber concept (the 16 dimensional conic [[sedenion]]s and certain subalgebras thereof), which is at times confused with the Musean hypernumber level concept itself. The current article separates this well-understood "M-algebra" from the remaining controversial hypernumbers, and lists certain applications envisioned by the inventor.

=="M-algebra" and "hypernumber levels"==

Musès was convinced that the basic laws of [[arithmetic]] on the reals are in direct correspondence with a concept where numbers could be arranged in "levels", where fewer arithmetical laws would be applicable with increasing level number.<ref name="Muses1978" /> However, this concept was not developed much further beyond the initial idea, and defining relations for most of these levels have not been constructed.

Higher dimensional numbers built on the first three levels were called "M-algebra"<ref name="Muses1980">
{{cite journal
| last = Musès
| first = Charles A.
| title = Hypernumbers and quantum field theory with a summary of physically applicable hypernumber arithmetics and their geometries
| journal = Applied Mathematics and Computing
| volume = 6
| pages = 63–94
| year = 1980
| doi = 10.1016/0096-3003(80)90016-8 }}
</ref><ref name="Muses1980a">
{{cite journal
| last = Musès
| first = Charles A.
| title = Erratum to "Hypernumbers and quantum field theory"
| journal = Applied Mathematics and Computing
| volume = 6
| pages = 3694
| year = 1980 }}
</ref> by Musès if they yielded a [[Distributivity|distributive]] multiplication, unit element, and multiplicative [[Norm (mathematics)|norm]]. It contains kinds of [[octonion]]s and historical [[quaternion]]s (except A. MacFarlane's [[hyperbolic quaternion]]s) as subalgebras. A proof of completeness of M-algebra has not been provided.

==Conic sedenions / "16 dimensional M-algebra"==
The term "M-algebra" (after C. Musès<ref name="Muses1980" />) refers to number systems that are [[vector space]]s over the [[real number|reals]], whose bases consist in roots of −1 or +1, and which possess a multiplicative modulus. While the idea of such numbers was far from new and contains many known isomorphic number systems (like e.g. [[Split-complex number|split-complex]] numbers or [[tessarine]]s), certain results from 16 dimensional (conic) sedenions were a novelty. Musès demonstrated the existence of a logarithm and real powers in number systems built to non-real roots of +1.

===Multiplication table===
The conic sedenions<ref name="Carmody1988">
{{cite journal
| last = Carmody
| first = Kevin
| title = Circular and hyperbolic quaternions, octonions, and sedenions
| journal = Appl. Math. Comput.
| volume = 28
| pages = 47–72
| year = 1988
| doi = 10.1016/0096-3003(88)90133-6 }}
</ref><ref name="Carmody1997">
{{cite journal
| last = Carmody
| first = Kevin
| title = Circular and hyperbolic quaternions, octonions, and sedenions— further results
| journal = Appl. Math. Comput.
| volume = 84
| pages = 27–48
| year = 1997
| doi = 10.1016/S0096-3003(96)00051-3 }}
</ref> form an algebra with a non-[[commutative]], non-[[associative]], but [[Alternative algebra|alternative]] multiplication and a multiplicative modulus. It consists of one real axis (to basis <math>1</math>), eight imaginary axes (to bases <math>i_n</math> with <math>i_n^2=-1</math>), and seven counterimaginary<ref name=counterimagexp>The terms "counterimaginary" and "countercomplex" used by Musès are synonymous to the more common term [[split-complex number|split-complex]]</ref> axes (to bases <math>\varepsilon</math> with <math>\varepsilon{}_n^2=+1</math>).

The multiplication table is:

[[Image:ConicSedenionsMultTable.png]]

Similar to unity (1), the imaginary basis <math>i_0</math> is always commutative and associative under multiplication. Musès at times used the symbol <math>\varepsilon_0 := 1</math> to highlight this similarity.<ref name="Muses1980" /> In fact, conic sedenions are isomorphic to complex [[octonion]]s, i.e. octonions with [[complex number]] coefficients. By examining <math>\varepsilon_n</math> as bases to real number coefficients, however, Musès was able to show certain algebraic relations, including power and logarithm of <math>\varepsilon_n</math>.

