The boundary term

Plain-language counterpart: What happens at the edges.

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 $Δ x$ ,

$\int \dots \int f (y, v (y), \frac{\partial v}{\partial y}, \dots) d y = \int \dots \int f (x, v (x), \frac{\partial v}{\partial x}, \dots) d x + \int \dots \int D i v (f \cdot Δ x) d x (N 8)$

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 $Δ u$ , the variation

$\bar{δ} u_{i} = v_{i} (x) - u_{i} (x) = Δ u_{i} - \sum \frac{\partial u_{i}}{\partial x_{λ}} Δ x_{λ} (N 9)$

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

$Δ u_{i}$ is the total change in $u_{i}$ under the transformation. It compares the new function at the new point with the old function at the old point.

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

$δ u_{i}$ is the ordinary variation of the calculus of variations, the one appearing in (N2) to (N6) on the Lagrange-expressions page, where no transformation of the $x$ is in play at all.

The two differ by $\sum (\partial u_{i} / \partial x_{λ}) Δ x_{λ}$ , 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 notation page tabulates all three.

The condition becomes local

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

$0 = \int \dots \int {\bar{δ} f + D i v (f \cdot Δ x)} d x (N 10)$

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

$\bar{δ} f + D i v (f \cdot Δ x) = 0 (N 11)$

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 $I$ , 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 Sources.