Manifolds, Flows, Lie Groups and Lie Algebras
The following sections are included:
Dynamical Systems
Manifolds and Diffeomorphisms
Flows and Vector Fields
A steady flow and its velocity field
Tangent vector and differential operator
Tangent space
Time-dependent (unsteady) velocity field
Dynamical Trajectory
Fiber bundle (tangent bundle)
Lagrangian and Hamiltonian
Legendre transformation
Differential and Inner Product
Covector (1-form)
Inner (scalar) product
Mapping of Vectors and Covectors
Push-forward transformation
Pull-back transformation
Coordinate transformation
Lie Group and Invariant Vector Fields
Lie Algebra and Lie Derivative
Lie algebra, adjoint operator and Lie bracket
An example of the rotation group SO(3)
Lie derivative and Lagrange derivative
Diffeomorphisms of a Circle S1
Transformation of Tensors and Invariance
Transformations of vectors and metric tensors
Covariant tensors
Mixed tensors
Contravariant tensors
