Since my previous postings on this subject I've been re-reading chapters 10 and 12 again and again to try to get to what Penrose is saying.

I think I've finally understood the thing about the geometry of co-vectors. In diagram 10.8 (a) Penrose shows that

points along the contour line ie the line of constant

.

Previously I was trying to reconcile this with my "vector calculus" view of gradient whereby the gradient of a scalar field is in the line of steepest ascent, so that grad

(or

) would in fact be orthogonal to the contours.

However, in diagram 12.9 (b) Penrose defines a covector

as referring to the plane (in a 3-manifold) spanned by all other coordinate axes other than the

axis, ie

is a 2-dimensional (ie n-1 dimensional, for n=3) plane of constant

, specifically

is constant over the plane spanned by the

and

axes.

Now go back to diagram 10.8. We have

constant along a contour line so this has to represent the covector

. This is then consistent with the fact that

is actually increasing in a direction orthogonal to

, compare with

increasing in the direction orthogonal to the plane spanned by

and

.

It's like the vector gradient grad

is orthogonal to the covector

, so I suppose in this instance the vector gradient and the covector are duals.