===Select findings===
Musès showed that a countercomplex basis <math>\varepsilon{}_n</math> (<math>n = 1, \ldots, 7</math>) not only has an [[exponential function]]<ref name="Muses1977comp">
{{cite journal
| last = Musès
| first = Charles A.
| title = Applied hypernumbers: computational concepts
| journal = Appl. Math. Comput.
| volume = 3
| pages = 211–226
| year = 1977
| doi = 10.1016/0096-3003(77)90002-9
| issue = 3 }}
</ref>

: <math>e ^ { \varepsilon{}_n \alpha } = \cosh ~\alpha + \varepsilon{}_n ( \sinh ~\alpha )</math>

(<math>\alpha</math> real) but also possesses real powers:<ref name="Carmody1988" /><ref name="Muses1994">
{{cite journal
| last = Musès
| first = Charles A.
| title = Hypernumbers applied, or how they interface with the physical world
| journal = Appl. Math. Comput.
| volume = 60
| pages = 25–36
| year = 1994
| doi = 10.1016/0096-3003(94)90203-8 }}
</ref>

: <math>\varepsilon{}_n ^ \alpha = \frac{1}{2} [ (1 - \varepsilon{}_n ) + (1 + \varepsilon{}_n ) e^{- \pi i_n \alpha } ]</math>

This is referred to as "power orbit" of <math>\varepsilon{}_n</math> by Musès. Also, a logarithm

: <math>\ln \varepsilon{}_n = \frac{\pi }{2} ( i_0 - i_n )</math>

is possible in this arithmetic.<ref name="Carmody1988" /> Their multiplicative modulus <math>|z|</math> is<ref name="Carmody1997" />

: <math>|z| = |a + \sum{b_n i_n} + \sum{c_n \varepsilon_n } + d| := \sqrt[4]{ (a^2 + b_n^2 - c_n^2 - d^2)^2 + 4(ad - b_n c_n)^2 }</math>

===List of number types<ref name="Carmody1988" /> and their isomorphisms===

====Circular quaternions and octonions====
Circular quaternions and octonions from the Musean hypernumbers are identical to [[quaternions]] and [[octonions]] from [[Cayley–Dickson construction]]. They are built on imaginary bases <math>i_n</math> only.

====Hyperbolic quaternions====
Hyperbolic quaternions after Musès, to bases {<math>1, \varepsilon{}_1 , \varepsilon{}_2 , i_3</math>} are isomorphic to [[coquaternion]]s (split-quaternions). They are different from [[Alexander Macfarlane]]'s [[hyperbolic quaternion]]s (first mention in 1891), which are not [[associative]].

====Conic quaternions====
Conic quaternions are built on bases {<math>1, i, \varepsilon, i_0</math>} and form a [[commutative]], [[associative]], and [[distributive]] [[arithmetic]]. They contain non-trivial [[idempotent]]s and [[zero divisor]]s, but no [[nilpotent]]s. Conic quaternions are isomorphic to [[tessarine]]s, and also to [[bicomplex number]]s (from the [[multicomplex number]]s).

In contrast, circular and hyperbolic quaternions are not commutative, hyperbolic quaternions also contain nilpotents.

====Hyperbolic octonions====
Hyperbolic [[octonions]] are isomorphic to [[split-octonion]] algebra. They consist of one [[real number|real]], three [[imaginary number|imaginary]] (<math>\sqrt{-1}</math>), and four counterimaginary (<math>\varepsilon</math>) bases, e.g. {<math>1, i_1, i_2, i_3, \varepsilon{}_4 , \varepsilon{}_5, \varepsilon{}_6 , \varepsilon{}_7 </math>}.

====Conic octonions====
Conic octonions to bases <math>\{ 1, i_1, i_2, i_3,~i_0, \varepsilon{}_1, \varepsilon{}_2, \varepsilon{}_3 \} </math> form an associative, non-commutative octonionic number system. They are isomorphic to [[biquaternion]]s.

===External links===
* Mention in zero-divisor analysis by [http://arxiv.org/abs/math.GM/0011260 R. de Marrais] on arXiv.org
* Zero-divisor algebras on [http://www.tony5m17h.net/NDalg.html Tony Smith]'s personal home page (as of 12 Jan 2007)

==The hypernumber "level" concept==
In<ref name="Muses1978" /> Musès paired certain fundamental laws of arithmetic with suggested number ''levels'', where fewer of these laws would be applicable with increasing level number. Musès envisioned "... sensitivity to operational distinctions on the part of hypernumbers". In the absence of [[Rigour|rigorous]] mathematical treatment, however, Musès' hypernumber level concept has only been adapted for metaphysical or religious ideas.<ref name="MusaiosLionPath">
Musaios (a [[pseudonym]] of Musès'), "The Lion Path", House of Horus (1990)
</ref><ref name="HouseOfHorusSite1">
[http://www.house-of-horus.de House of Horus web site]
</ref><ref name="LionPathSite1">
Private [http://www.siriusrising.com/lionpath.htm Lion Path] web site
</ref>

