Continuity and Gauges - Revision history

Khanh: /* Continuity */

2022-01-09T23:44:38Z

Continuity

Khanh: /* Properties */

2022-01-09T23:27:22Z

Properties

Khanh: /* Gauge Integral */

2022-01-09T23:23:59Z

Gauge Integral

Khanh: /* Gauge Integral */

2022-01-09T23:23:12Z

Gauge Integral

Khanh: /* Gauge Integral */

2022-01-09T23:20:55Z

Gauge Integral

Khanh at 23:16, 9 January 2022

2022-01-09T23:16:04Z

Khanh at 22:55, 9 January 2022

2022-01-09T22:55:42Z

Khanh: Created page with "= Continuity = The integral form of the continuity equation states that: * The amount of {{math|''q''}} in a region increases when additional {{math|''q''}} flows inward thro..."

2022-01-09T22:52:37Z

Created page with "= Continuity = The integral form of the continuity equation states that: * The amount of {{math|''q''}} in a region increases when additional {{math|''q''}} flows inward thro..."

New page

= Continuity =

The integral form of the continuity equation states that:
* The amount of {{math|''q''}} in a region increases when additional {{math|''q''}} flows inward through the surface of the region, and decreases when it flows outward;
* The amount of {{math|''q''}} in a region increases when new {{math|''q''}} is created inside the region, and decreases when {{math|''q''}} is destroyed;
* Apart from these two processes, there is ''no other way'' for the amount of {{math|''q''}} in a region to change.

Mathematically, the integral form of the continuity equation expressing the rate of increase of {{math|''q''}} within a volume {{math|''V''}} is:
{{Equation box 1
|indent=:
|equation={{oiint | preintegral = <math>\frac{d q}{d t} + </math> | intsubscpt = <math>\scriptstyle S</math> | integrand = <math>\mathbf{j} \cdot d\mathbf{S} = \Sigma</math>}}
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA
}}

[[File:SurfacesWithAndWithoutBoundary.svg|right|thumb|250px|In the integral form of the continuity equation, {{math|''S''}} is any closed surface that fully encloses a volume {{math|''V''}}, like any of the surfaces on the left. {{math|''S''}} can ''not'' be a surface with boundaries, like those on the right. (Surfaces are blue, boundaries are red.)]]
where
* {{math|''S''}} is any imaginary closed surface, that encloses a volume {{math|''V''}},
* {{oiint | preintegral = | intsubscpt = {{math|''S''}} | integrand={{math|''d'''''S'''}}}} denotes a surface integral over that closed surface,
* {{math|''q''}} is the total amount of the quantity in the volume {{math|''V''}},
* {{math|'''j'''}} is the flux of {{math|''q''}},
* {{math|''t''}} is time,
* {{math|Σ}} is the net rate that {{math|''q''}} is being generated inside the volume {{math|''V''}} per unit time. When {{math|''q''}} is being generated, it is called a ''source'' of {{math|''q''}}, and it makes {{math|Σ}} more positive. When {{math|''q''}} is being destroyed, it is called a ''sink'' of {{math|''q''}}, and it makes {{math|Σ}} more negative. This term is sometimes written as <math>dq/dt|_\text{gen}</math> or the total change of q from its generation or destruction inside the control volume.

In a simple example, {{math|''V''}} could be a building, and {{math|''q''}} could be the number of people in the building. The surface {{math|''S''}} would consist of the walls, doors, roof, and foundation of the building. Then the continuity equation states that the number of people in the building increases when people enter the building (an inward flux through the surface), decreases when people exit the building (an outward flux through the surface), increases when someone in the building gives birth (a source, {{math|Σ > 0}}), and decreases when someone in the building dies (a sink, {{math|Σ < 0}}).

