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1. Please verify my differential geometry results

Homework Statement Q1) One way to define a system of coordinants for a Sphere S^2 given by x^2 + y^2 + (z-1)^2 = 1 is socalled stereographical projection \pi \thilde \{N} \rightarrow R^2 which carries a point p=(x,y,z) of the sphere minus the Northpole (0,0,2) onto the intersection...
2. A steographic problem(did I get it right?)

Homework Statement A given sphere S^2 is given by x^2 + y^2 + (z-1)^2 = 1 where stereographical projection \pi:\pi: S^2 \thilde \{N\} \rightarrow \mathbb{R}^2 which carries a point p = (x,y,z) of the sphere minus Northpole N = (0,0,2) onto the intersection of the xy plane which a straight...
3. Equivalence relations and connected components(Please look at my calculations)

Homework Statement Hi I have justed switched to a new subject and have some question. 1) Show that if X is a topology space then there exist an equivalence relation if and only if there exist a connected subset which contains both x and y. 2) Show that the connected components are a...
4. Question regarding n-space and inner product

Just lets take it from the top. My understand that M can considered at a subspace of R_infinity. That hopefully correct? So what You are simply saying is simply to show that this is infact the case then I say for any real scalar S in M then S \cdot \sum_{n=1}^\infty} r_n^2 < \infty ...
5. Question regarding n-space and inner product

Anyway I just thought about something after re-reading my linear algebra book and talked to my professor he said something like this: An infinite dimensional Vector Space is defined as \mathbb{R}^\infty = \{(v_1, v_2, \ldots)| v_n \in \mathbb{R}, \forall n\} Then if Our M is of...
6. Question regarding n-space and inner product

Hi Mark what I don't get here. You say M is not a Vector Space? How is that possible since its the premise for the whole assignment?
7. Question regarding n-space and inner product

My space M isn't that an Euclidian N-Space? Because that deals also with a sequence of real valued vectors. Thusly making into an Vector Space in R^n ?
8. Question regarding n-space and inner product

The question is as follows: Is says that Let M = \{\{r_n\}_{n=1}^{\infty}| r_n \in \mathbb{R}, \sum_{n=1}^{\infty} r_n^2 <\infty\} Show that M is a Vectorspace with the inner product \langle \{r_n\}_{n=1}^{\infty} \{s_n\}_{n=1}^{\infty} \rangle = \sum_{n=1}^{\infty} r_n \cdot s_n and...
9. Question regarding n-space and inner product

Hi Mark, Thank You for your reply. (Maybe this is a stupid question) But how can M be seen as a Vector Space if it doesn't belong to R^n? Would you say its enough to show that the three axioms of inner product true and then conclude "Hence that the inner product between (r_n, p_n) exist...
10. Question regarding n-space and inner product

Homework Statement I have posted simular questions a couple of times but now I feel I have a better understanding(hopefully). Given a Vectorspace M which is defined as a sequence of realnumber \{r_n\} and where \sum_{r=1}^{\infty} r_n < \infty Show that M has an innerproduct given by...
11. Applying Cauchy-Schwarz to a sum(Have I understood this correctly?)

I was simply trying to understand why this sum of squared elements of the series {x_j} is put into the assigment? And why its less than infinity? I understand that maybe this square respresents that the inner product between elements in the above series. Isn't the series the same as \langle...
12. Applying Cauchy-Schwarz to a sum(Have I understood this correctly?)

Homework Statement Given the vectorspace consisting of a realvalued sequences \{x_j\} where \sum_{j=1}^{\infty} x_j^2 < \infty . Show that M the vectorspace has an innerproduct given by \langle \{x_j\}, \{y_j\}\rangle = \sum_{j=1}^\infty x_j \cdot y_j Homework Equations Since...
13. Parameterized tangent line to a parameterized curve

Homework Statement I seem to remember that a parameterized a(t) curve in \mathbb{R}^3 that one can construct the tangent from the slope of a'(t) and the curve itself. such that the tangent line L = a(t) + s * a'(t) to a. This is supposedly a straight line in \mathbb{R}^3. To make a...
14. How can an infinity series be less than infinity?

Question. Does all axioms of the product have to shown in context with the desired space M in order to show that the inner product and x and y satisfies the definition?
15. How can an infinity series be less than infinity?