Providing defining relations for hypernumbers remains a fringe interest today,<ref>[http://groups.yahoo.com/group/hypercomplex "Hypercomplex" number discussion group on Yahoo (R)]</ref> though it could benefit description of physical law that is based on the lower, well-understood levels.<ref name="Koeplinger2006rel">
{{cite journal
| last = Köplinger
| first = Jens
| title = Hypernumbers and relativity
| journal = Appl. Math. Comput.
| year = 2006
| doi = 10.1016/j.amc.2006.10.051
| volume = 188
| pages = 954 }}
</ref><ref name="Koeplinger2006grav">
{{cite journal
| last = Köplinger
| first = Jens
| title = Gravity and electromagnetism on conic sedenions
| journal = Appl. Math. Comput.
| year = 2006
| doi = 10.1016/j.amc.2006.10.050
| volume = 188
| pages = 954 }}
</ref>

The following lists an overview of the levels as envisioned by Musès.

===Real, complex, and epsilon numbers===
The first two levels in hypernumber arithmetic correspond to [[real number|real]] and [[imaginary number]] arithmetic. The <math>\varepsilon</math> basis after Musès is identical to ''j'' from the [[split-complex numbers]], and is a non-real root of <math>+1</math>. Epsilon numbers are assigned the 3rd level in the hypernumbers program.

===''w'' arithmetic===
Beginning with ''w'' arithmetic,<ref name="Muses1972" /><ref name="Muses1979bio" /><ref name="Muses1994" /> Musès envisioned hypernumber types that are increasingly unfamiliar and speculative. While providing certain rules on how to use these numbers, many open questions remain to date. ''w'' numbers are assigned the 4th level in the hypernumbers program.

In the two-dimensional (real, ''w'') plane, the power orbit <math>~w^\alpha</math> (with <math>~\alpha</math> real) is periodic with <math>w^0 = w^6 = 1</math> and the following integral powers:

: <math>w^1 = ~w</math>

: <math>w^2 = ~-1 + w</math>

: <math>w^3 = ~-1</math>

: <math>w^4 = ~-w</math>

: <math>w^5 = ~1 - w.</math>

They offer a multiplicative modulus:

: <math>||a + bw|| = \sqrt{a^2 + ab + b^2}</math>

If ''a'' and ''b'' are [[real number]] coefficients, the arithmetic <(1,''w''), +, *> is a [[field (mathematics)|field]] (in fact the [[complex number]]s with basis 1 and a primitive sixth [[root of unity]] rather than the usual fourth). However, the dual base number to ''(w)'' is ''(-w)'', which is different from the conjugate of ''(w)'', which is ''1-(w)''. This is in contrast to e.g. the imaginary base <math>i := \sqrt{-1}</math>, for which both dual and conjugate are the same (-''i''). The resulting (-''w'') arithmetic is therefore distinct from -(''w'') arithmetic, while coexisting on the same number plane.

[[Image:HypernumbersPowerOrbitW.gif]]

===''p'' and ''q'' numbers===
So-called ''p'' and ''q'' numbers<ref name="Muses1979bio" /> are assigned the 5th level in the hypernumbers program, and form a nearly dual system. Each being [[nilpotent]] (<math>p^2 = q^2 = 0</math>), the arithmetic is envisioned to offer a multiplicative [[absolute value|modulus]], an [[Complex number#Complex plane|argument]], and a polar form.

The integral powers are:

: <math>p^0 = q^0 = p^2 = q^2 =~0</math>

: <math>p^1 =~p</math>

: <math>q^1 =~q</math>

: <math>p^3 =~q</math>

: <math>q^3 =~p</math>

In the {''p'', ''q''} plane, both <math>~p^\alpha</math> and <math>~q^\alpha</math> (with <math>~\alpha</math> real) lie on a two-leaved rose, described through <math>ap +~bq</math> with

: <math>(a^2 + b^2)^2 =~(a + b)(a - b)^2.</math>

[[Image:HypernumbersPowerOrbitPQ.gif]]

====Note on (−''p''), ''p'' −1, 1/''p''====
From:<ref name="Muses1979bio" />

"...Note that −''p'' is generated via ''w'', thus: <math>(qw)^3 = (wq)^3 = (w^3)(q^3) = (-1)p =~-p</math>. It must be remembered that because ''p'' is nilpotent (<math>p^2 = 0, p \ne 0</math>), its zeroth power cannot be 1; in fact <math>p^0 =~0</math>. Hence also <math>p^{-1} \ne 1/p</math>, and since <math>(1/p)(1/p) = 1/p^2 = \infty</math>, we see that <math>~1/p</math> is panpotent, i.e. a root of infinity. Compare <math>1/(1 \pm \varepsilon)</math>, which are a pair of divisors of infinity."

