Struggling with Proofs in Calculus/Analysis: Need Advice!

[trap101]

I need some advice on proofs in my calculus/analysis course. I just wrote a test and it didn't go anywhere near according to plan. My problem keeps on revolving around how to do proofs of certain claims. Now the proofs for these claims on the test at least were proofs for the most part straight from the textbook. I put in the hrs studying the material, but I never really "memorized" the proofs from the book. It's as if I just can't get over this hurdle as I try over and over again it's like a huge block that's preventing me from getting to another level. I'm just out and out frustrated at the moment. There's obviously something I'm doing wrong in terms of how I'm studying for this course but I can't seem to figure out what it is. It's not lack of enthusiasm or effort I put in the time, maybe it's approach or trying to understand the wrong things. I don't know. Some advice would be appreciated.

 

[tt2348]

Welcome to the transition from calculus to analysis! It really is a labor of love... But once you get it, you get it. Can you give some examples of certain proofs giving you trouble?

 

[tt2348]

In terms of advice. Be clever and concise. The less wordy the easier a proof is to follow. Also keep in mind all the facts

 

[HallsofIvy]

And, perhaps most important, know the definitions, not just a general idea of what a word means. Often we use the precise words of the definitions in proofs.

 

[trap101]

Welcome to the transition from calculus to analysis! It really is a labor of love... But once you get it, you get it. Can you give some examples of certain proofs giving you trouble?

Well one of the questions was: Suppose U is in ℝ^k and f: S-->^k is continuous. Where S is defined as:

S = {x in ℝⁿ: f(x) is in U } i.e the pre image.

Show that if U is open then S is open.



So I saw this on the test, and I know we had proven in class how to do it in the closed case. I had a visual picture in my mind of what to do between the two sets. I drew the picture in the corner to give me an idea of using the sequences, but this is where I came up short, I couldn't remember the exact characteristics that would allow me to show that the subsequnce {_kj} which I assumed to be in U was mapped from a subsequnce {x_k} that I would have created in S. See it's not that I'm looking at the page completely clueless, I just can't seem to formalize it properly.


Another one was showing that the gradient [itex]\nabla[/itex]F(x,y,z) was perpendicular to all tangent lines in the plane.

For this one my idea was to form a "general" line and some how dot product it to obtain the orthogonality.

Is there some way that I can get the concepts and proofs without it being a case of me just memorizing the proof, but not really knowing what I'm doing because that's what I'm trying to avoid making it mechanical because that won't help anybody. I was reading the "want to be a mathematician" post and one of the suggestions was going over the proofs until you do get it, but we all know the proofs in the textbooks are concise and leave out most of those "trivial" steps, but to a person first seeing it those steps are the bridge.

 

[tt2348]

Let's start with the first question.
Consider a continuous function [itex]f:X→Y , X\subset\mathbb{R}^n ,Y\subset\mathbb{R}^k[/itex]
Prove the preimage of an open subset of Y is open in X.
Do you remember the old school definition of continuity? i.e. for [itex]\forall\epsilon>0 , \exists\delta>0 .\ni. |x-a|<\delta \Rightarrow |f(x)-f(a)|<\epsilon[/itex]? For euclidean spaces, this becomes the basis of how we define open sets. IE [itex] B_δ(a)= \left\{ x: |x-a|<\delta\right\}=\left\{ x: a-\delta<x<a+\delta\right\}=\left\{x:x\in(a-\delta,a+\delta)\right\} [/itex] for our one dimensional case.
Have you guys done anything with the idea of a metric?

 

[tt2348]

You start out with the simpler cases, and build on that knowledge. knowing the definitions will save your life. The hardest part will be the more abstract thinking. There is a reason mathematicians always use weird looking blobs when doing proofs, it's to keep us from associating specifics examples with abstract concepts.

 

[trap101]

Let's start with the first question.
Consider a continuous function [itex]f:X→Y , X\subset\mathbb{R}^n ,Y\subset\mathbb{R}^k[/itex]
Prove the preimage of an open subset of Y is open in X.
Do you remember the old school definition of continuity? i.e. for [itex]\forall\epsilon>0 , \exists\delta>0 .\ni. |x-a|<\delta \Rightarrow |f(x)-f(a)|<\epsilon[/itex]? For euclidean spaces, this becomes the basis of how we define open sets. IE [itex] B_δ(a)= \left\{ x: |x-a|<\delta\right\}=\left\{ x: a-\delta<x<a+\delta\right\}=\left\{x:x\in(a-\delta,a+\delta)\right\} [/itex] for our one dimensional case.
Have you guys done anything with the idea of a metric?

The concept of metrics hasn't been covered, but everything else that you mentioned makes perfect sense. Now thinking about it, I think there was a final part to that question and I think it was to show it was uniformly continuous. Thinking about it again that might not make sense on an open set though...