Yes with respect to the definition that M is Vector Space consisting of real numbers with \sum_{n = 1}^{\infty} x_n^2 < \infty. This last part that means that the squared normed space on M is less than infinity? Thus the normed space is not infinitely large and then its possible to test the...
16. How can an infinity series be less than infinity?

I get it now but anyway. With the above then if I need to show that M has an inner product defined by \langle{\{x_n\}, \{y_n\}}\rangle = \sum_{n = 1}^{\infty} x_n y_n So what I need to do here isn't to show since the Vector Space M is defined as \sum_{n = 1}^{\infty} x_n^2 < \infty...
17. How can an infinity series be less than infinity?

What I would simply like to understand is why a sequence of real numbers defined as in my original post can be less than infinity? If M is a vectorspace is the "less than infinity" part because a Vectorspace is closed set?
18. How can an infinity series be less than infinity?

Then my above series can be written as \sum_{n = 1}^{\infty} x_{n}^2 = \lim_{n \to \infty} \int_{1}^{n} x_{n} ^2 dx < \infty
19. How can an infinity series be less than infinity?

Homework Statement I have a space M which is a sequeces of real numbers \{x_n\} where \sum_{n = 1}^{\infty} x_{n}^2 < \infty How can a series mentioned above be become than less than infinity?? Please explain :confused: Sincerely Cauchy
20. Can a curve with singular point be a regular curve?

Hi First of all its suppose be t = p and p \in I and the curve is defined as \alpha(t) = (x(t),y(t)) a parameter curve. having a singular point on a regular curve isn't that a contradiction? Sincerrely Cauchy
21. Can a curve with singular point be a regular curve?

Homework Statement Given a parameterized curve \alpha:(a,b)\rightarrow \mathbb{R}^2, show that this curve is regular except at t = a. Homework Equations I know that according to the defintion that a parameterized curve \alpha: I \rightarrow \mathbb{R}^3 is said to be regular if...
22. Arc length and straight lines(Need clarification please!)

Hi again and one time more thank you for your reply, What I am going to end up here just to clarify is that some of the terms on the lefthand side of inequality will eat each other and I will end up with a formula which resembles the arc-length formula? Sincrely Cauchy
23. Arc length and straight lines(Need clarification please!)

Hi again You mean \frac{\vec{t}-\vec{s}}{|\vec{t}-\vec{s}|} \cdot (t-s)= \int_a^b \vec{\omega}(t)'\cdot \frac{\vec{t}-\vec{s}}{|\vec{t}-\vec{s}|} dt \leq \int_a^b ||\vec{\omega}(t)'|| dt?? and then just expand?? Cauchy
24. Arc length and straight lines(Need clarification please!)

I surely is. So just to recap then proving part(1) I partion the equation into vector equtions, and then use the fact about th constant vector. Thank You. I hope I don't sound completely stupid, been away from the studies for 6 month do a irregular heartrytm, but that a longer story...
25. Arc length and straight lines(Need clarification please!)

Then thinking about it it must be: \int_a^b \vec{\omega}(t)'\cdot\vec{v}(t)dt \leq \int_a^b ||\vec{\omega}(t)'|| dt?? Sincerely Cauchy
26. Arc length and straight lines(Need clarification please!)

Thats its that \vec{\omega}(b)\cdot\vec{v}(b) - \vec{\omega}(a)\cdot\vec{v}(a) ? Cauchy
27. Arc length and straight lines(Need clarification please!)

Hi That must be \vec{\omega}(t)'\cdot\vec{v}(t)\leq \|\vec{w(t)'}\| ? Cauchy
28. Arc length and straight lines(Need clarification please!)

That must be then ||\vec{\omega}(t)'||\cos \theta\leq ||\vec{\omega}(t)'|| please excuse me if I am misunderstanding you but cos(theta) <= 1 shows then that inequality is true if the vector v is constant? Cauchy
29. Arc length and straight lines(Need clarification please!)

hi again and thanks for taking time :) cos has max in cos(0) = 1 and min in cos(pi) = -1. and this basicly proves that the inequality is true? Sincerely Cauchy
30. Arc length and straight lines(Need clarification please!)

Hi again, I am very sorry if my op has been unclear. I am told to show the later. I will repost it again now. Let \omega: I \rightarrow \mathbb{R}^3 be a parameterized curve. Let [a,b]\subset I and set \omega(b) = s and \omega(a) = t. 1) Show that, for any constant vector v, |v|...