What happens at the edges
Mathematical counterpart: The boundary term.
This is the hinge of the whole argument, and it turns on being careful about a question that sounds trivial: changed compared with what?
Two thermometers
A river is warming through the afternoon, and it is warmer downstream than up.
Drop a floating thermometer in the water. It drifts with the current and records a rising temperature. Some of that rise is the afternoon warming everything. Some of it is the thermometer arriving somewhere that was already warmer.
Now fix a second thermometer to the bridge. It also records a rise, but only the afternoon's. It never goes anywhere, so nothing it records is due to arriving somewhere new.
Two thermometers, two different numbers, both correct. Which one you want depends on the question, and confusing them is a mistake that shows up as a wrong sign rather than as obvious nonsense.
Noether keeps three, not two
Most treatments of her theorems use two symbols where she used three. The middle one is the one that disappears.
The drifting thermometer. The total change: the new situation at the new place, against the old situation at the old place. Everything that changed, from every cause.
The bridge thermometer. The change at a fixed place: the new situation compared with the old at the same spot. This is the total change with the effect of moving subtracted out.
The nudge. The ordinary change you make when you test a shape by pushing it, as on the nudging page. Here nothing is being transformed at all — you are just poking the system to see if it pushes back.
The first two differ by exactly the amount the drift accounts for. Noether writes that difference down as its own equation, and it is one of the most quoted lines in the paper. Compress the three into two and you lose the equation that subtracts the drift, which is precisely where sign errors come from.
Moving the frame
There is a second piece of bookkeeping on this page, and it produces the thing everything else is built on.
Photograph a scene. Now slide the frame slightly and photograph again. The two pictures differ, and the difference is entirely at the border: a strip has come in on one side, a strip has gone out on the other. The middle is untouched.
Transforming the variables moves the frame in just this way, and the cost of moving it is exactly a border term.
That border term is where conserved quantities come from. It is not an error term to be assumed away. It is the answer.
Losing the integral sign
The last step on this page is short and it changes everything after it.
You end up with a statement that some total, added up over a region, is zero. That would be a mild statement if it were about one region. But it holds for every region you choose — big, small, anywhere.
A quantity whose total is zero over every possible region has to be zero everywhere. There is nowhere for it to hide.
So the statement stops being about totals and becomes a statement about each point separately. From here to the end of both theorems there is no adding up at all, only algebra.
That is why the results hold point by point, and why they hold whether or not anything is obeying the laws of physics.
Reading the wrong thermometer
Here is the failure this page exists to prevent.
Suppose you are working out the conserved quantity for a shift in time, and you reach for the drifting thermometer when the argument wanted the bridge one.
The two differ by the drift term. So your answer differs from the right answer by that same term. And the drift term is not small, and it is not obviously out of place — it has the right units, it has a sensible size, it varies the way you would expect a piece of the energy to vary.
What you get is not an error message. It is a wrong conserved quantity that looks entirely reasonable, and which happens to be conserved in the special case where nothing is moving.
The most common symptom is a sign. The two comparisons differ by the drift, and the drift enters with a minus, so a treatment that muddles them tends to produce an answer that is right except for being backwards. A minus sign is easy to blame on a convention and hard to trace to its cause.
That is why Noether kept three symbols where two would have looked tidier, and why this volume does the same.
No mathematics is used on this page. The mathematical treatment is on the counterpart page named above.