## Vector fields in open domains of

Ordinary differential equations, differential operators of the first order, local analysis and global consequences.

Ordinary differential equations, differential operators of the first order, local analysis and global consequences.

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- Definition of differentiability at a point. Maps between open subspaces of the Euclidean spaces smooth on their domain.
- Tangent spaces , tangent bundle .
- Differential of a smooth map: .
- What is the derivative? (answer: exists only when ). Partial derivatives.
- How do we define functions “having more than one derivative”?

Algebraic formalism:

- Algebra of functions infinitely smooth in a domain
- Pullback morphism of algebras .

Vector fields: smooth maps , such that .

Lie (directional, flow) derivations . The Leibniz rule (algebra) and its meaning (“Any Leibniz linear map of to itself is a Lie derivative along some vector field).

Commutator of two vector fields (to be discussed more in the future).

Push-forward of vector fields by smooth invertible maps.

The differential of a smooth function is in a sense container which conceals all directional derivatives along all directions, and dependence on is linear.

If we consider the directional Lie derivative for a form of degree , then simple computations show that is no longer equal to . However, one can “correct” the Lie derivative in such a way that the result will depend on linearly. For instance, if and is a vector field, we can define the form by the identity and show that the 2-form is indeed bilinear antisymmetric.

The 2-form is called the exterior derivative of and denoted . The correspondence is an -linear operator which satisfies the Leibniz rule and for any function .

It turns out that this exterior derivation can be extended to all -forms preserving the above properties and is a nice (algebraically) derivation of the graded exterior algebra .

The lecture notes are available here.