It's good to hear that I'm not the only one who is confused with this - probably I was just too used to all the -things I'm doing right now for studies that I couldn't realize the subtle difference between gradient and grad-vector ;).

So Penrose uses gradient in the sense of a scalar, which is defined at a point and in direction of , and we were used to the gradient as a vector at a point which points to max increase of . Quite a confusion of definition, but I think I now got the idea of it. Seems that his way of looking at this is more general.

I'm really looking forward to the physics part of Penrose's book, waiting for all those technical terms to become useful.

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 [unparseable or potentially dangerous latex formula] 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.

Just corrected the Latex:

Thanks Vasco, not sure what I did wrong, but I'll get the hang of Latex one day !

Note that if is a scalar field, then


You can see that has the same components as what you would be used to calling the gradient vector from undergraduate vector calculus classes (and incidentally note the consistency in the notation of using the symbol to represent the covariant derivative). It is common to identify with where are the usual basis vectors in Euclidean space. Often in undergraduate classes they will even tell you that the gradient is a vector field, where in fact it is a covector field.

Remember a covector is just the dual of a vector where the dual space refers to the vector space of linear functionals on a vector space. In our case the vector space in question is the tangent space (strictly speaking we should say the tangent space evaluated at a particular point in a manifold) which has a basis

The dual space (covector space) is the cotangent space (again we should actually be evaluating at a given point) with basis

where we define the covector basis according to:


If you want to try to think geometrically, then a covector field is something which allows you to assign a numerical value to each point in a vector field - in other words it acts on the vector field to produce a scalar field.

I hope this helps to answer your question.

Hi.. I am still confused even after all this discussion. In fig 10.8 penrose shows a scalar field defined on a 2-manifold S. In 12.3 page 225, he writes alpha determines a (n-1)-plane element. In the particular case when alpha is the gradient d(phi) the (n-1)-plane elements will be tangential to the contours.

What plane elements is he referring to? alpha (when its the gradient vector) itself is actually a 1-dimensional plane, orthogonal to the contour lines on our 2-manifold. If I am to think of plane elements as the basis elements of alpha, then they might be parallel at some point of the contour but won't be parallel at the point at which @ is being considered. As @ and kii (he symbol like E) are orthogonal, there wont even be a component of @ along kii. What am I to make of this?

Also, I ve difficulty visualizing @ when its not a gradient vector (d(phi)). I dont understand the 'twisting' diagram in fig. 12.8. :(

Any help wud be well appreciated!!

A hyperplane can be specified as either the span of n-1 vectors tangent to it or eqivalently by one normal.

Note that the space of p-forms has the same dimension as the space of (n-p)-forms and they can be treated as duals. In particular the space of 1-forms and the space of (n-1)-forms both have dimension n. The 1-forms (covectors) can be thought of as normal elements, while the corresponding (n-1)-forms are hyperplane volume elements.