Total vector field

(Anderson_1992 Definition 2.3)
Definition
Let $E \to M$ be a vector bundle and $X \in X (M)$ . Then there is a unique vector field in the jet bundle $J^{\infty} (E)$ , called the total vector field of $X$ and denoted by $tot X$ , such that:

$X$ and $tot X$ agree on functions on $M$ .
$tot X$ annihilates all contact 1-forms, that is, if $ω$ is a contact form then

tot X ⌟ ω = 0.

$◼$

If $X$ is given by $X = X^{i} \partial x_{i}$ then

tot X = X^{i} D_{x_{i}}

where $D_{x_{i}}$ are the total derivative operators. Observe that this definition gives us an intrinsic definition of this total derivative operators.

Idea: A vector in a manifold $M$ is a kind of little displacement. In the jet space $J^{\infty} (E)$ we have little displacement, too. For example, you can go from $(x, u, u_{1}) \in J^{\infty} (E)$ to:

(x, u, u_{1}, \dots) \mapsto (x + 0.1, u + 0.3, u_{1} - 0.04, \dots) .

But if you think of $(x, u, u_{1}, \dots)$ as the class of functions defined on $M$ (sections of $E$ , indeed) such that $f (x) = u$ , $f^{'} (x) = u_{1}$ and so on, then the little displacement $x \mapsto x + 0.1$ induce a natural displacement in $(x, u, u_{1})$ since:

u \mapsto u + 0.1 u_{1}

u_{1} \mapsto u_{1} + 0.1 u_{2}

because of the definition of derivative:
$f (x + 0.1) \approx f (x) + 0.1 f^{'} (x)$ ,
$f^{'} (x + 0.1) = f^{'} (x) + 0.1 f^{″} (x)$
...
So the induced natural displacement is

(x, u, u_{1}, \dots) \mapsto (x + 0.1, u + 0.1 u_{1}, u_{1} + 0.1 u_{2}, \dots) .

This induced displacement would be a total vector field.