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## Main Question or Discussion Point

Hello Physicsforums!

It is quite often that I find the geometrical visualization of a covector field as a field where the covector for each point bends along the manifold, M. That is to say it is "contained" in the surface - in contrast to the tangent field in which the vectors streches out from M ... as just tangents. Is there any good site for the justification of this visualization? :/

I am now reading 'Introduction to Smooth Manifolds' by John M. Lee where he writes that every nonzero functional can be determined by its kernel and the hyperplane for which w_p(X) = 1. Futhermore he writes that the kernel is of "codimension-1" - what does this mean?

Let us take the functional [1, 0, 0, 0] for the cotangent space for a point in R^4. If I understood everything this will have a kernel of dimension 3 in the tangentspace spanned by [0, 1, 0, 0]^T, [0, 0, 1, 0]^T and [0, 0, 0, 1]^T. It is not of dimension 1.. Have I missunderstood something fundamental here?

Thanks soooo much!

// Daniel

It is quite often that I find the geometrical visualization of a covector field as a field where the covector for each point bends along the manifold, M. That is to say it is "contained" in the surface - in contrast to the tangent field in which the vectors streches out from M ... as just tangents. Is there any good site for the justification of this visualization? :/

I am now reading 'Introduction to Smooth Manifolds' by John M. Lee where he writes that every nonzero functional can be determined by its kernel and the hyperplane for which w_p(X) = 1. Futhermore he writes that the kernel is of "codimension-1" - what does this mean?

Let us take the functional [1, 0, 0, 0] for the cotangent space for a point in R^4. If I understood everything this will have a kernel of dimension 3 in the tangentspace spanned by [0, 1, 0, 0]^T, [0, 0, 1, 0]^T and [0, 0, 0, 1]^T. It is not of dimension 1.. Have I missunderstood something fundamental here?

Thanks soooo much!

// Daniel