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I'm currently working through Schutz's "A first course in general relativity" as a preparation for a graduate course in General Relativity based on Carroll's notes. I'm a little confused about vectors, one-forms and gradients.

Schutz says the gradient is not a vector but a one-form, because it maps vectors into the reals in a linear way and explains that you should represent a gradient (and a one-form in general) by a series of surfaces. When you contract a one-form with a vector, the number you get is the number of surfaces the vector crosses.

If this was all I knew about the subject, it would be quite clear, however, my advanced calculus says the gradient is a vector pointing in the direction of fastest increase and it's a vector that's, when evaluated at a point p, is perpendicular to the tangent plane of the surface at point p. Schutz says there's nothing really wrong with this definition, because in normal Euclidean space, vectors and one-forms are the same, but how can something be both a vector (an arrow) and a one-form (a series of surfaces)?

Later on, Schutz says that a vector can also be seen as a linear map from one-forms into the reals (Carroll also says the dual of a dual vector space is the original vector space), so this way it seems whether to call something a vector or a one-form is totally arbitrary (as long as you do so in a consistent way), whether you're dealing with Euclidean space or not... :grumpy:

When I turn to other texts I get confused only more, for example, in some text on Differential Geometry a vector is defined as a linear operator on a function space that produces a real number...? I find this a strange definition, wouldn't this imply that a function is actually a one-form (and a vector, since it's totally up to you what to call the original vector space and what it's dual).

I think it's all equivalent in some peculiar way, but I really don't see how...

Schutz says the gradient is not a vector but a one-form, because it maps vectors into the reals in a linear way and explains that you should represent a gradient (and a one-form in general) by a series of surfaces. When you contract a one-form with a vector, the number you get is the number of surfaces the vector crosses.

If this was all I knew about the subject, it would be quite clear, however, my advanced calculus says the gradient is a vector pointing in the direction of fastest increase and it's a vector that's, when evaluated at a point p, is perpendicular to the tangent plane of the surface at point p. Schutz says there's nothing really wrong with this definition, because in normal Euclidean space, vectors and one-forms are the same, but how can something be both a vector (an arrow) and a one-form (a series of surfaces)?

Later on, Schutz says that a vector can also be seen as a linear map from one-forms into the reals (Carroll also says the dual of a dual vector space is the original vector space), so this way it seems whether to call something a vector or a one-form is totally arbitrary (as long as you do so in a consistent way), whether you're dealing with Euclidean space or not... :grumpy:

When I turn to other texts I get confused only more, for example, in some text on Differential Geometry a vector is defined as a linear operator on a function space that produces a real number...? I find this a strange definition, wouldn't this imply that a function is actually a one-form (and a vector, since it's totally up to you what to call the original vector space and what it's dual).

I think it's all equivalent in some peculiar way, but I really don't see how...

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