Recent content by bobbarker

Homework Statement Show that if F is continuous on Rn and F(X+Y) = F(X) + F(Y) for all X and Y in Rn, then A is linear. Hint: Rational numbers are dense in the reals. Homework Equations A transformation A is linear iff A(X) = (a matrix) [ a11x1+...+a1nxn ] [... ...

Homework Statement Suppose that f=f(x1,x2,...,xn) is a homogeneous function of degree r with mixed partial derivative of all orders. Show that Can this be generalized? Homework Equations We say that a function f is homogeneous of degree r if there exists r such that f(tx1,tx2,...,txn) = tr...

So since hu is independent of v and hv is independent of u, and disregarding the possibility that the functions U,V are constants, then we must have h(u,v) = U(u) + V(v)? Since hu is not necessarily 0 but huv=0, and similarly hv is not necessarily 0 but hvu=0?

Homework Statement "Let huv = 0 \forall u, v. Show that h is of the form h(u,v) = U(u) + V(v)."Homework Equations n/aThe Attempt at a Solution The problem doesn't really seem that complex to me, in fact, from PDEs a few years ago I remember this quite readily. However, the proof/demonstration...

Homework Statement Find sup{\epsilon| N\epsilon(X0 \subset S} for X0 = (1,2,-1,3); S = open 4-ball of radius 7 about (0,3,-2,2).Homework Equations If X1 is in Sr(X0) and |X - X1| < \epsilon = r - |X - X0| then X is in Sr(X0) The Attempt at a Solution This is my first foray into...

Homework Statement Prove: If {Fn} converges to F on [a,b] and Fn is nondecreasing for each n\in N, then F is nondecreasing. Homework Equations n/aThe Attempt at a Solution First, it doesn't say if Fn converges pointwise or uniformly, so I'm not entirely sure how to deal with that. Just prove...

Let me run this by you... I played with it for a few minutes. (I put the problem's variables in place of yours, and defined a new quantity e\in \mathbb{Q} = \sqrt{d}+ \sqrt{b} ) So since \sqrt{b}+\sqrt{d} is rational, (\sqrt{b}+\sqrt{d})^{2} is rational, or d+b+2 \sqrt{db} is rational, or...

Hey all, I'm new here so I'm a little noobish at the formatting capabilities of PF. Trying my best though! :P Homework Statement Let a, b, c, d \in Q, where \sqrt{b} and \sqrt{d} exist and are irrational. If a + \sqrt{b} = c + \sqrt{d}, prove that a = c and b = d. Homework...