Derivatives

Let \(\mathrm{M}\) be a differential manifold with \(\dim \mathrm{M} = n \in \mathbb{N}\) used throughout the section. Let \(\mathrm{TM}\) and \(\mathrm{T^*M}\) denote the tangent and cotangent bundle, \(V\) and \(V^*\) the fiber and dual fiber bundle and \(\mathscr{B}\) the tensor fiber bundle.

Lie derivative

Definition 1: the Lie derivative on a section of a tangent bundle \(\mathscr{L}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to \Gamma(\mathrm{TM})\) is a map defined by

\[ \mathscr{L}_\mathbf{w} \mathbf{v} = \mathbf{w} \circ \mathbf{v} - \mathbf{v} \circ \mathbf{w} = [\mathbf{w}, \mathbf{v}], \]

for all \(\mathbf{w}, \mathbf{v} \in \Gamma(\mathrm{TM})\).

In which the bracket formulation is also referred to as the Lie bracket.

Proposition 1: the Lie derivative can be decomposed into

\[ \mathscr{L}_\mathbf{w} \mathbf{v} = \mathscr{L}_\mathbf{w}^i \mathbf{v} \partial_i = (w^j \partial_j v^i - v^j \partial_j w^i) \partial_i, \]

for all \(\mathbf{w}, \mathbf{v} \in \Gamma(\mathrm{TM})\).

Proof:

Will be added later.

Exterior derivative

Definition 2: the exterior derivative \(d: \Gamma \big(\bigwedge_k(\mathrm{T}\mathrm{M}) \big) \to \Gamma \big(\bigwedge_{k+1}(\mathrm{T}\mathrm{M}) \big)\) of a \(k\)-form field, \(k \in \mathbb{N}[k \leq n]\) is the \((k+1)\)-form field

\[ \begin{align*} d \bm{\omega} &= d \omega_{|i_1 \dots i_k|} \wedge dx^{i_1} \wedge \dots \wedge dx^{i_k}, \\ &= \partial_j \omega_{|i_1 \dots i_k|} dx^j \wedge dx^{i_1} \wedge \dots \wedge dx^{i_k}, \end{align*} \]

for all \(\bm{\omega} \in \Gamma \big(\bigwedge_k(\mathrm{T}\mathrm{M}) \big)\).

From the definition of the exterior definition the following results arises.

Theorem 1: we have that

1. \(\forall\bm{\omega} \in \Gamma \big(\bigwedge_n(\mathrm{T}\mathrm{M}) \big): d \bm{\omega} = \mathbf{0}\),
2. \(\forall\bm{\omega} \in \Gamma \big(\bigwedge_k(\mathrm{T}\mathrm{M}) \big), k \in \mathbb{N}[k \leq n]: d^2 \bm{\omega} = \mathbf{0}\).

Proof:

Will be added later.

Hodge star operator

Definition 3: the hodge star operator \(*: \Gamma \big(\bigwedge_k(\mathrm{T}\mathrm{M}) \big) \to \Gamma \big(\bigwedge_{n-k}(\mathrm{T}\mathrm{M}) \big)\) with \(k \in \mathbb{N}[k \leq n]\) has the following properties

1. \(\forall \bm{\omega} \in \Gamma \big(\bigwedge_0(\mathrm{T}\mathrm{M}) \big): * \bm{\omega} = \bm{\epsilon}\),
2. \(* (dx^{i_1} \wedge \dots \wedge dx^{i_k}) = \bm{\epsilon} \lrcorner \mathbf{g}^{-1}(dx^{i_1}) \lrcorner \dots \lrcorner \mathbf{g}^{-1}(dx^{i_k})\),

for all \(dx^{i_1} \wedge \dots \wedge dx^{i_k} \in \Gamma \big(\bigwedge_k(\mathrm{T}\mathrm{M}) \big)\) with \(\bm{\epsilon}\) the Levi-Civita tensor \(\bm{\epsilon} \in \big(\bigwedge_n(\mathrm{T}\mathrm{M}) \big)\) and \(\mathbf{g}^{-1}: \Gamma(\mathrm{T}^*\mathrm{M}) \to \Gamma(\mathrm{T}\mathrm{M})\) the dual metric.