Recent content by hellofolks

Well, yungman, if you had series solutions and transforms, then I'd say your course on ODE was not basic, but intermediate. In fact, usually I didn't have problems with analytical methods, but was rather clumsy when it came to drawing phase portraits.

Basic ODE and PDE themselves are not that hard because really hard equations just cannot be solved explictly and it is just a matter of either resorting to numerical methods or qualitative theory. Again, I think people get stuck because they try to memorize rather than understand. More advanced...

I've never really thought it is such a monster. In fact, usually multivariate calculus is much about repeating univariate concepts n times. The only problem here is that it demands other prerequisites. In fact, besides having properly learned univariate calculus, one should also have a deeper...

Could you please give an example of an exercise or problem that you have trouble solving so that I can try to guess where is the source of your doubts (or confusion)? I mean -- if it is in the concepts, ability to work with the formulae, lack of understanding of prerequisite topics, physical...

A good course to do after PDE is one that involves applied functional analysis, in which you can apply the theory of Banach and Hilbert spaces to problems involving ODE, PDE and distributions.

Any progress solving the problem?

Hello again. I've seen Hong Kong in a TV show. I liked that tall escalator which also appeared in a Scooby-Doo cartoon. Have you climbed it? I'm from Brazil and originally from Brazil, too. So here goes the example: Evaluate, for r<\sqrt{3}, the monster integral \int_0^{2...

I'm a little busy now, so I'll give the example later. Just don't close the thread. Now, a curiosity, what country are you from?

Thanks, but I already have a copy of that book. In fact, I even like it better than Strauss. Anyway I think now you understand there was nothing wrong in the proof by Strauss, except that he used a partial derivative when he should've used an ordinary one while proving that the integral does not...

You can think of the steps that seem incorrect as mnemonics to obtain a correct result. The justification is the one given above.

The completion detail is usually not described in textbooks. Do you have any idea why?

Well, first I'm glad you enjoyed my little example. I'm going to show (I don't know if I'm allowed) how to prove that u=r^2 \sin^2 \theta \cos 2 \phi is harmonic. There are two ways to do it. First method (easier): Remember that \cos 2 \phi=\cos^2 \phi - \sin^2 \phi so that u=(r\sin \theta...

Take u(r,\theta,\phi)=r^2\sin^2 \theta\cos 2\phi. The function u is not constant, depends on r and is unbounded. Now calculate the surface integral of u on a sphere of radius r. You'll see that it equals zero, independently of r. The same is true of the average of u on that sphere.(Remark: in...

But how can we describe that completion?

That's the beauty of the mean value property! It doesn't matter if you're calculating the average of a harmonic function in huge or in a tiny ball -- the average is always the same and equals the harmonic function evaluated at the center of the ball. This has a lot of consequences, besides...