 

[tt2348]

Uniform continuity falls under a lil more specified function type. A metric is basically a notion of distance, think the distance formula from geometry extended further. Properties of a metric are d(x,x)=0, d(x,y)=d(y,x)... And triangle inequality. But with that in mind, uniform continuity happens for an open set d(x,y)<delta => d(f(x),f(y))<epsilon... Delta depending only on epsilon... And no other points, that is your pre image is only dependent on the size of your image... Not the points in it

 

[trap101]

Uniform continuity falls under a lil more specified function type. A metric is basically a notion of distance, think the distance formula from geometry extended further. Properties of a metric are d(x,x)=0, d(x,y)=d(y,x)... And triangle inequality. But with that in mind, uniform continuity happens for an open set d(x,y)<delta => d(f(x),f(y))<epsilon... Delta depending only on epsilon... And no other points, that is your pre image is only dependent on the size of your image... Not the points in it

Oh that's metrics? the yes I've heard of metrics, guess I would have to since that's the only way to really compare points in ℝ² up to ℝⁿ. So are metric spaces something that is built up from metrics? Just wondering since I've read up on future courses and they cover metric spaces. Now when you say the pre-image is only dependent on the size of my image, is the size of my image expressed as the distance of the radius in the open ball? i.e B(r,x)?

 

[tt2348]

In spaces equipped with a metric (a way of measuring distance) , they are called metrizable and open sets are defined as a union among "open balls" centered at a point, you can take a collection of disjoint open sets, and for continuous functions, the preimage will also be a collection of open sets.
Think f(X)=x^2, take an open set of the image, ie from (0,2) you'll have an open preimage (-(-sqrt(2),0)u(0,sqrt(2))... x^2 is not invertible however... since open preimage doesn't imply an open image ... (-1,1) maps to [0,1)

 

[HallsofIvy]

So, as I suggested before, it all comes down to the definition of "continuous". In some courses, for example, "General Topology" "[itex]f^{-1}(A)[/itex] is open for all open sets A" is the definition of "continuous". That would make this problem trivial so I presume you are not using that definition. What definition of "continuous" are you using?

By the way, tt2348, "Spaces equipped with a metric" are NOT called "metrizable", they are called "metric spaces". A space is called "metrizable" if it does NOT have a metric but it is possible to define a metric on it that gives the same open sets.

 

[tt2348]

Ahh, my mistake. I was referring to the topology that arises in respect to the metric spaces.
IE the way you define open sets on the given metric space.
http://mathworld.wolfram.com/MetrizableTopology.html
I'll be more explicit with my definitions :-).

Anywho, I think that the course he is taking is not a general topology course, most likely an intro to analysis.

"Well one of the questions was: Suppose U is in ℝk and f: S-->k is continuous. Where S is defined as:

S = {x in ℝn: f(x) is in U } i.e the pre image.

Show that if U is open then S is open."

Since it's euclidean, I'd say it's safe to assume it's not asking for stringent topological definitions.

 

[HallsofIvy]

Yes, but I would still like trap101 to tell us exactly what definition of "continuous" he is using- if only to be sure he knows what it is! It may not be "asking for stringent topological defintions" but you still need to use the definitions they are using.

 

[tt2348]

I think that he doesn't quite grasp it at the necessary level. Which is why he has trouble. There is such an inundation of definitions of the same thing, it's ridiculous. But trap, what are the definitions you have to work with for continuity, and what class is it?

 

[trap101]

I think that he doesn't quite grasp it at the necessary level. Which is why he has trouble. There is such an inundation of definitions of the same thing, it's ridiculous. But trap, what are the definitions you have to work with for continuity, and what class is it?

My apologies for the delayed response. But your right it is an intro to analysis course. So we've only be given an introductory flavor to the topological environment that your referring to. As for the deifnintions:

regular (1 variable continuity): For every positive number "epsilon" and every a[itex]\in[/itex]U, where U[itex]\subset[/itex]ℝⁿ, there is a positive "delta" so that |f(x) - f(a) < "epsilon" whenever |x-a| < "delta"


Uniform continuity: a function f: S-->ℝ^m is said to be uniformly continuous on S if for every "epsilon" > 0 there is a "delta" > 0 so that |f(x) - f(y)| < "epsilon" whenever x,y [itex]\in[/itex] S and |x-y| < "delta"

 

[tt2348]

Tell you what, can you give me an example of a uniformly continuous function? And what topics are coming up for the next test?

 

[trap101]

Tell you what, can you give me an example of a uniformly continuous function? And what topics are coming up for the next test?

Well I'm trying to think of another uniformly continuous function besides the example in the book which is sin(x). Also I guess depending if we define the set as being compact, then that would allow me to define a lot more functions as uniformly continuous. Wouldn't even x² be uniformly continuous if the set was compact?

As for topics of next test: - Implicit Function Thm and its Applications
- Integral Calculus
- Line and Surface Integrals; Vector Analysis

The book we're using is Folland's "Advanced Calculus"

 

[tt2348]

Sorry , I accidentally unsubscribed this post.
Yes this is true, restriction of the domain of x^2 would give a uniformly continuous function , but is not necessarily true for all R.

The best uniformly continuous function is a simple line.
since |k(x)-a-(k(y)-a)|=|k|*|x-y|<e