Dr. Wang Xingbo

1
Mathematical Mechanical Method in Mechanical
Engineering
Dr. Wang Xingbo Fall,2005
2
Mathematical Mechanical Method in Mechanical
Engineering
An Introduction to Manifolds

1. Review of Topology
2. Concepts of manifolds

3
Mathematical Mechanical Method in Mechanical
Engineering
An Introduction to Manifolds
How can we describe it?
4
Mathematical Mechanical Method in Mechanical
Engineering
An Introduction to Manifolds

• Many engineering objects have a shape of
  complicated surface. These complicated surfaces
  can be described by manifolds. Theories of
  manifolds have exhibited their elegances and
  excellences in many aspects of engineering, e.g.
  in controlling of robots, in structural analysis
  of mechanical engineering. Manifold is regarded
  to be a powerful tool for a senior engineer or a
  researcher to master.

5
Mathematical Mechanical Method in Mechanical
Engineering
Review of Topology

• A topological space is a pair (X, T)
• where X is a set and T is a class of
• subsets of X, called topology, which
• satisfies the following three properties.
• (i) X, ??T.
• (ii) If X ii?I?T, then ?i?I X i ?T
• (iii) If X 1,