TT's Maths Thread

[kamil9876]

Thanks Mao!

Can someone show me how to use an proof to show

I've done some stuff, but it doesn't really lead anywhere... so I won't be bothered posting it.

Bit of a too magical solution, (the algebraic trickery comes from my proof about limit laws which is more natural).

let ,



Now the trick is to show that each of the terms can be made as small as one wishes(ie: each less than ).

[TrueTears]

How did you get all that inequality and stuff? It doesn't seem obvious to me at all, can you leave in all your working?

thx.

[kamil9876]

Well I tried to imitate a proof I once did of the following:



let and be the limits of and respectively.

Now intuitively we can represent as  (ie is some error term) and similairly

now the errors can be made as small as possible, and because we want to prove that theorem we might as well multiply:





Now plug back in and And you basically get an expression without the "error terms". This expression is what inspires:

Coz you think the and "ought to be" (Nonetheless, the expression can be verified simply by expanding RHS right? So it is technically independant of this post, but still good to understand where the idea comes from).

As for the inequality, well that is just the triangle inequality; an indespensible tool in epsilon delta proofs.

[TrueTears]

For the differentiability of a function of 2 variables, say we have , then if and exist near and are continuous at then is differentiable at .

For continuous I know you just have to prove

But how do you prove , then if and exist near ?

(Btw as a side note, that condition sounds so informal, "exist NEAR" wdf)

[TrueTears]

The definition of a closed set in R^2 is one that contains all its boundary points, ie if the set was D, then a boundary point of D is a point (a,b) such that every disk with centre (a,b) contains points in D and also points not in D.

Is this a correct interpretation of the definition?

[humph]

Yup.

[TrueTears]

Thanks humph, also a closed set can be infinite in extent right? For example the closed set between the lines x = -3 and x = 3.

[humph]

Yup again. In fact, the entire plane is an open and closed set. The only other subset of the plane with this property is the empty set.

[TrueTears]

Ahh thanks humph!

[Mao]

For the differentiability of a function of 2 variables, say we have , then if and exist near and are continuous at then is differentiable at .

For continuous I know you just have to prove

But how do you prove , then if and exist near ?

(Btw as a side note, that condition sounds so informal, "exist NEAR" wdf)

and exist near means the partial derivatives are continuous at (a,b). By the definition of limits, this continuity ensures any path taken to (a,b) will yield the same gradient (which eliminates any possibilities that a partial derivative in another direction is undefined, because the partial derivative in any direction won't be continuous in that case). Hence the graph must be 'smooth' at (a,b) and is differentiable.

[TrueTears]

can be interpreted as the volume of a solid that lies above a bounded region (on the -plane) and under the surface if

So what is the geometric interpretation of a triple integral. Say:



Where is a bounded region in three dimensional space and is the projection of onto the -plane. and are the upper boundary and lower boundary surfaces of the solid , respectively.

[kamil9876]

Don't be afraid of dimension>3. Physics probably has a lot of examples. Such as let f(x,y,z) be the density of a solid at the point (x,y,z). Triple integral can be used to find it's mass probably.

[TrueTears]

Yeah I ain't afraid haha, I just wanted to know some meaning attached to it.

[/0]

The triple integral gives you the 4-dimensional hypervolume of a shape, but that's not easy to visualise. There is another way of thinking about it which avoids pure geometry and I think is easier to visualise.

If we go back to double integrals, gives you the height of the solid about the x-y plane. But could also describe other things, like the probability density of finding a particle at a point on the x-y plane, in which case would be the probability of finding the particle in a region D. (Just a 2D analogue to 1D probability density functions)

Likewise, in three dimensions, could be the probability density of finding a particle at a point in space, and the probability of finding it in a region E.

could also describe other things, like temperature, concentration, field strength etc.
When combined with the stoke's and the divergence theorem, the triple integral becomes even more powerful.

[TrueTears]

Ahh yeah, and when you set f(x,y,z) = 1 in triple integrals you do get a volume interpretation!