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Noether theorems volume, initial import (F-06)

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{{DISPLAYTITLE:The boundary term}}

''Plain-language counterpart: [[Noether's theorems/Explainers/What happens at the edges|What happens at the edges]].''

[[File:NT-F-14.png|thumb|right|360px|alt=A solid rectangle and a shifted dashed copy of it, with the strip that enters shaded on one side and the strip that leaves shaded on the other. The overlapping interior is marked unchanged. A caption says the whole cost of moving the region is a border term, and that this term becomes the conserved current.|The origin of the boundary term]]

Condition (N7) compares integrals over two different regions. To do anything
with it, both integrals have to be over the same region. Making that conversion
produces a boundary term, and introduces the third of Noether's three
variations. This page is the hinge of the whole proof.

== Moving the region ==

For infinitesimal <math>\Delta x</math>,

<math>\int\dots\int f\left(y,\ v(y),\ \frac{\partial v}{\partial y},\dots\right)dy=\int\dots\int f\left(x,\ v(x),\ \frac{\partial v}{\partial x},\dots\right)dx+\int\dots\int\mathrm{Div}(f\cdot\Delta x)\,dx\qquad\qquad\mathrm{(N8)}</math>

The added term is the whole cost of moving the region of integration. It is a
divergence, so it is a boundary term, and it is the term that carries the
conserved current in the end.

== The three variations ==

Noether now introduces, in place of the infinitesimal transformation
<math>\Delta u</math>, the variation

<math>\bar{\delta}u_{i}=v_{i}(x)-u_{i}(x)=\Delta u_{i}-\sum\frac{\partial u_{i}}{\partial x_{\lambda}}\Delta x_{\lambda}\qquad\qquad\mathrm{(N9)}</math>

'''This equation is why the volume insists on three symbols and not two.'''

<math>\Delta u_{i}</math> is the total change in <math>u_{i}</math> under the
transformation. It compares the new function at the new point with the old
function at the old point.

<math>\bar{\delta}u_{i}</math> is the change at fixed argument. It compares the
new function with the old at the '''same''' point <math>x</math>, and it is what
remains after the motion of the independent variables has been subtracted out.

<math>\delta u_{i}</math> is the ordinary variation of the calculus of
variations, the one appearing in (N2) to (N6) on
[[Noether's theorems/First theorem/The Lagrange expressions and the first variation|the
Lagrange-expressions page]], where no transformation of the <math>x</math> is in
play at all.

The two differ by <math>\sum(\partial u_{i}/\partial x_{\lambda})\Delta x_{\lambda}</math>,
which is exactly (N9). Compressing them is where sign errors in derivations of
Noether's theorem come from, and a treatment that writes two symbols where she
wrote three has already lost the distinction the proof turns on. The
[[Noether's theorems/Notation|notation page]] tabulates all three.

== The condition becomes local ==

With (N9), the relations (N7) and (N8) combine to give

<math>0=\int\dots\int\left\{\bar{\delta}f+\mathrm{Div}(f\cdot\Delta x)\right\}dx\qquad\qquad\mathrm{(N10)}</math>

whose right-hand side is the standard formula for simultaneous variation of the
dependent and independent variables.

Now the key step, and it is one line of reasoning. Relation (N10) holds for
integration over '''every arbitrary region'''. An integral that vanishes over
every region has a vanishing integrand. Therefore

<math>\bar{\delta}f+\mathrm{Div}(f\cdot\Delta x)=0\qquad\qquad\mathrm{(N11)}</math>

'''The integral sign is gone, and it does not come back.''' Everything from here
to the end of both theorems is algebra on identities between expressions. This
is what makes the results hold pointwise and unconditionally, and it is the
reason the theorems can say anything before the field equations are imposed.

Relation (N11) is Lie's differential equation for the invariance of
<math>I</math>, in the form Noether needs.

''Provenance: (N8) to (N11) are Noether's equations (8) to (11), reproduced in
her notation from the public-domain German original. Her prose is paraphrased.
See [[Noether's theorems/Sources|Sources]].''

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