I asked this question in math.stackexchange few days ago.  

Unfortunately, I haven't seen any simple answer.


One can say that the Stiefel-Whitney classes is dual classes to the locus of linearly dependence of generic sections. What means "generic"? I want to see some relation in local coordinates.


The same question about the Chern classes and a complex generic sections.



Stiefel classes and generic sections
Consider the vector fields (let say $r$ vector fields on an $n$ dimensional manifold $M$) as a map from the trivial $r$-dimensional vector bundle $\varepsilon^r$ over $M$ into the tangent bundle $TM.$
It gives a section $\alpha$ of the bundle $HOM(\varepsilon^r, TM).$ The fiber over $x \in M$ of this later bundle is the space of linear maps from $R^r$ to $T\_xM = R^n.$ Let $\Sigma \_x$ be the set of non-injective linear maps in the fiber over $x$, and let $\Sigma$ be the union of all $\Sigma\_x$ as $x$ runs over the points of $M.$
The set $\Sigma$ is a stratified subset in the total space $HOM(\varepsilon^r, TM)$. The vector fields are generic if the corresponding section $\alpha$ is transverse to $\Sigma.$


