Vector calculus problem from Walter Strauss Intro to PDE?
In the appendix review of Intro to PDE Walter Strauss says:
Let f(x,y,z) be continuous of function in the closure of D_0 where D_0 is a bounded domain such that the triple integral \int\int\int_D f(x,y,z) dx dy dz = 0 for all domains D that are a subset of D_0. THen f(x,y,z) is identically zero.
Proof (still in the book): Let D be a ball and let its radius shrink to zero.
Could someone elaborate on this proof for me? I don't see what that does!
Update:Does it have something to do with differentiating the integral? Anyone?
Comments
Does it have something to do with differentiating the integral?
No.
If what you've posted is all that's in the proof, I understand your confusion----not very enlightening!
It's all about continuity. Suppose that there exists a point P(x, y, z) in D_0 at which f(P) ≠ 0. We can assume without loss of generality that f(P) > 0.
Because f is continuous at P, we can find an open ball centered at P such that f(x, y, z) > 0 everywhere in this ball. This is the key!
(1) D_0 is a domain, hence open, hence we can find a ball about P.
(2) f is continuous, so f(P) > 0 means that f is positive on some ball.
Now take D to be this ball. You'd get ∫∫∫_D f dV > 0 contrary to the hypothesis. This disproves the existence of such a point P---i.e. it proves that f(x, y, z) = 0 identically.
BTW: This is a wonderfully powerful observation. It's used to obtain PDE models of various physical phenomena built from conservations laws. One constructs a model by considering what happens in an arbitrary control volume. This gives an integral statement. Then the above is used to deduce a point-wise (a.k.a. PDE) statement.