In section 2.2 on "What is a manifold?" after equation (2.9) Sean Carroll says check the equation for yourself. The equation is

The diagram shows a section through the sphere
which has radius 1. The section is in the plane of

Read more at Commentary 2.2 sphere manifold.pdf with additional material working up to the 'Northern plane'

This was my first attempt at a 3-D diagram. They improve!

ϕ

_{1 }is the stereographic mapping from coordinates on a sphere to the 'southern plane' as shown below. (Similar to Carroll's Figure 2.16)*(y*and the X^{1}, y^{2})^{3}axis, so it looks oval. The point*(x*on the sphere is projected onto^{1}, x^{2}, x^{3})*(y*on the southern plane which is at^{1}, y^{2})*x*. We then drop a perpendicular (dashed line) from^{3}=-1*(x*onto the southern plane to^{1}, x^{2}, x^{3})*(x*and from there to the Y^{1}, x^{2})^{1}axis. It hits at*(x*. We also drop a perpendicular from^{1}, 0)*(y*to the Y^{1}, y^{2})^{1}axis, it hits at*(y*. From there we draw a line back up to the North Pole and draw a line straight up from^{1}, 0)*(x*to intersect it. The intersection point is inside the sphere. We now have two right angled, similar triangles in Y^{1}, 0)^{1}X^{3}plane. These give us the equation below. The left hand side is from the sides of the larger triangle......Read more at Commentary 2.2 sphere manifold.pdf with additional material working up to the 'Northern plane'

This was my first attempt at a 3-D diagram. They improve!

## No comments:

## Post a Comment