= Licensing =
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Continuity_equation Continuity equation, Wikipedia] under a CC BY-Sa license

@@ Line 100: / Line 100: @@
 :<math>F(x) - F(a) = \int_a^x F'(t) \,dt.</math>
-Conversely, the Lebesgue differentiation theorem continues to hold for the Henstock–Kurzweil integral: if ''f'' is Henstock–Kurzweil integrable on {{closed-closed|''a'', ''b''}}, and
+Conversely, the Lebesgue differentiation theorem continues to hold for the Henstock–Kurzweil integral: if ''f'' is Henstock–Kurzweil integrable on [''a'', ''b''], and
 :<math>F(x) = \int_a^x f(t)\,dt,</math>

@@ Line 23: / Line 23: @@
 = Gauge Integral =
-In mathematics, the '''Henstock–Kurzweil integral''' or '''generalized Riemann integral''' or '''gauge integral''' – also known as the (narrow) '''Denjoy integral''' (pronounced [dɑ̃ˈʒwa])), '''Luzin integral''' or '''Perron integral''', but not to be confused with the more general wide Denjoy integral – is one of a number of definitions of the integral of a function. It is a generalization of the Riemann integral, and in some situations is more general than the Lebesgue integral.  In particular, a function is Lebesgue integrable if and only if the function and its absolute value are Henstock–Kurzweil integrable.
+In mathematics, the '''Henstock–Kurzweil integral''' or '''generalized Riemann integral''' or '''gauge integral''' – also known as the (narrow) '''Denjoy integral''' (pronounced [dɑ̃ˈʒwa]), '''Luzin integral''' or '''Perron integral''', but not to be confused with the more general wide Denjoy integral – is one of a number of definitions of the integral of a function. It is a generalization of the Riemann integral, and in some situations is more general than the Lebesgue integral.  In particular, a function is Lebesgue integrable if and only if the function and its absolute value are Henstock–Kurzweil integrable.
 This integral was first defined by Arnaud Denjoy (1912). Denjoy was interested in a definition that would allow one to integrate functions like

@@ Line 29: / Line 29: @@
 :<math>f(x)=\frac{1}{x}\sin\left(\frac{1}{x^3}\right).</math>
-This function has a singularity at 0, and is not Lebesgue integrable. However, it seems natural to calculate its integral except over the interval {{closed-closed|−''ε'', ''δ''}} and then let {{math|ε, δ → 0}}.
+This function has a singularity at 0, and is not Lebesgue integrable. However, it seems natural to calculate its integral except over the interval [−''ε'', ''δ''] and then let {{math|ε, δ → 0}}.
 Trying to create a general theory, Denjoy used transfinite induction over the possible types of singularities, which made the definition quite complicated. Other definitions were given by Nikolai Luzin (using variations on the notions of absolute continuity), and by Oskar Perron, who was interested in continuous major and minor functions. It took a while to understand that the Perron and Denjoy integrals are actually identical.
@@ Line 36: / Line 36: @@
 ==Definition==
-Given a tagged partition {{math|''P''}} of {{closed-closed|''a'', ''b''}}, that is,
+Given a tagged partition {{math|''P''}} of [''a'', ''b''], that is,
 :<math>a = u_0 < u_1 < \cdots < u_n = b </math>
@@ Line 62: / Line 62: @@
 :<math> \left \vert \sum_P f - I \right \vert < \varepsilon. </math>
-If such an {{math|''I''}} exists, we say that {{math|''f''}} is Henstock–Kurzweil integrable on {{closed-closed|''a'', ''b''}}.
+If such an {{math|''I''}} exists, we say that {{math|''f''}} is Henstock–Kurzweil integrable on [''a'', ''b''].
 Cousin's theorem states that for every gauge <math>\delta</math>, such a <math>\delta</math>-fine partition ''P'' does exist, so this condition cannot be satisfied vacuously. The Riemann integral can be regarded as the special case where we only allow constant gauges.
@@ Line 68: / Line 68: @@
 ==Properties==
-Let {{math|''f'': {{closed-closed|''a'', ''b''}} → '''R'''}} be any function.
+Let {{math|''f'': [''a'', ''b''] → '''R'''}} be any function.
-Given {{math|''a'' < ''c'' < ''b''}}, {{math|''f''}} is Henstock–Kurzweil integrable on {{closed-closed|''a'', ''b''}} if and only if it is Henstock–Kurzweil integrable on both {{closed-closed|''a'', ''c''}} and {{closed-closed|''c'', ''b''}}; in which case,
+Given {{math|''a'' < ''c'' < ''b''}}, {{math|''f''}} is Henstock–Kurzweil integrable on [''a'', ''b''] if and only if it is Henstock–Kurzweil integrable on both [''a'', ''c''] and [''c'', ''b'']; in which case,
 :<math>\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx.</math>
@@ Line 104: / Line 104: @@
 :<math>F(x) = \int_a^x f(t)\,dt,</math>
-then ''F''′(''x'') = ''f''(''x'') almost everywhere in {{closed-closed|''a'', ''b''}} (in particular, ''F'' is differentiable almost everywhere).
+then ''F''′(''x'') = ''f''(''x'') almost everywhere in [''a'', ''b''] (in particular, ''F'' is differentiable almost everywhere).
-The space of all Henstock–Kurzweil-integrable functions is often endowed with the [[Alexiewicz norm]], with respect to which it is barrelled but incomplete.
+The space of all Henstock–Kurzweil-integrable functions is often endowed with the Alexiewicz norm, with respect to which it is barrelled but incomplete.
 ==McShane integral==

