Calculus Problem

[tristanreid]

Thanks, guys!  This is really great. I'm going to try to master using the Leibniz rule, it seems really useful.



I got a response from my prof, he says:



At this point in this course, you should just limit your "why does this work this way" to "look, cool, here is another way to solve these second order differential equations.  I'm not sure why it works, but hey look - it does work."



As for your specific question, P[t] is a function of t.  Look that the integration is with

respect to x.

So when you differentiate, using an advanced calculus theorem called Fubuini's theorem (which Mma knows, of course), you can play games with commuting the differential operator into the integral, and integrate the insides.  The inside of that step is due to the chain rule.



But, once again, I would suggest trying to smile and say, "look, there is some wild stuff going on here" and consider this section an exposure to a topic, rather than a complete analysis.



I can't just memorize techniques, I'm just not built that way.  For me, this is dark side of online courses.  Pardon my going all warm and fuzzy, but I'm really glad you guys helped me understand this. 



-t.

[Graeme]

IMO, a poor response from your professor. One can figure this stuff out, to an extent; it isn't Alchemy. 



What he says basically suggests that you can take the differentiation sign inside free of charge. In fact, the charge is managing the limits in the way that has been indicated. And what this has to do with Fubini's theorem I cannot say.



I also cannot memorise anything. I can't remember this Leibnitz rule. Every time I use it I fumble with a cloudy memory, and a sense of what the answer should look like. Every time one uses something, and rederives the correct formula - either formally or informally - you are in a better position to use it naturally in the future.



I'd be happy to help you with this stuff in the future. Of course the usual caveat applies, although you seem to be fully aware of it (unlike many students I have had): one doesn't build muscle by watching people doing gym.

[AVt]

Graeme: "one doesn't build muscle by watching people doing gym" ... hm, depends on the girls and what muscle you have in mind ...



edited: i also cant memorize ... but for that questions one can assume an antiderivate to find out the formula ... that covers most cases

[Cheng]

I am with Graeme here. IMO it is much better to have a good intuition than memorizing every single formula without knowing what it actually does. By intuition I mean that you look at a problem and have a rough idea what tools you might need to solve it. The rest can be looked up.



Feel free to post similar stuff in the future, I will try to have a look at it if time permits.

[athletico]

I think I can recite the quadratic formula on demand and that's about it.  Thank God for this: CRC Standard Mathematical Tables and Formulae



By the way the Leibniz differentiation rule was one of Feynman's famous tricks.  From "Surely You're Joking", where Feynman was describing his experience at Los Alamos working on the bomb:



I was sent to Chicago with the instructions to go to each group, tell them I was going to work with them, and have them tell me about a problem in enough detail that I could actually sit down and start to work on it. As soon as I got that far, I was to go to another guy and ask for another problem. That way I would understand the details of everything.



It was a very good idea, but my conscience bothered me a little bit because they would all work so hard to explain things to me, and I'd go away without helping them. But I was very lucky. When one of the guys was explaining a problem, I said, "Why don't you do it by differentiating under the integral sign?" In half an hour he had it solved, and they'd been working on it for three months.

[saffron]

I don't think nnja was wrong. Say we have a causal LTI ( linear & time invariant) system with an impulse response g(x) defined on x >= 0. Elec. Engs like to use the heaviside step function to write the system as p(x) = u(x)g(x). Writing the output of the system on an input signal q(x),  [img]/User%20Files/1554/Latex-Equation-7196.gif[/img] we have:



[img]/User%20Files/1554/Latex-Equation-7197.gif[/img]



so I wouldn't say this isn't a convolution integral. The boundaries matter when we want to use the Fourier transform. Also I don't see why periodical continuation is needed for making sense of convolution integrals (I think you may be confusing this with Fourier series expansion of periodical functions... If the functions are not periodical we can just use a Fourier transform ).



Finally to kevink, I think we can just use a mean value theorem on that second limit.

[kevink]

saffron, yes you're right, the mean value theorem works just fine. Lebesgues differentiation theorem was overkill.

[Dimatrix]

So, guys, I'm struggling with this one:



[img]/User%20Files/1559/Latex-Equation-7218.gif[/img]



v is an integer and f is not differentiable. It's not an exercise with a solution, so I don't know if its solvable. Any ideas? Looks like substitution rule (change of variables), but the vx in the sinus is problematic.

[kevink]

The answer is 0 since your integrand is an odd function and you are integrating over a domain that is symmetric about the origin.

[Dimatrix]

Is there a more formal proof for this? The odd integrand thing is well known and true, but I'm trying to work with this integral, since I have to solve the same guy but with f(cos(x))cos(vx) instead of f(cos(x))sin(vx). I'm currently trying to work with addition-theorems, such as: sin(x+y)=...., where I set: sin(vx)=sin( x(v-1)+x)... But this is tedious and I don't have a result yet. There might be an easier solution, which I don't see yet.