Properties of Differentials, Smooth Manifolds.

In summary, John M. Lee's book discusses smooth manifolds and introduces the concept of smooth maps between them. The proposition states that the tangent map of a smooth map is linear. This means that the image of a linear combination of tangent vectors is equal to the same linear combination of the tangent map of those vectors.
  • #1
BrainHurts
102
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I'm reading the second edition of John M. Lee's Introduction to Smooth Manifolds and he has a proposition that I'd like to understand better

Let M, N, and P be smooth manifolds with or without boundary, let F:M→N and G:N→P be smooth maps and let p[itex]\in[/itex]M

Proposition: TpF : TpM → TF(p) is linear

ok I know that v[itex]\in[/itex]TpM means that

v:C∞(M)→ℝ is a derivation and that TpM is a vector space.

Does this mean that the image of (av+bw) under TpF where v,w [itex]\in[/itex] TpM and a,b [itex]\in[/itex] ℝ

is aTpF(v) + bTpF(w) which means TpF is linear?
 
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  • #2
Yes, that's what it means.
 

Related to Properties of Differentials, Smooth Manifolds.

1. What is a differential?

A differential is a mathematical object that describes a small change or variation in a function or variable. It can be thought of as the slope or rate of change of a function at a specific point.

2. What are the properties of differentials?

Some properties of differentials include linearity, the product rule, and the chain rule. Linearity means that the differential of a sum is equal to the sum of the differentials. The product rule states that the differential of a product is equal to the first function times the differential of the second function plus the second function times the differential of the first function. The chain rule states that the differential of a composite function is equal to the product of the differentials of the individual functions in the composition.

3. What is a smooth manifold?

A smooth manifold is a mathematical object that can be described locally by smooth coordinates, similar to how a flat surface can be described by a map with longitude and latitude lines. It is a topological space that is locally homeomorphic to Euclidean space, meaning that it looks like a piece of a plane or space when zoomed in enough.

4. What are the properties of smooth manifolds?

Some properties of smooth manifolds include differentiability, orientability, and dimensionality. Differentiability means that the manifold is locally smooth and can be described by smooth functions. Orientability means that there is a consistent way to assign a direction to every point on the manifold. Dimensionality refers to the number of coordinates needed to describe the manifold locally.

5. How are differentials and smooth manifolds related?

Differentials are used to describe the local behavior of smooth manifolds. They are essential in defining the tangent space of a manifold, which is necessary for calculating derivatives and understanding the local geometry of the manifold. Additionally, the properties of differentials, such as linearity and the chain rule, are crucial in studying the behavior of functions on smooth manifolds.

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