Let's look at the first link in the chain, which is straightforward, but still deserving its clear treatment.

Each point in the manifold is called a phase, or state. So the manifold _is_ the state space.

In this whole discussion, we are assuming a deterministic system. That means that the "direction of motion" of the system is uniquely determined by the state. It's not random, and it doesn't depend upon time - just the state.

In a manifold, the notion of direction of motion at a point is expressed by a tangent vector. The space of all tangent vectors at a point is the tangent space at that point. It is a vector space.

Here we have a manifold \\(M\\) consisting of states. For a given state \\(s \in M\\), the tangent space \\(\mathit{Tangent}(s)\\) consists of all the vectors for the possible directions of motion out of that state.

That the system is deterministic means that for each \\(s\\), there is one uniquely determined vector in \\(M(s)\\) that gives its direction of motion. Let \\(\Gamma(s) \in \mathit{Tangent}(s)\\) be this uniquely determined vector.

\\(\Gamma(s)\\) is a field of tangent vectors on the manifold.

This tangent vector field is a function that completely specifies the law of motion for the system.

Each point in the manifold is called a phase, or state. So the manifold _is_ the state space.

In this whole discussion, we are assuming a deterministic system. That means that the "direction of motion" of the system is uniquely determined by the state. It's not random, and it doesn't depend upon time - just the state.

In a manifold, the notion of direction of motion at a point is expressed by a tangent vector. The space of all tangent vectors at a point is the tangent space at that point. It is a vector space.

Here we have a manifold \\(M\\) consisting of states. For a given state \\(s \in M\\), the tangent space \\(\mathit{Tangent}(s)\\) consists of all the vectors for the possible directions of motion out of that state.

That the system is deterministic means that for each \\(s\\), there is one uniquely determined vector in \\(M(s)\\) that gives its direction of motion. Let \\(\Gamma(s) \in \mathit{Tangent}(s)\\) be this uniquely determined vector.

\\(\Gamma(s)\\) is a field of tangent vectors on the manifold.

This tangent vector field is a function that completely specifies the law of motion for the system.