Saint-Venant's conditions of compatibility (contd...):

Let us focus on the second integral:

By partial integration:

(1.38)

Now , so that,

(1.39a)

And , so that

(1.39b)

Now substituting these expressions into

and , and equation 1.37 we obtain

(1.40)

Where,

(1.41)

Since and are known at , then, can be found out using 1.40.

However, it is imperative that be functions only of and do not depend upon the path of integration.

The necessary and sufficient condition that the integral be path independent is that,

Or, by Green's theorem for integration over a closed curve,

(1.42)

Analogous conditions are obtained for other equations. These equations reduce to the following six equations,

(1.43)

These equations are called the conditions of compatibility of Barré de Saint-Venant.