@@ Line 23: / Line 23: @@
 = Gauge Integral =
-In mathematics, the '''Henstock–Kurzweil integral''' or '''generalized Riemann integral''' or '''gauge integral''' – also known as the (narrow) '''Denjoy integral''' (pronounced {{IPA-fr|dɑ̃ˈʒwa|}}), '''Luzin integral''' or '''Perron integral''', but not to be confused with the more general [[Khinchin integral|wide Denjoy integral]] – is one of a number of definitions of the [[integral]] of a [[function (mathematics)|function]]. It is a generalization of the [[Riemann integral]], and in some situations is more general than the [[Lebesgue integral]].  In particular, a function is Lebesgue integrable if and only if the function and its [[absolute value]] are Henstock–Kurzweil integrable.
+In mathematics, the '''Henstock–Kurzweil integral''' or '''generalized Riemann integral''' or '''gauge integral''' – also known as the (narrow) '''Denjoy integral''' (pronounced [dɑ̃ˈʒwa])), '''Luzin integral''' or '''Perron integral''', but not to be confused with the more general wide Denjoy integral – is one of a number of definitions of the integral of a function. It is a generalization of the Riemann integral, and in some situations is more general than the Lebesgue integral.  In particular, a function is Lebesgue integrable if and only if the function and its absolute value are Henstock–Kurzweil integrable.
-This integral was first defined by [[Arnaud Denjoy]] (1912). Denjoy was interested in a definition that would allow one to integrate functions like
+This integral was first defined by Arnaud Denjoy (1912). Denjoy was interested in a definition that would allow one to integrate functions like
 :<math>f(x)=\frac{1}{x}\sin\left(\frac{1}{x^3}\right).</math>
-This function has a [[singularity (mathematics)|singularity]] at 0, and is not Lebesgue integrable. However, it seems natural to calculate its integral except over the interval {{closed-closed|−''ε'', ''δ''}} and then let {{math|ε, δ → 0}}.
+This function has a singularity at 0, and is not Lebesgue integrable. However, it seems natural to calculate its integral except over the interval {{closed-closed|−''ε'', ''δ''}} and then let {{math|ε, δ → 0}}.
-Trying to create a general theory, Denjoy used [[transfinite induction]] over the possible types of singularities, which made the definition quite complicated. Other definitions were given by [[Nikolai Luzin]] (using variations on the notions of [[absolute continuity]]), and by [[Oskar Perron]], who was interested in continuous major and minor functions. It took a while to understand that the Perron and Denjoy integrals are actually identical.
+Trying to create a general theory, Denjoy used transfinite induction over the possible types of singularities, which made the definition quite complicated. Other definitions were given by Nikolai Luzin (using variations on the notions of absolute continuity), and by Oskar Perron, who was interested in continuous major and minor functions. It took a while to understand that the Perron and Denjoy integrals are actually identical.
-Later, in 1957, the Czech mathematician [[Jaroslav Kurzweil]] discovered a new definition of this integral elegantly similar in nature to [[Riemann]]'s original definition which he named the '''gauge integral'''; the theory was developed by [[Ralph Henstock]]. Due to these two important contributions, it is now commonly known as the '''Henstock–Kurzweil integral'''. The simplicity of Kurzweil's definition made some educators advocate that this integral should replace the Riemann integral in introductory calculus courses.<ref>{{cite web|title=An Open Letter to Authors of Calculus Books|url=https://math.vanderbilt.edu/schectex/ccc/gauge/letter/|access-date=27 February 2014}}</ref>
+Later, in 1957, the Czech mathematician Jaroslav Kurzweil discovered a new definition of this integral elegantly similar in nature to Riemann's original definition which he named the '''gauge integral'''; the theory was developed by Ralph Henstock. Due to these two important contributions, it is now commonly known as the '''Henstock–Kurzweil integral'''. The simplicity of Kurzweil's definition made some educators advocate that this integral should replace the Riemann integral in introductory calculus courses.
 ==Definition==
-Given a [[tagged partition]] {{math|''P''}} of {{closed-closed|''a'', ''b''}}, that is,
+Given a tagged partition {{math|''P''}} of {{closed-closed|''a'', ''b''}}, that is,
 :<math>a = u_0 < u_1 < \cdots < u_n = b </math>
@@ Line 64: / Line 64: @@
 If such an {{math|''I''}} exists, we say that {{math|''f''}} is Henstock–Kurzweil integrable on {{closed-closed|''a'', ''b''}}.
-[[Cousin's theorem]] states that for every gauge <math>\delta</math>, such a <math>\delta</math>-fine partition ''P'' does exist, so this condition cannot be satisfied [[vacuous truth|vacuously]]. The Riemann integral can be regarded as the special case where we only allow [[constant function|constant]] gauges.
+Cousin's theorem states that for every gauge <math>\delta</math>, such a <math>\delta</math>-fine partition ''P'' does exist, so this condition cannot be satisfied vacuously. The Riemann integral can be regarded as the special case where we only allow constant gauges.
 ==Properties==
@@ Line 78: / Line 78: @@
 :<math>\int_a^b \left(\alpha f(x) + \beta g(x)\right) dx = \alpha \int_a^bf(x)\,dx + \beta \int_a^b g(x)\,dx.</math>
