At the top of the page, Penrose writes:

Quote:

The field equations then arise from the assertion that the quantity

S is stationary with respect to variations of all the variables (so it gives the analogue of a geodesic; see Fig. 20.3), which means that the variational derivative of

with respect to all the constituent fields and their derivatives has to vanish. This condition is written

.

I'm thinking that he should have written

S instead of

in the above; i.e. that

the variational derivative of S with respect to all the constituent fields and their derivatives has to vanish.

Of course, what Penrose wrote

implies this (i.e. if the derivatives of

vanish everywhere on

D, then so do the derivatives of

S). But if this were true, there'd be no Langrange equations! (Or rather, the Lagrange equations would degnerate to 0=0). That's because the Lagrange equations are composed of just such derivatives!

...or have I misinterpreted something?