In the context of differential geometry and general relativity, the gradient of a scalar function is indeed considered a 1-form rather than a vector. This might seem counterintuitive at first, but it is a consequence of the mathematical formalism used in these fields.
To understand why the gradient is considered a 1-form, let's delve into some mathematical concepts. In differential geometry, vectors and 1-forms are dual to each other. Given a vector space, the dual space consists of linear functionals that take vectors as inputs and return scalars. In other words, 1-forms can be thought of as "direction detectors" that assign a scalar value to each vector.
The gradient of a scalar function is defined as the vector that, at each point, points in the direction of maximum increase of the function. However, in differential geometry, it is more appropriate to think of the gradient as a 1-form that acts on vectors and produces scalar values.
Formally, if we have a scalar function f, the gradient of f, denoted as ∇f or df, is a 1-form defined by the equation:
(df)(v) = v(f)
Here, v is a vector, and (df)(v) represents the action of the 1-form df on the vector v. It yields the directional derivative of the function f in the direction of v.
This distinction between the gradient being a 1-form rather than a vector is related to the differential of the scalar function. The differential, denoted as df, is indeed a 1-form. It represents the linear approximation of the function near a given point. In contrast, the gradient, which is the "dual" of the differential, is the object that acts on vectors to produce the derivative of the function in a specific direction.
So, while it might seem counterintuitive from a more classical vector calculus perspective, in the mathematical formalism of differential geometry and general relativity, the gradient is considered a 1-form, and the differential is the 1-form associated with the scalar function.