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...
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...
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...
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...
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...
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...
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
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...
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...
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...