===''m'' numbers===
The 6th level in the Musean hypernumbers is governed by cassinoids or Cassinian ovals,<ref name="Muses1979bio" /> which geometrically describe their multiplication.

In the {real, ''m''} plane, they offer the following relations:

: <math>m^2 =~m</math>

: <math>(\sqrt{2} m )^2 =~0</math>

: <math>(\sqrt{3} m )^2 =~-1</math>

It is speculated that a number system like this would use coefficients such as <math>\sqrt{2}</math> in the expression <math>\sqrt{2} m</math>, that are not actually real numbers. Instead, one would need to look at +1, -1, +''m'', and -''m'' as units, and the coefficients as ''absolute numbers'' which are distinct from real numbers and are never negative.

[[Image:HypernumbersPowerOrbitM.gif]]

The Cassinian ovals are described by:

: <math>s^4 :=~(a^2 + b^2)^2 + 2(a^2 - b^2) + 1</math>

===The remaining levels===
In the 7th level, Musès pictured a number <math>\Omega</math> where <math>\Omega^n = \Omega</math> for any finite ''n'', <math>\Omega^\infty = 0</math>, but <math>\Omega^{\infty - n}</math> would be a number of the form <math>a + b \Omega</math> (with ''a'', ''b'' real).<ref name="Muses1979bio" />

The 8th level, <math>\upsilon</math> is envisioned as unifying concept to allow to transition between all the lower hypernumber types.<ref name="Muses1983" />

The 9th level, <math>\sigma</math> is envisioned as the creator of axes, and has somewhat the characteristic of an operator (rather than a number). The product <math>\sigma \upsilon</math> is proposed to be the unit step function.<ref name="Muses1983" />

The 10th level consists of 0 and antinumbers. Antinumbers are envisioned to be numbers beyond positive and negative infinity. With use of <math>\upsilon</math> one would be able to span entire spaces consisting of axes of zeros, and connect numbers beyond positive and negative infinity.<ref name="Muses1983" />

==Visions of applicability==
The range of applications envisioned by Musès of his hypernumber concept can be considered grandiose: A full and complete understanding of all laws of physics (in particular quantum mechanics<ref name="Muses1980" /><ref name="Muses1984">
{{cite journal
| last = Musès
| first = Charles A.
| title = Some current dilemmas in applied physical mathematics with some solutions
| journal = Appl. Math. Comput.
| volume = 14
| pages = 207–211
| year = 1984
| doi = 10.1016/0096-3003(84)90038-9
| issue = 2 }}
</ref>), a description of consciousness in terms of physical formulations,<ref name="Muses1972" /><ref name="Muses1979bio" /><ref name="Muses1983" /> spiritual growth, religious enlightenment, the solution of well-known mathematical problems (including the [[Riemann hypothesis]]), and the exploration of para-psychological phenomena (e.g.<ref name="thinkingAllowed">
Charles Musès - "Time and destiny", Thinking Allowed Productions (#S460) [http://www.thinking-allowed.com/2cmuses.html online])
</ref>). Many of Musès' own writings combine mathematical content with one or more of these speculative projects,.<ref name="MusesYoung">
"''The nature of hypernumbers can reveal the projection process ... (and) on the source of the hologram world or ordinary bodily experience ... to be able to go between the image world and the source world at will (time travel).''" (from C. Musès, A. M. Young: "Consciousness and reality: the human pivot point", Outerbridge & Lazard, New York, 1972)
</ref> The secondary literature on Musès devotes itself more to his speculative thought than to his mathematics.

==See also==
* [[Absolute infinite]]
* [[Biquaternion]]
* [[Hyperbolic quaternion]] (per A. MacFarlane)
* [[Hypercomplex number]]
* [[Nilpotent]]s
* [[Octonion]]
* [[Quaternion]]
* [[Sedenion]]
* [[Split-complex number]]
* [[Split-octonion]]s
* [[Split-quaternion]] / [[Coquaternion]]
* [[Tessarine]]s
* [[Zero divisor]]

==References==
{{reflist|2}}

==External links==
*[http://web.archive.org/web/20060505224444/http://www.kevincarmody.com/math/hypernumbers.html Kevin Carmody's website on hypernumbers] at the [[Wayback Machine]]

{{Number Systems}}

[[Category:Hypercomplex numbers]]
[[Category:Non-associative algebra]]

Talk:Common subexpression elimination

2013-05-29T19:22:58Z

67.164.143.201:

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2012-06-11T03:46:17Z

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