random thread

[Ahmad]

I agree about Pugh's book being excellent. It's really well motivated. I love the diagrams and the reasons he gives for doing things which are usually given before proofs. I haven't read it all but I've come back to it many many times and would say that it's a fantastic second real analysis book, it's more advanced than Spivak's "Calculus" though. Here's a link to its amazon page.

[kamil9876]

so I assume your objection comes from the following:

*intergration by parts is proved by differentiation and fundamental theorem of calculus.
*therefore using integration by parts means we are assuming differentiable => continous
*but function is not continous therefore we don't have a right to use integration by parts.

This is true if you have derived integration by parts using differentiation. But if you do it without differentiation and only integration then you have a right to use it.

[Ahmad]

Dirac delta function isn't actually a function, it's a generalized function (distribution) which I don't know much about at all. That said it so happens that you can treat it as though it were a function in lots of ways and you still get correct answers, that's not rigorous but it works and is probably good enough for physicists and engineers (who probably use it the most).

[/0]

ah ok, thanks kamil and ahmad

[/0]

Hmm I came across these equations for del in a book on electromagnetics which omitted proof. I'm wondering... how could they be derived?

Cylindrical:



Spherical:



Where is the polar angular and the zenith angle.

[zzdfa]

in xyz coordinates right?

have you tried just subbing the defintions of cylindrical/spherical coordinates in there? like, get the 3 equations that define cyclindrical unit vectors in terms of cartesian unit vectors, and sub the expressions for in terms of into the above and do all the calculations.

I don't think stewart covers unit vectors in different coordinate systems.

[humph]

Nah don't use it, stick to Stewart. If you want to learn calculus from a physics-y point of view anyway then you're best off getting a book on electromag, because they're great for vector calculus (I can't remember what the set textbook for the ANU electromag course is, but its treatment of vector calculus is far superior to that of the maths course about it).

[/0]

Thanks zzdfa, I eventually got it to work that way
@ humph, I'm having a read of an electromagnetics text at the moment (Introduction to Electrodynamics - Griffiths) and it seems to use vector calculus a lot more casually than even stewart. Thanks for the book recommendation - although, I can't find it on the website. I might ask them by email


Just another question, say you have two different coordinate systems and you want to dot or cross vectors from them, do you have to convert all the vectors to the same coordinate system first?

[humph]

Thanks zzdfa, I eventually got it to work that way
@ humph, I'm having a read of an electromagnetics text at the moment (Introduction to Electrodynamics - Griffiths) and it seems to use vector calculus a lot more casually than even stewart. Thanks for the book recommendation - although, I can't find it on the website. I might ask them by email


Just another question, say you have two different coordinate systems and you want to dot or cross vectors from them, do you have to convert all the vectors to the same coordinate system first?

Yeah sorry, can't remember the book at all. It might be that Griffiths one, I'm not sure - I never did electrodynamics at uni anyway so I'm basing this off the words of my friends.

As for your other question, yes, you can only combine vectors in that way when you're in a set coordinate system. This becomes very important when you start looking at differential vector operators like grad, div, and curl, which do have coordinate-free definitions, but generally when applied to any situation are calculated differently depending on coordinates.

[/0]

Ah thanks humph XD

Just wondering, in spherical coordinates, what is considered 'more' standard:
 
Radial distance: r
Azimuth angle:
Zenith angle:

or

Radial distance: r
Azimuth angle:
Zenith angle:

If not, are any of them preferred in Australian universities?



Stewart and Griffiths use different notation so it's really confusing xD

I kind of prefer the first one since is the azimuth angle in polar coordinates anyway. I don't see the logic in assigning to it a different role

Wikipedia and hyperphysics seem to have it in form too 
But wolfram has it in form 

[humph]

So if you get bored of Spivak, try Abbotts' or Pugh's book. Both do a good job of motivating the rigor, but apparently Abbott is better suited for someone who has not done proofs before .

Thank zzdfa, I'll might investigate those books
And also addikaye03, Yeah seems like it could work for those functions too, I'll give it a go



Another question, for the dirac delta function,

and

The derivative is meant to be defined as:



Just wondering, does this make sense mathematically? The above expression can be derived with integration by parts, but the function isn't even continuous :/

Oh I didn't notice this. Yeah, there are ways to make sense of this mathematically, as Ahmad said in the sense of distributions. It comes in handy a lot in the area of Fourier analysis (which I know a bit about), where distributions pop up naturally as Fourier transforms of functions that are locally integrable but not globally integrable (e.g. makes sense over any finite interval, but not over an infinite interval). Anyway, the basic idea is to look at distributions in the sense of their effect on other suitably nice functions, that is, to look at for "nice" functions . If you're lucky, you can then extend the result to lots more functions , and you can then deal with distributions in certain situations without anything bad happening. Of course, mathematicians are always careful to only use them when they know that the theory behind them is consistent, whereas engineers and physicists will just blast away... 

Ah thanks humph XD

Just wondering, in spherical coordinates, what is considered 'more' standard:
 
Radial distance: r
Azimuth angle:
Zenith angle:

or

Radial distance: r
Azimuth angle:
Zenith angle:

If not, are any of them preferred in Australian universities?



Stewart and Griffiths use different notation so it's really confusing xD

I kind of prefer the first one since is the azimuth angle in polar coordinates anyway. I don't see the logic in assigning to it a different role

Wikipedia and hyperphysics seem to have it in form too 
But wolfram has it in form 

Hmmm I'm not sure what is more standard. I guess I'd use the first one for the same reason as you, but to be honest it doesn't really matter too much, and bear in mind that you'll find plenty of cases where you'll use different coordinate systems but with familiar labels, which can be quite confusing (see e.g. this thread).

[/0]

Thanks humph XD

[/0]

Ned help with a DE from electrodynamics :/





Where and

The answer is





But I don't understand how they get that, because the discrimant I get for the second order ODE doesn't show any clear indication of being negative thx

[Ahmad]

One way (not necessarily the best way) to go about this is to take derivatives of both sides of the first equation which will allow you to eliminate z'' from the second equation. Let u = y' then the resulting equation will be linear second order in u so you should be able to solve it using standard methods.

[Ilovemathsmeth]

Just curious, do I see these things in Actuarial?