My son was given a proof of Pascal's Law and told to learn it for his course in Fluid Mechanics.
a. It was done with a prism. I cannot type in a link, but google: pascal law prism and pick the first link and scroll down half a page to pressure and you will see this proof.
b. For the life of me, I cannot understand how this is possibly a proof since the direction of the pressure is NOT arbitrary.
c. So I attempted my own proof... However, I am a mathematician - not an engineer. I fixed the arbitrary direction problem. But I still do not understand something critical.
I have posted my proof on scribd (remove the spaces):
scribd .com /doc /94146673
However, in my opinion my "proof" still has a mathematical flaw.
The Attempt at a SolutionLet M be a point in a static fluid and let p(n,M) be the pressure at a point M in the direction of an arbitrary vector n to M . Presumably - to prove Pascal's Law - I must show that p(n,M) does not depend on n.
Now, using the techniques of the prism proof, I "show" geometrically that p(n,M)=p(x,M)=p(y,M)=p(z,M) where p(x,M) is the pressure at point M in the direction of "positive x-axis", p(y,M) is the pressure at point M in the direction of positive y-axis, ...
Question: Mathematically, it seems to me that I need only ONE of these 3 equalities, e.g. p(n,M)=p(x,M). Using it, I could conclude that given another vector t to M p(t,M)=p(x,M) and conclude that p(t,M)=p(x,M)=p(n,M) and thus p(M) does not depend on my choice of vector.
I am certain that I need all 3 equalities, but WHY? Maybe something with the limits or even with the definition of p(n,M)?
Thanks for any help!!