Editing Total variation (section)

===Total variation of differentiable functions===
The total variation of a <math>C^1(\overline{\Omega})</math>  function <math>f</math> can be expressed as an [[integral]] involving the given function instead of as the [[supremum]] of the [[functional (mathematics)|functional]]s of definitions {{EquationNote|1|1.1}} and {{EquationNote|2|1.2}}.

====The form of the total variation of a differentiable function of one variable====
{{EquationRef|5|Theorem 1.}} The '''total variation''' of a [[differentiable function]] <math>f</math>, defined on an [[interval (mathematics)|interval]] <math> [a , b] \subset \mathbb{R}</math>, has the following expression if <math>f'</math> is Riemann integrable

:<math> V_a^b(f) = \int _a^b |f'(x)|\mathrm{d}x</math>

If <math> f</math> is differentiable and [[Monotonic function|monotonic]], then the above simplifies to
:<math> V_a^b(f) = |f(a) - f(b)|</math>

For any differentiable function <math>f</math>, we can decompose the domain interval <math>[a,b]</math>, into subintervals <math>[a,a_1], [a_1,a_2], \dots, [a_N,b]</math> (with <math>a<a_1<a_2<\cdots<a_N<b </math>) in which <math>f</math> is locally monotonic, then the total variation of <math> f</math> over <math>[a,b]</math> can be written as the sum of local variations on those subintervals:
:<math>
\begin{align}
V_a^b(f) &= V_a^{a_1}(f) + V_{a_1}^{a_2}(f) + \, \cdots \, +V_{a_N}^b(f)\\[0.3em]
&=|f(a)-f(a_1)|+|f(a_1)-f(a_2)|+ \,\cdots \, + |f(a_N)-f(b)|
\end{align}</math>

====The form of the total variation of a differentiable function of several variables====
{{EquationRef|6|Theorem 2.}} Given a <math>C^1(\overline{\Omega})</math> function <math>f</math> defined on a [[bounded set|bounded]] [[open set]] <math>\Omega \subseteq \mathbb{R}^n</math>, with <math>\partial \Omega </math> of class <math>C^1</math>,  the '''total variation of <math>f</math>''' has the following expression

:<math>V(f,\Omega) = \int_\Omega \left|\nabla f(x) \right| \mathrm{d}x</math> .

=====Proof=====
The first step in the proof is to first prove an equality which follows from the [[Gauss–Ostrogradsky theorem]].

=====Lemma=====
Under the conditions of the theorem, the following equality holds:
: <math> \int_\Omega f\operatorname{div}\varphi = -\int_\Omega\nabla f\cdot\varphi </math>

======Proof of the lemma======
From the [[Gauss–Ostrogradsky theorem]]:
: <math> \int_\Omega \operatorname{div}\mathbf R = \int_{\partial\Omega}\mathbf R\cdot \mathbf n </math>
by substituting <math>\mathbf R:= f\mathbf\varphi</math>, we have:

:<math> \int_\Omega\operatorname{div}\left(f\mathbf\varphi\right) =
\int_{\partial\Omega}\left(f\mathbf\varphi\right)\cdot\mathbf n </math>
where <math>\mathbf\varphi </math> is zero on the border of <math>\Omega</math> by definition:
:<math> \int_\Omega\operatorname{div}\left(f\mathbf\varphi\right)=0</math>
:<math> \int_\Omega \partial_{x_i} \left(f\mathbf\varphi_i\right)=0</math>
:<math> \int_\Omega \mathbf\varphi_i\partial_{x_i} f + f\partial_{x_i}\mathbf\varphi_i=0</math>
:<math> \int_\Omega f\partial_{x_i}\mathbf\varphi_i = - \int_\Omega \mathbf\varphi_i\partial_{x_i} f </math>
:<math> \int_\Omega f\operatorname{div} \mathbf\varphi = - \int_\Omega \mathbf\varphi\cdot\nabla f </math>

=====Proof of the equality=====
Under the conditions of the theorem, from the lemma we have:
:<math> \int_\Omega f\operatorname{div} \mathbf\varphi
= - \int_\Omega \mathbf\varphi\cdot\nabla f
\leq \left| \int_\Omega \mathbf\varphi\cdot\nabla f \right|
\leq \int_\Omega \left|\mathbf\varphi\right|\cdot\left|\nabla f\right|
\leq \int_\Omega \left|\nabla f\right| </math>
in the last part <math>\mathbf\varphi</math> could be omitted, because by definition its essential supremum is at most one.

On the other hand, we consider <math>\theta_N:=-\mathbb I_{\left[-N,N\right]}\mathbb I_{\{\nabla f\ne 0\}}\frac{\nabla f}{\left|\nabla f\right|}</math> and <math>\theta^*_N</math> which is the up to <math>\varepsilon</math> approximation of <math>\theta_N</math> in <math> C^1_c</math> with the same integral. We can do this since <math> C^1_c</math> is dense in <math> L^1 </math>. Now again substituting into the lemma:

:<math>\begin{align}
&\lim_{N\to\infty}\int_\Omega f\operatorname{div}\theta^*_N \\[4pt]
&= \lim_{N\to\infty}\int_{\{\nabla f\ne 0\}}\mathbb I_{\left[-N,N\right]}\nabla f\cdot\frac{\nabla f}{\left|\nabla f\right|} \\[4pt]
&= \lim_{N\to\infty}\int_{\left[-N,N\right]\cap{\{\nabla f\ne 0\}}} \nabla f\cdot\frac{\nabla f}{\left|\nabla f\right|} \\[4pt]
&= \int_\Omega\left|\nabla f\right|
\end{align}</math>
This means we have a convergent sequence of <math display="inline">\int_\Omega f \operatorname{div} \mathbf\varphi</math> that tends to <math display="inline">\int_\Omega\left|\nabla f\right|</math> as well as we know that <math display="inline">\int_\Omega f\operatorname{div}\mathbf\varphi \leq \int_\Omega\left|\nabla f\right| </math>. [[Q.E.D.]]

It can be seen from the proof that the supremum is attained when
: <math>\varphi\to \frac{-\nabla f}{\left|\nabla f\right|}.</math>

The [[Function (mathematics)|function]] <math>f</math> is said to be of [[bounded variation]] precisely if its total variation is finite.