# Search results

1. ### Geometrical interpretation of this property

Geometrically, think of f as a vector field. Imagine it as a collection of arrows describing the velocity of a fluid; that is, for each x, f(x) is the velocity of the fluid at the point x. Now, imagine a small pebble moving in this fluid. At each point x, the trajectory of the pebble must be...
2. ### Distributional derivative of one-parameter family of distributions

Suppose, for a suitable class of real-valued test functions T(\mathbb{R}^n), that \{G_x\} is a one-parameter family of distributions. That is, \forall x \in \mathbb{R}^n, G_x: T(\mathbb{R}^n) \to \mathbb{R}. Now, suppose L is a linear differential operator. That is, \forall g \in...
3. ### How to numerically solve a PDE with delta function boundary condition?

Oh okay, that makes sense. So that seems to solve that problem. Now I just need to find some software to do this in since apparently with the Matlab toolboxes I have I can only solve 1-dimensional PDEs
4. ### How to numerically solve a PDE with delta function boundary condition?

I have a PDE of the following form: f_t(t,x,y) = k f + g(x,y) f_x(t,x,y) + h(x,y) f_y(t,x,y) + c f_{yy}(t,x,y) \\ \lim_{t\to s^+} f(t,x,y) = \delta (x-y) Here k and c are real numbers and g, h are (infinitely) smooth real-valued functions. I have been trying to learn how to do this...
5. ### Is every smooth simple closed curve a smooth embedding of the circle?

That was very helpful. Thank you!
6. ### ODE textbook recommentation (Arnold or other?)

Thanks for the suggestion, but this looks like a book almost entirely devoted to linear differential equations. I don't think this is what I'm looking for, unfortunately.
7. ### ODE textbook recommentation (Arnold or other?)

Hi everyone. I have a copy of Ordinary Differential Equations by Vladimir Arnold. I'm hoping to learn more about differential equations, building up to differential equations on manifolds. I've heard that this is a great book, but I've also heard Arnold sometimes leaves out important details...
8. ### Is every smooth simple closed curve a smooth embedding of the circle?

Thanks everyone! I'm not very comfortable with Lie groups yet, but I think I understand what you're saying. Basically, I just need to require that \dot{\gamma}(1) = \dot{\gamma}(0), right? It sounds like requiring the curve to be a periodic function ensures this, and eliminates the issues of...
9. ### Is every smooth simple closed curve a smooth embedding of the circle?

Suppose I have a smooth curve \gamma:[0,1] \to M, where M is a smooth m-dimensional manifold such that \gamma(0) = \gamma(1), and \hat{\gamma}:=\gamma|_{[0,1)} is an injection. Suppose further that \gamma is an immersion; i.e., the pushforward \gamma_* is injective at every t\in [0,1]. Claim...
10. ### What is the solution to this ODE (and SDE)?

Thanks a lot for the answer. Can you explain why \theta(t) = \theta(0) + \int_0^t b_\theta(s)\,ds, \\ \phi(t) = \phi(0) + \int_0^t \frac{b_\phi(s)}{\sin \theta(s)}\,ds, becomes invalid if ever \theta(t) < 0 or \theta(t) > \pi?
11. ### What is the solution to this ODE (and SDE)?

I'm trying to analyze the following Ito stochastic differential equation: $$dX_t = \|X_t\|dW_t$$ where X_t, dX_t, W_t, dW_t \in \mathbb{R}^n. Here, dW_t is the standard Wiener process and \|\bullet\| is the L^2 norm. I'm not sure if this has an analytical solution, but I am hoping to at...
12. ### Prove that Locally Lipschitz on a Compact Set implies Lipschitz

Interesting. I'd never heard of sequential compactness before, but I just found some notes online and read about it. Did you mean to say "assume f is not Lipschitz"? We are given in the problem that f is locally Lipschitz, so I don't know why you would assume otherwise. If I'm right in...
13. ### Prove that Locally Lipschitz on a Compact Set implies Lipschitz

