This article was nominated for deletion on 24 July 2025. The result of the discussion was Transversality (mathematics) moved to Transversality.

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If someone's more capable than I, a picture of transverse and tangent plane curves would be an excellent addition to this article! David Farris 21:30, 4 December 2005 (UTC)[reply]

To answer this implied question: Two vector subspaces V and W of a vector space X are said to "generate" X if every vector x ∈ X can be expressed as the sum

  x  =  v + w

of some vectors v ∈ V and w ∈ W. This is possible — but not necessarily the case — only when dim(V) + dim(W) ≥ dim(X).

If dim(V) + dim(W) = dim(X), then given x ∈ X, the representation x = v + w will be unique. Otherwise, there will be many ways to express x as v + w for some v ∈ V and w ∈ W.

For instance, the xy-plane V and the yz-plane W are 2-dimensional subspaces of the 3-dimensional real vector space R³ that generate R³.

The Optimal control section begins as follows:

"In fields utilizing the calculus of variations or the related Pontryagin maximum principle, the transversality condition is frequently used to control the types of solutions found in optimization problems. For example, it is a necessary condition for solution curves to problems of the form:

Minimize ${\displaystyle \int {F(x,y,y^{\prime })}dx}$ where one or both of the endpoints of the curve are not fixed."

But "the curve" referred to here is never mentioned, not in the above passage and not later in this section.

Also (and no doubt related to this) is that in the integration above, the region of integration is unfortunately never mentioned.

I hope that someone familiar with this subject, and with how to write clearly, can fix this.