Is the Uniqueness Theorem in EM Always Proven by Contradiction?

[cosmicash]

isnt the field in any given volume always uniquely determined
to prove by contradiction assume E1 and E2
div(E₁)=charge density/epsilon ₀ =div(E ₂)
by fundamental theorem of divergences
integral of E₁.da = integral of E₂.da over entire surface
thus E₁=E₂

then why does griffith in his book(intro to em) state the requirement of total charge on each conductor

 

[xepma]

Could you please state the exact page on which he makes that claim? Just to put things into context here..

 

[Meir Achuz]

The uniqueness theorem requires boundary conditions that make the surface integrals vanish. In the presence of conductors, either the charge or the potential of each conductor must be given.

 

[cosmicash]

it was on pg 118 of the 3rd edition

 

[cosmicash]

The uniqueness theorem requires boundary conditions that make the surface integrals vanish. In the presence of conductors, either the charge or the potential of each conductor must be given.

but what i tried to prove was without these conditions
so what is wrong with my proof

 

[Meir Achuz]

What is wrong with your proof it is that it doesn't make sense.
Proving two integrals equal does not mean the integrands are equal.
If G does not give a good enough proof, look at a more advanced text.

 

[Cantab Morgan]

Two integrands are sure to be equal only if their integrals are equal over all possible surfaces. You're taking the integrals over a particular surface. So you've established only a necessary but insufficient condition for $$E_1 = E_2$$.

However, I would eschew proofs by contradiction anyway except for really trivial theorems. Direct proofs usually convey (or require) more insight into what's going on. I don't have my Griffith's in front of me. Is his a direct proof?

 

[cosmicash]

Two integrands are sure to be equal only if their integrals are equal over all possible surfaces. You're taking the integrals over a particular surface. So you've established only a necessary but insufficient condition for $$E_1 = E_2$$.

However, I would eschew proofs by contradiction anyway except for really trivial theorems. Direct proofs usually convey (or require) more insight into what's going on. I don't have my Griffith's in front of me. Is his a direct proof?

it isn't that tough but i was trying an alternative
i get it
so i have to integrate over each possible surface thus needing the charge density at each surface

 

[Meir Achuz]

However, I would eschew proofs by contradiction anyway except for really trivial theorems. Direct proofs usually convey (or require) more insight into what's going on. I don't have my Griffith's in front of me. Is his a direct proof?

Proofs by contradiction are a standard part of math and physics.
I know of no proof of any uniqueness theorem that is not by contradiction.
I don't know how you could start.

 

[Cantab Morgan]

Hi, Clem, nice to meet you. Great post!

Proofs by contradiction are a standard part of math and physics.
I know of no proof of any uniqueness theorem that is not by contradiction.
I don't know how you could start.

Of course you are correct. Proofs by contradiction are used and relied upon every day. When a short proof by contradiction is possible, it often represents a very efficient and natural way to shore up a theorem. Humans often think in proofs by contradiction. ("Hmm. It must be Saturday today because on Sunday the bank I see open would be closed.")

However, mathematicians prefer direct proofs for a couple of reasons. First, when one arrives at a contradiction, one has little insight into which of the premises "broke" the consistency. It's entirely possible that the initial axioms were themselves inconsistent before adding in the converse of the theorem to be proved. A proof by contradiction doesn't easily reveal that. For this reason, proofs by contradiction are less trustworthy as the number of axioms relied upon grows, and they're less suitable to prove nontrivial theorems. ("Oh, it's really Monday? I had assumed it was the weekend.")

Second, which is what I was really driving at, is that direct proofs often require greater exploration of the mathematical structures under study. While this makes it harder, there can be real payoff in comprehension. If I recall correctly, the first proof I ever saw of the Intermediate Value Theorem was a proof by contradiction. Years later, I saw a direct proof that relied upon the Heine-Borel theorem, which rests on the deep properties of the real numbers. Now if my goal was to use the Intermediate Value Theorem, then the quicker proof suffices. But if my goal was to understand why it's true, and to find other sets besides the real numbers where it still might be true, then the direct proof was harder but worth it.

None of which, of course, takes anything away from your insight that proofs by contradiction are a standard part of math and physics. But, if I could put on my computer programming hat for a moment, they are the "goto" statements of mathematics: Economical at the assembly code level, but considered harmful in higher order languages.

I'm trying to think of a direct uniqueness proof, and what comes to mind is that the inverse of an element of a group is unique. If x is an element of a group, then let y and z be its inverse, meaning that xy = xz = 1. You could do a proof by contradiction, assume that $$y \neq z$$, and then show that leads to a contradiction. Or you could just use the group axioms to prove that y = z directly.

xy = xz
yxy = yxz
1y = 1z
y = z

On the other hand, I can't think of a direct proof of the irrationality of $$\sqrt 2$$ off the top of my head, but the reductio ad absurdam proof is elegant and accessible.