# Differential Geometry Question

• latentcorpse
In summary, the one-sheeted hyperboloid is a surface of revolution described by the equation x^2+y^2-z^2=1, with a positive function f(u)=\sqrt{1+u^2} parameterizing it as x=f(u)*cos(v), y=f(u)*sin(v), and z=v. This was determined by solving for f(u) using the equations for x, y, and z in terms of u and v, and setting them equal to the equation for the hyperboloid.
latentcorpse
Problem1.3. Describe the one-sheeted hyperboloid as a surface of revolution;
that is, find a positive function $f : R \rightarrow R$ such that
x(u, v)= \left[ \begin {array}{c} f \left( u \right) {\it cos}\nu \\\noalign{\medskip}f \left( u \right) {\it sin}\nu \\\noalign{\medskip}\nu\end {array} \right] parameterises the hyperboloid.

So far all I have is the equaiton of the hyperboloid is $x^2+y^2-z^2=1$ and no clue how to proceed. Help please?

How about putting your u,v expressions for x, y and z into the xy form and trying to solve for f(u)?

i'm not too sure what you mean, in particular by the "xy form" you talk about?

could $f(u)=x^2+y^2-z^2-1$. if so then what is $\nu$

or am i waffling?

Waffling. x=f(u)*cos(v), y=f(u)*sin(v) etc. Put them into x^2+y^2-z^2=1.

ok ill give that a bash. it's pretty of confusing of them to call that matrix x, no?

latentcorpse said:
ok ill give that a bash. it's pretty of confusing of them to call that matrix x, no?

Probably should have called it X with a vector on it.

ok. working that through i get $f(u)=\sqrt{1+u^2}$. is that the end of the question? it seems awfully short and yet i appear to have found an f as required.

also, how did you know to procede this way?

Sure. I knew because X(u,v) and x^2+y^2-z^2=1 are supposed to describe the same surface. X(u,v) had better satisfy the second equation.

## 1. What is the definition of differential geometry?

Differential geometry is a branch of mathematics that studies the properties of curves and surfaces in higher-dimensional spaces. It uses tools from calculus and linear algebra to understand the geometric properties of these objects.

## 2. What are some real-world applications of differential geometry?

Differential geometry has many real-world applications, such as in physics, engineering, computer graphics, and robotics. It is used to study the shape of objects and surfaces, understand the motion of particles in curved spaces, and design efficient and stable structures.

## 3. Can you explain the concept of curvature in differential geometry?

Curvature is a measure of how much a curve or surface deviates from being a straight line or a flat plane. In differential geometry, it is represented by a mathematical quantity called the curvature tensor, which describes how much a space is curved at a particular point.

## 4. How does differential geometry differ from classical geometry?

Classical geometry deals with the properties of shapes in a flat, two-dimensional space, while differential geometry extends these concepts to higher-dimensional spaces that are curved. Differential geometry also uses tools from calculus, such as derivatives and integrals, to study these curved spaces.

## 5. What are some key theorems in differential geometry?

Some key theorems in differential geometry include the Gauss-Bonnet theorem, which relates the curvature of a surface to its topology, and the fundamental theorem of surface theory, which characterizes surfaces by their first and second fundamental forms. Other important theorems include the Gauss-Codazzi equations and the Ricci flow equation.

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