I was thinking along these lines earlier. Then d_N(f(x),f(y)) \leq d_N(f(x),f(z)) + d_N(f(z),f(y)) \leq L_1*d_M(x,z) + L_2*d_M(z,y)\\ \leq \max(L_1,L_2)*(d_M(x,z)+d_M(z,y)) But I'm stuck here. How would I use x_i and x_j? Also, how can I show that there is always a "chain" of overlapping balls...
14. ### Prove that Locally Lipschitz on a Compact Set implies Lipschitz

Homework Statement Let M and N be two metric spaces. Let f:M \to N. Prove that a function that is locally Lipschitz on a compact subset W of a metric space M is Lipschitz on W. A similar question was asked here...
15. ### How can I prove this theorem on differential inclusions?

Consider the following differential equations with initial conditions at time $t_0$ specified: \dot{x}_1 = f_1(x_1,t) ; \,\,\,x_1: [t_0,T]\to\mathbb{R}^n, f_1:\mathbb{R}^n\times[t_0,T]\to\mathbb{R}^n \\ \vdots \\ \dot{x}_k = f_k(x_k,t); \,\,\,x_1: [t_0,T]\to\mathbb{R}^n...
16. ### Generalization of Comparison Theorem

Sorry -- I was a little hasty in writing this and meant to say the functions f and g map \Re^2 to \Re, or, in the second possible theorem, map \Re^{n+1} to \Re^n.
17. ### Generalization of Comparison Theorem

I was wondering if there is a generalization of the following (roughly stated) theorem to n-dimensional systems: Given some restrictions on the functions f and g \in \Re, let y_s(t) and x_s(t) \in \Re be solutions to the initial value problems: \dot{x}(t) = f(x,t), x(t_0) = x_0\\...
18. ### Moving limits in and out of functions

Ahh... Thank you for the detailed proof! I was able to use the ideas from your proof to prove the result in my original post (basically the exact same proof, but mine involved a function ##g(x)##, whereas yours involved the sequence ##\{x_n\}##) Anyway, thanks again! And thanks to everyone else :)
19. ### Moving limits in and out of functions

Interesting. But how would you show that this definition is equivalent to, for example, the epsilon-delta definition?
20. ### Moving limits in and out of functions

What exactly do you mean by the sequence x_n here?
21. ### Moving limits in and out of functions

Thanks for the specific example. Can you prove (or point me to the proof) of the general case?
22. ### Moving limits in and out of functions

When is the following equivalence valid? $$\lim_{x \to a} f(g(x)) = f(\lim_{x \to a} g(x))$$ I was told that continuity of f is key here, but I'm not positive. This question comes up, for instance in one proof showing the equivalence of the limit definition of the number e to the...
23. ### Generalization of Mean Value Theorem for Integrals Needed

That makes sense. Thanks a lot for the help.
24. ### Generalization of Mean Value Theorem for Integrals Needed

I'm actually fairly certain now that my conjecture is false...
25. ### Generalization of Mean Value Theorem for Integrals Needed

Hi all, I'm having trouble finding a certain generalization of the mean value theorem for integrals. I think my conjecture is true, but I haven't been able to prove it - so maybe it isn't. Is the following true? If F: U \subset \mathbb{R}^{n+1} \rightarrow W \subset \mathbb{R}^{n}...
26. ### Inequality problem

That makes sense to me. Thanks again for the help.
27. ### Inequality problem

It tells me that ##1 - K + x > -(y+1)## which I actually had written on my paper before, but thanks to you I think I've realized the connection... Since I already had from the first assumption ##1-K+x<1+y##, I now have ##-(y+1)<(1-K+x)<(y+1)##, which seems to imply ##|1-K+x| < (y+1)## as...
28. ### Inequality problem

Homework Statement Given: |x-y| < K x+y > K - 2 0 < K < 1 Prove: \frac{|1-K+x|}{|1+y|} < 1 The Attempt at a Solution I have tried using the fact that |x-y| < K \Rightarrow -K < x-y < K \Rightarrow y-K < x < y+K to write \frac{1-K+x}{1+y} < \frac{1+y}{1+y} = 1 But I can't figure...