# Search results

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

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...
5. ### 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...
6. ### 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...
7. ### 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
8. ### 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...
9. ### Arc length and straight lines(Need clarification please!)

Homework Statement Here is difficult one guys, Lets imagine that an object movement along a curve is described by the parameterized function called \omega: I \rightarrow \mathbb{R}^3 which moves on the interval [a,b]\subset I. and this depended on motor which supplies the constant...
10. ### The behavior of the tracetrix(need help to verify properties)

The behavior of the tracetrix(need help to verify properties) :( Homework Statement Howdy Given the parametric function \beta(t) = (sin(t), cos(t) + ln(tan(t/2)) where t is the angle between the tangent vector and the y-axis and where \beta: (o,\pi) \rightarrow \mathbb{R}^2...