Continuity and Gauges

Continuity

The integral form of the continuity equation states that:

• The amount of q in a region increases when additional q flows inward through the surface of the region, and decreases when it flows outward;
• The amount of q in a region increases when new q is created inside the region, and decreases when q is destroyed;
• Apart from these two processes, there is no other way for the amount of q in a region to change.

Mathematically, the integral form of the continuity equation expressing the rate of increase of q within a volume V is:

${\frac {dq}{dt}}$ + ∯_{$\scriptstyle S$} $\mathbf {j} \cdot d\mathbf {S} =\Sigma$

In the integral form of the continuity equation, S is any closed surface that fully encloses a volume V, like any of the surfaces on the left. S can not be a surface with boundaries, like those on the right. (Surfaces are blue, boundaries are red.)

where

• S is any imaginary closed surface, that encloses a volume V,
• ∯_{$\scriptstyle S$} d$\mathbf {S}$ denotes a surface integral over that closed surface,
• q is the total amount of the quantity in the volume V,
• j is the flux of q,
• t is time,
• Σ is the net rate that q is being generated inside the volume V per unit time. When q is being generated, it is called a source of q, and it makes Σ more positive. When q is being destroyed, it is called a sink of q, and it makes Σ more negative. This term is sometimes written as $dq/dt|_{\text{gen}}$ or the total change of q from its generation or destruction inside the control volume.

In a simple example, V could be a building, and q could be the number of people in the building. The surface 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, Σ > 0), and decreases when someone in the building dies (a sink, Σ < 0).

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.

This integral was first defined by Arnaud Denjoy (1912). Denjoy was interested in a definition that would allow one to integrate functions like

$f(x)={\frac {1}{x}}\sin \left({\frac {1}{x^{3}}}\right).$

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 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.

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 P of [a, b], that is,

$a=u_{0}<u_{1}<\cdots <u_{n}=b$

together with

$t_{i}\in [u_{i-1},u_{i}],$

we define the Riemann sum for a function $f\colon [a,b]\to \mathbb {R}$ to be

$\sum _{P}f=\sum _{i=1}^{n}f(t_{i})\Delta u_{i}.$

where $\Delta u_{i}:=u_{i}-u_{i-1}.$

Given a positive function

$\delta \colon [a,b]\to (0,\infty ),$

which we call a gauge, we say a tagged partition P is $\delta$-fine if

$\forall i\ \ [u_{i-1},u_{i}]\subset [t_{i}-\delta (t_{i}),t_{i}+\delta (t_{i})].$

We now define a number I to be the Henstock–Kurzweil integral of f if for every ε > 0 there exists a gauge $\delta$ such that whenever P is $\delta$-fine, we have

$\left\vert \sum _{P}f-I\right\vert <\varepsilon .$

If such an I exists, we say that f is Henstock–Kurzweil integrable on [a, b].

Cousin's theorem states that for every gauge $\delta$, such a $\delta$-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

Let f: [a, b] → R be any function.

Given a < c < b, 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,

$\int _{a}^{b}f(x)\,dx=\int _{a}^{c}f(x)\,dx+\int _{c}^{b}f(x)\,dx.$

Henstock–Kurzweil integrals are linear. Given integrable functions f, g and real numbers α, β, the expression αf + βg is integrable; for example,

$\int _{a}^{b}\left(\alpha f(x)+\beta g(x)\right)dx=\alpha \int _{a}^{b}f(x)\,dx+\beta \int _{a}^{b}g(x)\,dx.$

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

$\int _{a}^{b}f(x)\,dx=\lim _{c\to b^{-}}\int _{a}^{c}f(x)\,dx$

whenever either side of the equation exists, and likewise symmetrically for the lower integration bound. This means that if f is "improperly Henstock–Kurzweil integrable", then it is properly Henstock–Kurzweil integrable; in particular, improper Riemann or Lebesgue integrals of types such as

$\int _{0}^{1}{\frac {\sin(1/x)}{x}}\,dx$

are also proper Henstock–Kurzweil integrals. To study an "improper Henstock–Kurzweil integral" with finite bounds would not be meaningful. However, it does make sense to consider improper Henstock–Kurzweil integrals with infinite bounds such as

$\int _{a}^{\infty }f(x)\,dx:=\lim _{b\to \infty }\int _{a}^{b}f(x)\,dx.$

For many types of functions the Henstock–Kurzweil integral is no more general than Lebesgue integral. For example, if f is bounded with compact support, the following are equivalent:

• f is Henstock–Kurzweil integrable,
• f is Lebesgue integrable,
• f is Lebesgue measurable.

In general, every Henstock–Kurzweil integrable function is measurable, and f is Lebesgue integrable if and only if both 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 g(x) ≤ f_n(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:

$F(x)-F(a)=\int _{a}^{x}F'(t)\,dt.$

Conversely, the Lebesgue differentiation theorem continues to hold for the Henstock–Kurzweil integral: if f is Henstock–Kurzweil integrable on [a, b], and

$F(x)=\int _{a}^{x}f(t)\,dt,$

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.

McShane integral

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

$t_{i}\in [u_{i-1},u_{i}],$

then we get a definition of the McShane integral, which is equivalent to the Lebesgue integral. Note that the condition

$\forall i\ \ [u_{i-1},u_{i}]\subset [t_{i}-\delta (t_{i}),t_{i}+\delta (t_{i})]$

does still apply, and we technically also require ${\textstyle t_{i}\in [a,b]}$ for ${\textstyle f(t_{i})}$ to be defined.

Licensing

Content obtained and/or adapted from:

• Continuity equation, Wikipedia under a CC BY-Sa license
• Henstock–Kurzweil integral, Wikipedia under a CC BY-SA license