Limits, Continuity and Calculus in 3D Space

[Aftermarth]

Ok, i am hoping we can (owing to the large amounts of questions in the homework help on stuff like this) create a nice guide for Calculus in 3D including definitions, practise questions and general examples.

Firstly, i know something i really do not like is Limits.
My textbook gives the definition of:

"Suppose l is a real number. Then the limit of f(x), as x approaches c, is equal to l if for any number e > 0 there exists a number d >0 such that |f(x) - l | < e whenever 0 < | x - c | < d"

now, in terms that i can understand ( i hate the way that is all shoved into one sentence there has to be someone who can break it up so that it makes sense!) i have found this definition:

I tell you how close I need to be, and you can tell me what I have to do to be that close. (from http://www.karlscalculus.org/calc2.html )

ok now. this needs to be related back.

lim f(x) = L
x > a

This is the limit of f(x) as x approaches a. No worries there. Now what is L?

 

[Hootenanny]

lim f(x) = L
x > a

This is the limit of f(x) as x approaches a. No worries there. Now what is L?

Thanks for taking the time to write a tutorial.

Are you asking a question here, or is this part of your tutorial?

 

[Aftermarth]

its part of the tutorial :) haha.
I am learning this stuff now, and i thought the best way to understand it myself would be to provide a large thread here where everyone can contribute to it.

so, as i am still learning this myself i am guaranteed to make mistakes, so please fix them up for me ( but remember, keep the jargon to use only where it is needed and necessary)

The idea of limits is quite annoying and abstract, but when you understand it, it fits perfectly.
c is a value for which x is approaching (but does not necessarily need to touch as the function may be undefined at c)
L is the limit of the function (what the function is approaching at the value of x = c)

(again correct me if i am wrong)

But functions do not always have a limit. For there to be a limit, the function must approach c from both sides. The function does not have to be defined at c, but either side of it, it must be.

If the calculated limit from the positive direction is different to the calculated limit from the negative direction then the lim (x -> c) of f(x) does not exist. An example of this is the absolute value graph. |x| at c = 0.
The calculated limit from the negative direction is -1, and from the positive direction it is 1.
(can someone help me here with diagrams and equations please)
therefore lim (x -> 0) |x| does not exist.


Continuity

A function is continuous at any point a in its domain if:
Lim (x -> a) f(x) = f(a)

that is, the limit you calculate for x tends toward a is equal to the value of simply subbing in x = a into the equation f(x)

 

[Aftermarth]

Partial Derivatives

Take a function of 2 variables:
F(x, y) = 3x^2 + 2xy^3 + 3y^2 + 2xy + 2x + 5y + 10

The partial derivatives are obtained by fixing one variable at a point for example (x, b) or (a, y) where a,b are the fixed points.
So, pick one variable and fix it. If you want Fx(x , y) then fix y. If you want Fy (x, y) then fix x. Once u have fixed the value, treat it like any normal constant and derive in terms of x or y.
eg.
Fx(x,y) for the function above is:
6x + 2y^3 + 2y + 2
and F y (x,y) is:
6xy^2 + 6y + 2x + 5

(its hard to integrate in terms of y cause I am so used to doing it in terms of x, but just pretend x is a constant, so any x terms to any power are a constant, and if alone will disappear)

To calculate the second derivatives just do the same again, but this time there are four second derivatives
Fxy(x,y), Fyx(x,y), Fxx(x,y), Fyy(x,y)
where Fxy is the function derived in terms of y and then again in terms of x
and Fyx is the function derived in terms of x and then agian in terms of y
and Fxx is the function derived twice in terms of x
and Fyy is the function derived twice in terms of y