**Introduction**

In its most elementary form, the theorem addresses the problem of finding a maximal set of independent solutions of a regular system of first-order linear homogeneous partial differential equations. Suppose that *f*_{k}i(*x*) are a collection of real-valued *C*1 functions on *R**n*, for *i* = 1, 2, ..., *n*, and *k* = 1, 2, ..., *r*, where *r* < *n*, such that the matrix (*f*_{k}*i*) has rank *r*. Consider the following system of partial differential equations for a real-valued *C*2 function *u* on *R**n*:

- (1)

One seeks conditions on the existence of a collection of solutions *u*_{1}, ..., *u*_{n−r} such that the gradients

are linearly independent.

The Frobenius theorem asserts that this problem admits a solution locally if, and only if, the operators *L*_{k} satisfy a certain integrability condition known as *involutivity*. Specifically, they must satisfy relations of the form

for *i*, *j* = 1, 2,..., *r*, and all *C*2 functions *u*, and for some coefficients *c**k*_{ij}(*x*) that are allowed to depend on *x*. In other words, the commutators must lie in the linear span of the *L*_{k} at every point. The involutivity condition is a generalization of the commutativity of partial derivatives. In fact, the strategy of proof of the Frobenius theorem is to form linear combinations among the operators *L*_{i} so that the resulting operators do commute, and then to show that there is a coordinate system *y*_{i} for which these are precisely the partial derivatives with respect to *y*_{1}, ..., *y*_{r}.

Read more about this topic: Frobenius Theorem (differential Topology)

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