-If ''f'' is Riemann or Lebesgue integrable, then it is also Henstock–Kurzweil integrable, and calculating that integral gives the same result by all three formulations. The important [[Hake's theorem]] states that
+If ''f'' is Riemann or Lebesgue integrable, then it is also Henstock–Kurzweil integrable, and calculating that integral gives the same result by all three formulations. The important Hake's theorem states that
 :<math>\int_a^b f(x)\,dx = \lim_{c\to b^-} \int_a^c f(x)\,dx</math>
-whenever either side of the equation exists, and likewise symmetrically for the lower integration bound. This means that if {{math|''f''}} is "[[improper integral|improperly]] Henstock–Kurzweil integrable", then it is properly Henstock–Kurzweil integrable; in particular, improper Riemann or Lebesgue integrals of types such as
+whenever either side of the equation exists, and likewise symmetrically for the lower integration bound. This means that if {{math|''f''}} is "improperly Henstock–Kurzweil integrable", then it is properly Henstock–Kurzweil integrable; in particular, improper Riemann or Lebesgue integrals of types such as
 :<math>\int_0^1 \frac{\sin(1/x)}x\,dx</math>
@@ Line 93: / Line 93: @@
 *{{math|''f''}} is Henstock–Kurzweil integrable,
 *{{math|''f''}} is Lebesgue integrable,
-*{{math|''f''}} is [[measurable function|Lebesgue measurable]].
+*{{math|''f''}} is Lebesgue measurable.
-In general, every Henstock–Kurzweil integrable function is measurable, and {{math|''f''}} is Lebesgue integrable if and only if both {{math|''f''}} and |''f''| are Henstock–Kurzweil integrable. This means that the Henstock–Kurzweil integral can be thought of as a "[[absolute convergence|non-absolutely convergent]] version of the Lebesgue integral". It also implies that the Henstock–Kurzweil integral satisfies appropriate versions of the [[monotone convergence theorem#Lebesgue monotone convergence theorem|monotone convergence theorem]] (without requiring the functions to be nonnegative) and [[dominated convergence theorem]] (where the condition of dominance is loosened to {{math|''g''(''x'') ≤ ''f<sub>n</sub>''(''x'') ≤ ''h''(''x'')}} for some integrable ''g'', ''h'').
+In general, every Henstock–Kurzweil integrable function is measurable, and {{math|''f''}} is Lebesgue integrable if and only if both {{math|''f''}} and |''f''| are Henstock–Kurzweil integrable. This means that the Henstock–Kurzweil integral can be thought of as a "non-absolutely convergent version of the Lebesgue integral". It also implies that the Henstock–Kurzweil integral satisfies appropriate versions of the monotone convergence theorem (without requiring the functions to be nonnegative) and dominated convergence theorem (where the condition of dominance is loosened to {{math|''g''(''x'') ≤ ''f<sub>n</sub>''(''x'') ≤ ''h''(''x'')}} for some integrable ''g'', ''h'').
-If ''F'' is differentiable everywhere (or with countably many exceptions), the derivative ''F''′ is Henstock–Kurzweil integrable, and its indefinite Henstock–Kurzweil integral is ''F''.  (Note that ''F''′ need not be Lebesgue integrable.) In other words, we obtain a simpler and more satisfactory version of the [[second fundamental theorem of calculus]]: each differentiable function is, up to a constant, the integral of its derivative:
+If ''F'' is differentiable everywhere (or with countably many exceptions), the derivative ''F''′ is Henstock–Kurzweil integrable, and its indefinite Henstock–Kurzweil integral is ''F''.  (Note that ''F''′ need not be Lebesgue integrable.) In other words, we obtain a simpler and more satisfactory version of the second fundamental theorem of calculus: each differentiable function is, up to a constant, the integral of its derivative:
 :<math>F(x) - F(a) = \int_a^x F'(t) \,dt.</math>
-Conversely, the [[Lebesgue differentiation theorem]] continues to hold for the Henstock–Kurzweil integral: if ''f'' is Henstock–Kurzweil integrable on {{closed-closed|''a'', ''b''}}, and
+Conversely, the Lebesgue differentiation theorem continues to hold for the Henstock–Kurzweil integral: if ''f'' is Henstock–Kurzweil integrable on {{closed-closed|''a'', ''b''}}, and
 :<math>F(x) = \int_a^x f(t)\,dt,</math>
@@ Line 106: / Line 106: @@
 then ''F''′(''x'') = ''f''(''x'') almost everywhere in {{closed-closed|''a'', ''b''}} (in particular, ''F'' is differentiable almost everywhere).
-The space of all Henstock–Kurzweil-integrable functions is often endowed with the [[Alexiewicz norm]], with respect to which it is [[barrelled space|barrelled]] but [[complete space|incomplete]].
+The space of all Henstock–Kurzweil-integrable functions is often endowed with the [[Alexiewicz norm]], with respect to which it is barrelled but incomplete.
 ==McShane integral==
-[[Lebesgue integral]] on a line can also be presented in a similar fashion.
+Lebesgue integral on a line can also be presented in a similar fashion.
 If we take the definition of the Henstock–Kurzweil integral from above, and we drop the condition
@@ Line 115: / Line 115: @@
 :<math>t_i \in [u_{i-1}, u_i],</math>
-then we get a definition of the [[McShane integral]], which is equivalent to the Lebesgue integral. Note that the condition
+then we get a definition of the McShane integral, which is equivalent to the Lebesgue integral. Note that the condition
 :<math>\forall i \ \ [u_{i-1}, u_i] \subset [t_i-\delta(t_i), t_i + \delta (t_i)]</math>

