What does it mean that a distribution is integrable? I am studying geometric control theory, and I am focusing on the Frobenius theorem. I have seen that it gives sufficient and necessary conditions for integrability of a distribution, but I am having troubles understanding well the concept.
The Frobenius theorem states that a distribution is integrable if and only if it is involutive. I have clear the concept of involutivity, but what I have not clear is the concept of integrability.
From the notes of my professor, I have that a distribution $\Delta (x)$ of rank $k$ is integrable if there exist the functions $\lambda_1(x) .... \lambda_{k-n}(x) $ such that:
$\frac{\partial \lambda }{\partial x}\Delta (x)=0$
but what does it mean? I cannot understand clearly the concept of integrability of a distribution. Moreover, I don't understand the meaning of the fact that the product of the derivative of $\lambda$ with respect to $x$ with  the distribution gives zero. I know that this implies orthogonality, but what does it mean that these two are orthogonal?
Can somebody please help me?
 A: Integrability of a distribution of rank $k$ can be seen as the extension of the notion of integrability of a vector field. 
A vector field is integrable if you can find a curve to which the field is pointwise tangent. The same happens for $k-$distributions but of course you do not have integral curves but integral submanifolds.
So the distribution $D$ on $M$ is integrable if, said $D_x$ its element specified on $x\in M$, there exists a submanifold $N\subset M$ such that $T_xN=D_x$ for any $x\in N$.
An example of non-integrable distribution is the one defined over the manifold $M=\mathbb{R}^3$ by linear combinations of the two vector fields:
$$ X = \partial_x + y\partial_z $$
$$ Y = \partial_y$$
in fact
$$[X,Y] = \partial_z $$
which does not belong to $D=\langle X,Y\rangle$.
So by Froebenius theorem you can conclude that $D$ is not integrable, which geometrically means that there is no $2-$dimensional manifold $N$ embedded in $\mathbb{R}^3$ which is pointwise tangent to this distribution.
I suggest you to check Lee book on "Introduction to smooth manifolds" to understand better this concept.
Moreover, it is even possible to relate the concept of integrability of a distribution to a dynamical approach if the manifold $M$ is for example the phase space of a mechanical system. 