@@ Line 8: / Line 8: @@
 Mathematically, the integral form of the continuity equation expressing the rate of increase of {{math|''q''}} within a volume {{math|''V''}} is:
-:<math>\frac{d q}{d t} </math> + <math> \oiint_{S} </math> <math>\mathbf{j} \cdot d\mathbf{S} = \Sigma</math>
+:<math>\frac{d q}{d t} </math> + &#8751;<sub><math>\scriptstyle S</math></sub> <math>\mathbf{j} \cdot d\mathbf{S} = \Sigma</math>
 [[File:SurfacesWithAndWithoutBoundary.svg|right|thumb|250px|In the integral form of the continuity equation, {{math|''S''}} is any closed surface that fully encloses a volume {{math|''V''}}, like any of the surfaces on the left. {{math|''S''}} can ''not'' be a surface with boundaries, like those on the right. (Surfaces are blue, boundaries are red.)]]
 where
 * {{math|''S''}} is any imaginary closed surface, that encloses a volume {{math|''V''}},
-* <math> \oiint_{S} dS </math> denotes a surface integral over that closed surface,
+* &#8751;<sub><math>\scriptstyle S</math></sub> d<math>\mathbf{S}</math> denotes a surface integral over that closed surface,
 * {{math|''q''}} is the total amount of the quantity in the volume {{math|''V''}},
 * {{math|'''j'''}} is the flux of {{math|''q''}},

@@ Line 7: / Line 7: @@
 Mathematically, the integral form of the continuity equation expressing the rate of increase of {{math|''q''}} within a volume {{math|''V''}} is:
-<blockquote style="background: white; border: 1px solid blue; padding: 1em;">
-:<math>\frac{d q}{d t} + </math> | intsubscpt = <math>\scriptstyle S</math> | integrand = <math>\mathbf{j} \cdot d\mathbf{S} = \Sigma</math>
+:<math>\frac{d q}{d t} </math> + <math> \oiint_{S} </math> <math>\mathbf{j} \cdot d\mathbf{S} = \Sigma</math>
 [[File:SurfacesWithAndWithoutBoundary.svg|right|thumb|250px|In the integral form of the continuity equation, {{math|''S''}} is any closed surface that fully encloses a volume {{math|''V''}}, like any of the surfaces on the left. {{math|''S''}} can ''not'' be a surface with boundaries, like those on the right. (Surfaces are blue, boundaries are red.)]]
 where
 * {{math|''S''}} is any imaginary closed surface, that encloses a volume {{math|''V''}},
-* {{oiint | preintegral = | intsubscpt = {{math|''S''}} | integrand={{math|''d'''''S'''}}}} denotes a surface integral over that closed surface,
+* <math> \oiint_{S} dS </math> denotes a surface integral over that closed surface,
 * {{math|''q''}} is the total amount of the quantity in the volume {{math|''V''}},
 * {{math|'''j'''}} is the flux of {{math|''q''}},
@@ Line 21: / Line 20: @@
 In a simple example, {{math|''V''}} could be a building, and {{math|''q''}} could be the number of people in the building. The surface {{math|''S''}} would consist of the walls, doors, roof, and foundation of the building. Then the continuity equation states that the number of people in the building increases when people enter the building (an inward flux through the surface), decreases when people exit the building (an outward flux through the surface), increases when someone in the building gives birth (a source, {{math|Σ > 0}}), and decreases when someone in the building dies (a sink, {{math|Σ < 0}}).
 = Licensing =
 Content obtained and/or adapted from:
 * [https://en.wikipedia.org/wiki/Continuity_equation Continuity equation, Wikipedia] under a CC BY-Sa license